INFINITE FREE RESOLUTIONS

LUCHEZAR L. AVRAMOV

Purdue University

West Lafayette, Indiana 47907, U.S.A.

                       avramov@math.purdue.edu

This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matemàtica, Institut d’Estudis Catalans, July 15–26, 1996.

Joan Elias, José Giral, Rosa Maria Miró-Roig, and Santiago Zarzuela were successively fantastic organizers, graceful hosts, and tactful editors. I am extremely grateful to them for everything.

It was a pleasure to address the receptive audience, and to be part of it for the lectures of Mark Green, Craig Huneke, Peter Schenzel, Tito Valla, Wolmer Vasconcelos, and the talks of many others. I learned a lot, both from and at the blackboard. Dave Jorgensen supplied a list of typos and corrections, and Alberto Corso introduced me to XY-pic; their help is much appreciated.

Srikanth Iyengar has read more versions than he might care to recall. For all those corrected errors and pertinent suggestions, he has my special thanks.

Finally, I wish to acknowledge my most obvious debt: these notes would not have been written without Zoya’s understanding and support.

The author was partially supported by a grant from the National Science Foundation.

Contents

Introduction 3 1. Complexes 6 1.1. Basic constructions 6 1.2. Syzygies 8 1.3. Differential graded algebra 11 2. Multiplicative structures on resolutions 15 2.1. DG algebra resolutions 15 2.2. DG module resolutions 18 2.3. Products versus minimality 21 3. Change of rings 24 3.1. Universal resolutions 24 3.2. Spectral sequences 28 3.3. Upper bounds 30 4. Growth of resolutions 33 4.1. Regular presentations 33 4.2. Complexity and curvature 36 4.3. Growth problems 39 5. Modules over Golod rings 42 5.1. Hypersurfaces 42 5.2. Golod rings 44 5.3. Golod modules 47 6. Tate resolutions 51 6.1. Construction 51 6.2. Derivations 55 6.3. Acyclic closures 57 7. Deviations of a local ring 61 7.1. Deviations and Betti numbers 61 7.2. Minimal models 62 7.3. Complete intersections 66 7.4. Localization 67 8. Test modules 70 8.1. Residue field 70 8.2. Residue domains 71 8.3. Conormal modules 77 9. Modules over complete intersections 82 9.1. Cohomology operators 82 9.2. Betti numbers 86 9.3. Complexity and Tor 89 10. Homotopy Lie algebra of a local ring 93 10.1. Products in cohomology 93 10.2. Homotopy Lie algebra 96 10.3. Applications 99 References 103 Index 109

Introduction

A free resolution of a module extends a presentation in terms of generators and relations, by choosing generators for the relations, then generators for the relations between relations, ad infinitum. Two questions arise:

How to write down a resolution?

How to read the data it contains?

These notes describe results and techniques for finite modules over a commutative noetherian ring R. An extra hypothesis that R is local (meaning that it has a unique maximal ideal) is often imposed. It allows for sharper formulations, due to the existence of unique up to isomorphism minimal free resolutions; in a global situation, the information can be retrieved from the local case by standard means (at least when all finite projective modules are free).

∗ ∗ ∗

A finite free resolution is studied ‘from the end’, using linear algebra over the ring(!) R. The information carried by matrices of differentials is interpreted in terms of the arithmetic of determinantal ideals. When R is a polynomial ring over a field each module has a finite free resolution; in this case, progress in computer algebra and programming has largely moved the construction of a resolution for any concrete module from algebra to hardware.

The monograph of Northcott [125] is devoted to finite free resolutions. Excellent accounts of the subject can be found in the books of Hochster [89], Roberts [136], Evans and Griffith [61], Bruns and Herzog [46], Eisenbud [58].

∗ ∗ ∗

Here we concentrate on modules that admit no finite free resolution.

There is no single approach to the construction of an infinite resolution and no single key to the information in it, so the exposition is built around several recurring themes. We describe them next, in the local case.

∗ ∗ ∗

To resolve a module of infinite projective dimension, one needs to solve an infinite string of iteratively defined systems of linear equations. The result of each step has consequences for eternity, so a measure of control on the propagation of solutions is highly desirable. This may be achieved by endowing the resolution with a multiplicative structure (a natural move for algebraists, accustomed to work with algebras and modules, rather than vector spaces). Such a structure can always be put in place, but its internal requirements may prevent the resolution from being minimal. Craft and design are needed to balance the diverging constraints of multiplicative structure and minimality; determination of cases when they are compatible has led to important developments.

Handling of resolutions with multiplicative structures is codified by Differential Graded homological algebra. Appearances notwithstanding, this theory precedes the familiar one: Homological Algebra [51] came after Cartan, Eilenberg, and MacLane [56], [50] developed fundamental ideas and constructions in DG categories to compute the homology of Eilenberg-MacLane spaces. An algebraist might choose to view the extra structure as an extension of the domain of rings and modules in a new, homological, dimension.

DG homological algebra is useful in change of rings problems. They arise in connection with a homomorphism of rings Q ⟶ R and an R–module M, when homological data on M, available over one of the rings, are needed over the other. A typical situation arises when R and M have finite free resolutions over Q, for instance when Q is a regular ring. It is then possible to find multiplicative resolutions of M and R over Q, that are ‘not too big’, and build from them resolutions of M over R. Although not minimal in general, such constructions are often useful, in part due to their functoriality and computability.

Multiplicative structures on a resolution are inherited by derived functors. It is a basic observation that the induced higher structures in homology do not depend of the choice of the resolution, and so are invariants of the R–module (or R–algebra) M. They suggest how to construct multiplicative structures of the resolution, or yield obstructions to its existence. Sometimes, they provide criteria for a ringor module-theoretic property, e.g. for R to be a complete intersection. All known proofs that this property localizes depend on such characterizations—as all proofs that regularity localizes use the homological description of regular rings.

The behavior of resolutions at infinity gives rise to intriguing results and questions. New notions are needed to state some of them; for instance, complexity and curvature of a module are introduced to differentiate between the two known types of asymptotic growth. A striking aspect of infinite resolutions is the asymptotic stability that they display, both numerically (uniform patterns of Betti numbers) and structurally (high syzygies have similar module-theoretic properties). In many cases this phenomenon can be traced to a simple principle: the beginning of a resolution is strongly influenced by the defining relations of the module, that can be arbitrarily complicated; far from the source, the relations of the ring (thought of as a residue of some regular local ring) take over. In other words, the singularity of the ring dominates asymptotic behavior.

Part of the evidence comes from information gathered over specific classes of rings. At one end of the spectrum sit the complete intersections, characterized by asymptotically polynomial growth of all resolutions. The other end is occupied by the Golod rings, defined by an extremal growth condition on the resolution of the residue field; all resolutions over a Golod ring have asymptotically exponential growth; higher order homology operations, the Massey products, play a role in constructions. Results on complete intersections and Golod rings are presented in detail; generalizations are described or sketched.

A basic problem is whether some form of the polynomial/exponential dichotomy extends to all modules over local rings. No intermediate growth occurs for the residue field, a case of central importance. This result and its proof offer a glimpse at a mutually beneficial interaction between local algebra and rational homotopy theory, that has been going on for over a decade.

A major link in that connection is the homotopy Lie algebra of a local ring; it corresponds to the eponymous object attached to a CW complex, and has a representation in the cohomology of every R–module. Its structure affects the asymptotic patterns of resolutions, and is particularly simple when R is a complete intersection: each cohomology module is then a finite graded module over a polynomial ring, that can be investigated with all the usual tools. This brings up a strong connection with modular representations of finite groups.

∗ ∗ ∗

Proving a result over local rings, we imply that a corresponding statement holds for graded modules over graded rings. Results specific to the graded case are mostly excluded; that category has a life of its own: after all, Hilbert [88] introduced resolutions to study graded modules over polynomial rings.

∗ ∗ ∗

The notes assume a basic preparation in commutative ring theory, including a few homological routines. Modulo that, complete proofs are given for all but a couple of results used in the text. Most proofs are second or third generation, many of them are new. Constructions from DG algebra are developed from scratch. A bonus of using DG homological algebra is that spectral sequences may be eliminated from many arguments; we have kept a modest amount, for reasons of convenience and as a matter of principle.

∗ ∗ ∗

The only earlier monographic exposition specifically devoted to infinite resolution is the influential book of Gulliksen and Levin [83], which concentrates on the residue field of a local ring. The overlap is restricted to Sections 6.1 and 6.3, with some differences in the approach. Sections 6.2, 7.2, and 8.2 contain material that has not been presented systematically before.

1. Complexes

This chapter lays the ground for the subsequent exposition. It fixes terminology and notation, and establishes some basic results.

All rings are assumed^{1} commutative. No specific references are made to standard material in commutative algebra, for which the books of Matsumura [117], Bruns and Herzog [46], or Eisenbud [58], provide ample background.

For technical reasons, we choose to work throughout with algebras over a ubiquitous commutative ring 𝕜, that will usually be unspecified and even unmentioned^{2} (think of 𝕜 = ℤ, or 𝕜 = k, a field).

1.1. Basic constructions. Let R be a ring.

A (bounded below) complex of R–modules is a sequence

⋯ ⟶ F_{n} ⟶^{∂_{n}} F_{n−1} ⟶^{∂_{n−1}} F_{n−2} ⟶ ⋯

of R–linear maps with ∂_{n−1}∂_{n} = 0 for n ∈ ℤ (and F_{n} = 0 for n ≪ 0). The underlying R–module {F_{n}}_{n∈ℤ} is denoted F^{♮}. Modules over R are identified with complexes concentrated in degree zero (that is, having F_{n} = 0 for n ≠ 0); |x| denotes the degree of an element x; thus, |x| = n means x ∈ F_{n}.

Operations on complexes. Let E, F, G be complexes of R–modules.

A degree d homomorphism β: F ⟶ G is simply a collection of R-linear maps {β_{n}: F_{n} ⟶ G_{n+d}}_{n∈ℤ}. All degree d homomorphisms from F to G form an R– module, Hom_{R} (F, G)_{d}. This is the degree d component of a complex of R–modules Hom_{R} (F, G) , in which the boundary of β is classically defined by

∂(β) = ∂^{G}∘β − (−1)^{|β|}β∘∂^{F} .

The power of −1 is a manifestation of the sign that rules over homological algebra: Each transposition of homogeneous entities of degrees i, j is ‘twisted’ by a factor (−1)^{ij}, and permutations are signed accordingly.

The cycles in Hom_{R} (F, G) are those homomorphisms β that satisfy ∂∘β = (−1)^{|β|}β∘∂; they are called chain maps. Chain maps β, β′: F ⟶ G are homotopic if there exists a homomorphism σ: F ⟶ G of degree |β| + 1, called a homotopy from β to β′, such that

β − β′ = ∂∘σ + (−1)^{|β|}σ∘∂;

equivalently, β − β′ is the boundary of σ in the complex Hom_{R} (F, G) .

A chain map β induces a natural homomorphism H(β): H(F) ⟶ H(G) of degree |β|; homotopic chain maps induce the same homomorphism. Chain maps of degree zero are the morphisms of the category of complexes. A quasi-isomorphism is a morphism that induces an isomorphism in homology; the symbol ≃ next to an arrow identifies a quasi^{3}-isomorphism.

The tensor product E ⊗_{R} F has (E ⊗_{R} F)_{n} = ∑_{i+j=n} E_{i} ⊗_{R} F_{j} and

∂(e ⊗ f) = ∂^{E}(e) ⊗ f + (−1)^{|e|}e ⊗ ∂^{F} (f) .

1Following a grand tradition of making blanket statements, to be violated down the road.

2Thus, the unqualified word ‘module’ stands for ‘𝕜–module’; homomorphisms are 𝕜-linear; writing ‘ring’ or ‘homomorphism of rings’, we really mean ‘𝕜–algebra’ or ‘homomorphism of 𝕜– algebras’. The convention is only limited by your imagination, and my forgetfulness.

3The symbol ≅ is reserved for the real thing.

The transposition map τ(e ⊗ f) = (−1)^{|e||f|}f ⊗ e is an isomorphism of complexes τ: E ⊗_{R} F ⟶ F ⊗_{R} E.

A degree d homomorphism β: F ⟶ G induces degree d homomorphisms

Hom_{R} (E, β) : Hom_{R} (E, F) ⟶ Hom_{R} (E, G) ,
Hom_{R} (E, β) (α) = β∘α;
Hom_{R} (β, E) : Hom_{R} (G, E) ⟶ Hom_{R} (F, E) ,
Hom_{R} (β, E) (γ) = (−1)^{|β||γ|}γ∘β;

\begin{matrix}β ⊗_{R} E: F ⊗_{R} E ⟶ G ⊗_{R} E , & (β ⊗_{R} E)(f ⊗ e) = β(f) ⊗ e; \\ E ⊗_{R} β: E ⊗_{R} F ⟶ E ⊗_{R} G , & (E ⊗_{R} β)(e ⊗ f) = (−1)^{|β||e|}e ⊗ β(f) , \end{matrix}

with signs determined by the Second Commandment^{4}. All maps are natural in both arguments. If β is a chain map, then so are the induced maps.

The shift ΣF of a complex F has (ΣF)_{n} = F_{n−1} for each n. In order for the degree 1 bijection Σ^{F} : F ⟶ ΣF, sending f ∈ F_{n} to f ∈ (ΣF)_{n+1}, to be a chain map the differential on ΣF is defined by ∂^{ΣF} (Σ^{F} (f)) = −Σ^{F} (∂^{F} (f)).

The mapping cone of a morphism β: F ⟶ G is the complex C(β) with under-

lying module G^{♮} ⊕ (ΣF)^{♮}, and differential (\begin{matrix}∂^{G} & (Σ^{G})^{−1}Σ(β) \\ 0 & ∂^{ΣF} \end{matrix}). The connecting

homomorphism defined by the exact mapping cone sequence

0 ⟶ G ⟶ C(β) ⟶ ΣF ⟶ 0

is equal to H(β). Thus, β is a quasi-isomorphism if and only if H(C(β)) = 0.

A projective (respectively, free) resolution of an R–module M is a quasiisomorphism ϵ^{F} : F ⟶ M from a complex F of projective (respectively, free) modules with F_{n} = 0 for n < 0. The projective dimension of M is defined by pd_{R} M = inf{p |M a projective resolution with F_{n} = 0 for n > p}.

Example 1.1.1. In the Koszul complex K = K(g; R) on a sequence g = g_{1}, …, g_{r} of elements of R the module K_{1} is free with basis x_{1}, …, x_{r}, the module K_{n} is equal to ⋀_{R}^{n} K_{1} for all n, and the differential is defined by

∂(x_{i_{1}} ∧ ⋯ ∧ x_{i_{n}}) = ∑_{j=1}^{n}(−1)^{j−1}g_{i_{j}} x_{i_{1}} ∧ ⋯ ∧ x_{i_{j−1}} ∧ x_{i_{j+1}} ∧ ⋯ ∧ x_{i_{n}} .

For each R–module M, set K(g; M) = K(g; R) ⊗_{R} M, and note that H_{0}(K(g; M)) = M/(g)M. A crucial property of Koszul complexes is their depth^{5}- sensitivity: If M is a finite module over a noetherian ring R, then

sup{i | H_{i}(K(g; M)) ≠ 0} = r − depth_{R}((g), M) .

In particular, if g_{1}, …, g_{r} is an R–regular sequence, then K(g; R) is a free resolution of R′ = R/(g) over R, and H(K(g; M)) ≅ Tor^{R} (R′, M) .

Minimal complexes. A local ring (R, 𝔪, k) is a noetherian ring R with unique maximal ideal 𝔪 and residue field k = R/𝔪. A complex of R–modules F, such that ∂_{n}(F_{n}) ⊆ 𝔪F_{n−1} for each n, is said to be minimal. Here is the reason:

4Obey the Sign! While orthodox compliance is a nuisance, transgression may have consequences ranging anywhere from mild embarrassment (confusion of an element and its opposite) to major disaster (creation of complexes with ∂^{2} ≠ 0).

5Recall that the depth of an ideal I ⊆ R on M, denoted depth_{R}(I, M), is the maximal length of an M-regular sequence contained in I.

Proposition 1.1.2. If F is a bounded below complex of finite free modules over a local ring (R, 𝔪, k), then the following conditions are equivalent.

(i) F is minimal.

(ii) Each quasi-isomorphism α: F ⟶ F is an isomorphism.

(iii) Each quasi-isomorphism β: F ⟶ F′ to a bounded below minimal complex of

free modules is an isomorphism. (iv) Each quasi-isomorphism β: F ⟶ G to a bounded below complex of free mod-

ules is injective, and G = Im β ⊕ E for a split-exact subcomplex E.

Proof. (i) ⟹ (iii). The mapping cone C(β) is a bounded below complex of free modules with H(C(β)) = 0. Such a complex is split-exact, hence so is C(β) ⊗_{R} k ≅ C(β ⊗_{R} k). Thus, H(β ⊗_{R} k) is an isomorphism. Since both F ⊗_{R} k and F′ ⊗_{R} k have trivial differentials, each β_{n} ⊗_{R} k is an isomorphism. As F_{n} and F′_{n} are free modules, β_{n} is itself an isomorphism by Nakayama.

(iii) ⟹ (iv). Choose a subset Y ⊆ G such that ∂(Y) ⊗_{R} 1 is a basis of the vector space ∂(G⊗_{R} k). As Y ∪∂(Y) is linearly independent modulo 𝔪G, it extends to a basis of the R–module G^{♮}. Thus, the complex

E: ⨁_{y∈Y} (0 ⟶ Ry ⟶ R∂(y) ⟶ 0)  (∗)

is split-exact and a direct summand of G, hence G ⟶ G/E = G′ is a quasiisomorphism onto a bounded below free complex, that is minimal by construction. As the composition β′: F ⟶ G ⟶ G′ is a quasi-isomorphism, it is an isomorphism by our hypothesis. The assertions of (iv) follow.

(iv) ⟹ (ii): the split monomorphisms α_{n}: F_{n} ⟶ F_{n} are bijective.

(ii) ⟹ (i). Assume that (i) fails, and form a surjective quasi-isomorphism β: F ⟶ F/F′ = G onto a bounded below free complex as in the argument above. There is then a morphism γ: G ⟶ F with βγ = id^{G}, cf. Proposition 1.3.1. Such a γ is necessarily a quasi-isomorphism, hence γβ = α is a quasi-isomorphism F ⟶ F with Ker α ⊇ F′ ≠ 0, a contradiction. □

1.2. Syzygies. In this section (R, 𝔪, k) is a local ring. Over such rings, projectives are free; for finite modules, this follows easily from Nakayama.

A minimal resolution F of an R–module M is a free resolution, that is also a minimal complex. If m_{1}, …, m_{r} minimally generate M, then the Third Commandment^{6} prescribes to start the construction of a resolution by a map F_{0} = R^{r} ⟶ M with (a_{1}, …, a_{r}) ↦ a_{1}m_{1} +⋯+a_{r}m_{r}. This is a surjection with kernel in 𝔪R^{r}; iterating the procedure, one sees that M has a minimal resolution. As any two resolutions of M are linked by a morphism that induces the identity of M, Proposition 1.1.2 completes the proof of the following result of Eilenberg from [55], where minimal resolutions are introduced.

Proposition 1.2.1. Each finite R–module M has a minimal resolution, that is unique up to isomorphism of complexes. A minimal resolution F is isomorphic to a direct summand of any resolution of M, with complementary summand a splitexact free complex. In particular, pd_{R} M = sup{n | F_{n} ≠ 0}. □

6Resolve minimally!

The ‘uniqueness’ of a minimal resolution F implies that each R-module Syz_{n}^{R} (M) = Coker(∂_{n+1} : F_{n+1} ⟶ F_{n}) ≅ ∂_{n}(F_{n}) is defined uniquely up to a (noncanonical) isomorphism; it is called the n’th syzygy of M; note that Syz_{0}^{R} (M) = M, and Syz_{n}^{R} (M) = 0 for n < 0.

The number β_{n}^{R}(M) = rank_{R} F_{n} is called the n’th Betti number of M (over R). The complexes F ⊗_{R} k and Hom_{R} (F, k) have zero differentials, so Tor_{n}^{R} (M, k) ≅ F_{n} ⊗_{R} k and Ext_{R}^{n} (M, k) ≅ Hom_{R} (F_{n}, k) ; in other words:

Proposition 1.2.2. If M is a finite R–module, then

β_{n}^{R}(M) = ν_{R}( Syz_{n}^{R} (M)) = dim_{k} Tor_{n}^{R} (M, k) = dim_{k} Ext_{R}^{n} (M, k)

and pd_{R} M = sup{n | β_{n}^{R}(M) ≠ 0}. □

Syzygies behave well under certain base changes by local homomorphisms, that is, homomorphisms of local rings φ(R, 𝔪) ⟶ (R′, 𝔪′) with φ(𝔪) ⊆ 𝔪′.

Proposition 1.2.3. If M is a finite R–module and φ: R ⟶ R′ is a local homomorphism such that Tor_{n}^{R} (R′, M) = 0 for n > p, then

Syz_{n−p}^{R′} (R′ ⊗_{R} Syz_{p}^{R} (M)) ≅ R′ ⊗_{R} Syz_{n}^{R} (M) for n ≥ p .

Proof. Let F be a minimal free resolution of M over R.

Since H_{n}(R′ ⊗_{R} F) ≅ Tor_{n}^{R} (R′, M) = 0 for n > p, the complex of R′–modules (R′⊗_{R}F)_{⩾ p} is a free (and obviously minimal) resolution of the module Coker(R′⊗_{R} ∂_{p+1}) ≅ R′ ⊗_{R} Coker ∂_{p+1} ≅ R′ ⊗_{R} Syz_{p}^{R} (M). □

Corollary 1.2.4. Let M be a finite R–module.

(1) If R ⟶ R′ is a faithfully flat homomorphism of local rings, then

Syz_{n}^{R′} (R′ ⊗_{R} M) ≅ R′ ⊗_{R} Syz_{n}^{R} (M) for n ≥ 0 .

(2) If a sequence g_{1}, …, g_{r} ∈ R is both R–regular and M–regular, then

Syz_{n}^{R′} (M′) ≅ R′ ⊗_{R} Syz_{n}^{R} (M) for n ≥ 0

with R′ = R/(g_{1}, …, g_{r})R, and M′ = M/(g_{1}, …, g_{r})M. □

Proof. The proposition applies with p = 0, twice: by definition in case (1), and by Example 1.1.1 in case (2). □

When R is local, depth_{R} M = depth_{R}(𝔪, M) and depth R = depth_{R} R.

Corollary 1.2.5. If R is a direct summand of Syz_{n}^{R} (M), then n ≤ m, where m = max{0, depth R − depth M}.

Proof. Let n > m, and assume that Syz_{n}^{R} (M) has R as a direct summand. Choose a maximal (R ⊕ M)–regular sequence, and complete it (if necessary) to a maximal R–regular sequence g. By Example 1.1.1 and Proposition 1.2.3, R′ = R/(g) is a direct summand of a syzygy of the R′–module R′ ⊗_{R} Syz_{m}^{R} (M), hence sits in 𝔪F′, where F′ is a free R′–module. As depth R′ = 0, the ideal (0:_{R′} 𝔪) ≠ 0 annihilates R′; this is absurd. □

Depth can be computed cohomologically, by the formula

depth_{R} M = inf{n | Ext_{R}^{n} (k, M) ≠ 0} .

The following well known fact is recorded for ease of reference.

Lemma 1.2.6. If M is a finite R–module, then

depth_{R} Syz_{1}^{R} (M) = \begin{cases}depth_{R} M + 1 & when depth_{R} M < depth R; \\ g ≥ depth R & when depth_{R} M = depth R; \\ depth R & when depth_{R} M > depth R . \end{cases}

Proof. Track the vanishing of Ext_{R}^{n} (k, −) through the long exact cohomology sequence induced by the exact sequence 0 ⟶ Syz_{1}^{R} (M) ⟶ F_{0} ⟶ M ⟶ 0. □

Proposition 1.2.7. If M is a finite R–module with pd_{R} M < ∞, then (1) pd_{R} M + depth_{R} M = depth R.

(2) (0:_{R} M) = 0, or (0:_{R} M) contains a non-zero-divisor on R.

(3) ∑_{n ⩾ 0}(−1)^{n}β_{n}^{R}(M) ≥ 0, with equality if and only if (0:_{R} M) ≠ 0.

Proof. Set m = pd_{R} M, g = depth_{R} M, and d = depth R.

(1) As Syz_{m}^{R} (M) ≠ 0 is free, Corollary 1.2.5 yields m + g ≤ d. Thus, g ≤ d, and if g = d, then M is free and (1) holds. If g < d, then assume by descending induction that (1) holds for modules of depth > g, and use the lemma:

m + g = (pd_{R} Syz_{1}^{R} (M) + 1) + (depth_{R} Syz_{1}^{R} (M) − 1) = d .

(2) and (3). If F is a minimal resolution of M and 𝔭 ∈ Spec R, then

0 ⟶ (F_{m})_{𝔭} ⟶ (F_{m−1})_{𝔭} ⟶ … ⟶ (F_{1})_{𝔭} ⟶ (F_{0})_{𝔭} ⟶ M_{𝔭} ⟶ 0

is exact. If 𝔭 ∈ Ass R, then depth R_{𝔭} = 0, hence M_{𝔭} is free by (1).

Counting ranks, we get ∑_{n}(−1)^{n}β_{n}^{R}(M) = rank_{R_{𝔭}} M_{𝔭} ≥ 0.

If ∑_{n}(−1)^{n}β_{n}^{R}(M) = 0, then M_{𝔭} = 0 for all 𝔭 ∈ Ass R, so (0:_{R} M) ⊈ ⋃_{𝔭∈Ass R} 𝔭, that is, (0:_{R} M) contains a non-zero-divisor.

If (0 :_{R} M) ≠ 0, then (0 :_{R_{𝔭}} M_{𝔭}) = (0 :_{R} M)_{𝔭} ≠ 0 for 𝔭 ∈ Ass(0 :_{R} M) ⊆ Ass R; as M_{𝔭} is free, this implies M_{𝔭} = 0, and so ∑_{n}(−1)^{n}β_{n}^{R}(M) = 0. □

The arguments for (2) and (3) are from Auslander and Buchsbaum’s paper [17]; there is a new twist in the proof of their famous equality (1). It computes depths of syzygies when pd_{R} M < ∞; otherwise, Okiyama [126] proves:

Proposition 1.2.8. If M is a finite R–module with pd_{R} M = ∞, then

depth Syz_{n}^{R} (M) ≥ depth R for n ≥ max{0, depth R − depth_{R} M}

with at most one strict inequality, at n = 0 or at n = depth R − depth_{R} M + 1.

Proof. Set M_{n} = Syz_{n}^{R} (M) and d = depth R. Iterated use of Lemma 1.2.6 yields the desired inequality, and reduces the last assertion to proving that inequalities for n and n + 1 imply equality for n + 2.

Break down a minimal free resolution F of M into short exact sequences E^{i}: 0 ⟶ M_{i+1} ⟶^{ι_{i+1}} F_{i} ⟶^{π_{i}} M_{i} ⟶ 0. If depth_{R} N_{n+2} > d, then the cohomology exact sequence of E^{n+1} implies that the homomorphism

Ext_{R}^{d} (k, π_{n+1}) : Ext_{R}^{d} (k, F_{n+1}) ⟶ Ext_{R}^{d} (k, M_{n+1})

is injective. As depth_{R} M_{n} ≥ d, the cohomology sequence of E^{n} shows that

Ext_{R}^{d} (k, ι_{n+1}) : Ext_{R}^{d} (k, M_{n+1}RM) ⟶ Ext_{R}^{d} (k, F_{n})

is injective. Since ι_{n}∘π_{n+1} = ∂_{n+1} : F_{n+1} ⟶ F_{n}, we see that the map

Ext_{R}^{d} (k, ∂_{n+1}) = Ext_{R}^{d} (k, ι_{n}) ∘ Ext_{R}^{d} (k, π_{n+1})

is injective as well. But ∂_{n+1} : F_{n+1} ⟶ F_{n} is a matrix with elements in 𝔪, so Ext_{R}^{d} (k, ∂_{n+1}) = 0, hence Ext_{R}^{d} (k, F_{n+1}) = 0. Since depth_{R} F_{n+1} = d this is impossible, so depth M_{n} ≤ d, as desired. □

Remark 1.2.9. By the last two results, there exists an R–regular sequence g of length d = depth R, that is also regular on N = Syz_{m}^{R} (M), where m = max{0, d − depth M}. Proposition 1.2.4.2 yields β_{n+m}^{R}(M) = β_{n}^{R}(N) = β_{n}^{R/(g)}(N/(g)N) for n ≥ 0. When k is infinite, a sequence g may be found that also preserves multiplicity: mult(R) = mult(R/(g)); if k is finite, then R′ = R[t]_{𝔪[t]} has mult(R′) = mult(R), and β_{n}^{R}(N) = β_{n}^{R′}(N ⊗_{R} R′), for n ≥ 0.

Remark 1.2.10. The name graded ring is reserved^{1} for rings equipped with a direct sum decomposition R = ⨁_{i ⩾ 0} R_{i}, and having R_{0} = k, a field. For such a ring we denote 𝔪 the irrelevant maximal ideal ⨁_{i > 0} R_{i}. An R–moduleM is graded if M = ⨁_{j∈ℤ} M_{j} and R_{i}M_{j} ⊆ M_{i+j} for all i, j ∈ ℤ. To minimize confusion with gradings arising from complexes, we say that a ∈ M_{i} has internal degrees i, and write deg(a) = i. The d’th translate of M is the graded R–module M(d) with M(d)_{j} = M_{j+d}. A degree zero homomorphism α: M ⟶ N of graded R–modules is an R–linear map such that α(M_{j}) ⊆ N_{j} for all j.

The free objects in the category of graded modules and degree zero homomorphisms are isomorphic to direct sums of modules of the form R(d). Each graded R–module M has a graded resolution by free graded modules with differentials that are homomorphisms of degree zero. If M_{j} = 0 for j ≪ 0 (in particular, if M is finitely generated), then such a resolution F exists with ∂(F_{n}) ⊆ 𝔪F_{n} for all n. This minimal graded resolution is unique up to isomorphism of complexes of graded R– modules, so the numbers β_{nj} appearing in isomorphisms F_{n} ≅ ⨁_{j∈ℤ} R(−j)^{β_{nj}} are uniquely defined, and finite if M is a finite R–module; these graded Betti numbers of M over R are denoted β_{nj}^{R}(M).

1.3. Differential graded algebra. The term refers to a hybrid of homological algebra and ring theory. When describing the progeny^{7}, we systematically replace the compound ‘differential graded’ by the abbreviation DG.

DG algebras. A DG algebra A is a complex (A, ∂), with an element 1 ∈ A_{0} (the unit), and a morphism of complexes (the product)

A ⊗_{𝕜} A → A , a ⊗ b ↦ ab ,

that is unitary: 1a = a = a1, and associative: a(bc) = (ab)c. In addition, we assume the A is (graded) commutative:

ab = (−1)^{|a||b|}ba for a, b ∈ A  and  a^{2} = 0 when |a| is odd ,

and that A_{i} = 0 for i < 0; without them, we speak of associative DG algebras. The fact that the product is a chain map is expressed by the Leibniz rule:

∂(ab) = ∂(a)b + (−1)^{|a|}a∂(b) for a, b ∈ A ,

Its importance comes from a simple observation: The cycles Z(A) are a graded subalgebra of A, the boundaries ∂(A) are an ideal in Z(A), hence the canonical projection Z(A) ⟶ H(A) makes the homology H(A) into a graded algebra. In particular, each H_{n}(A) is a module over the ring H_{0}(A).

7It is not that exotic: a commutative ring is precisely a DG algebra concentrated in degree zero, and a DG module over it is simply a complex. A prime example of a ‘genuine’ DG algebra is a Koszul complex, with multiplication given by wedge product.

A morphism of DG algebras is a morphism of complexes ϕ: A ⟶ A′, such that ϕ(1) = 1 and ϕ(ab) = ϕ(a)ϕ(b); we say that A′ is a DG algebra over A.

If A and A′ are DG algebras, then the tensor products of complexes A ⊗_{𝕜} A′ is a DG algebra with multiplication (a ⊗ a′)(b ⊗ b′) = (−1)^{|a′||b|}(ab ⊗ a′b′).

A graded algebra is a DG algebra with zero differential, that is, a family^{8} {A_{n}}, rather than a direct sum ⨁_{n} A_{n}.

A DG module U over the DG algebra A is a complex together with a morphism A ⊗ U → U, a ⊗ u ↦ au, that satisfies the Leibniz rule

∂(au) = ∂(a)u + (−1)^{|a|}a∂(u) for a ∈ A and u ∈ U

and is unitary and associative in the obvious sense. A module is a DG module with zero differential; U^{♮} is a module over A^{♮}, and H(U) is a module over H(A).

Let U and V be DG modules over A.

A homomorphism β: U ⟶ V of the underlying complexes is A–linear if β(au) = (−1)^{|β||a|}aβ(u) for all a ∈ A and u ∈ U. The A–linear homomorphisms form a subcomplex Hom_{A} (U, V) ⊆ Hom_{𝕜} (U, V) . The action

(aβ)(u) = a(β(u)) = (−1)^{|a||β|}β(au)

turns it into a DG module over A. Two A–linear chain maps β, β′: U ⟶ V that are homotopic by an A–linear homotopy are said to be homotopic over A. Thus, H_{d}(Hom_{A} (U, V) ) is the set of homotopy classes of A–linear, degree d chain maps. DG modules over A and their A–linear morphisms are, respectively, the objects and morphisms of the category of DG modules over A.

The residue complex U ⊗_{A}V of U ⊗_{𝕜}V by the subcomplexspanned by all elements au ⊗_{𝕜} v − (−1)^{|a||u|}u ⊗_{𝕜} av, has an action

a(u ⊗_{A} v) = au ⊗_{A} v = (−1)^{|a||u|}u ⊗_{A} av .

It is naturally a DG module over A, and has the usual universal properties.

The shift ΣU becomes a DG module over A by setting aΣ^{U}(u) = (−1)^{|a|}Σ^{U}(au); the map Σ^{U}: U ⟶ ΣU is then an A–linear homomorphism.

The mapping cone of a morphism β: U ⟶ V of DG modules over A is a DG module over A, and the maps in the mapping cone sequence are A–linear.

Semi-free modules. A bounded below DG module F over A is semi-free if its underlying A^{♮}–module F^{♮} has a basis^{9} {e_{λ}}_{λ∈Λ}. Thus, for each f ∈ F there are unique a_{λ} ∈ A with f = ∑_{λ∈Λ} a_{λ}e_{λ}; we set Λ_{n} = {λ ∈ Λ : |e_{λ}| = n}.

Note that F is not a free object on {e_{λ}} in the category of DG modules over A: As ∂(e_{λ}) is a linear combination of basis elements e_{μ} of lower degree, the choice for the image of e_{λ} is restricted by the choices already made for the images of the e_{μ}. What freeness remains is in the important lifting property:

8This convention reduces the length of the exposition by 1.713%, as it trims from each argument all sentences starting with ‘We may assume that the element x is homogeneous’; note that by Remark 1.2.10 above, a graded ring is the usual thing.

9Over a ring, a such a DG module is simply a bounded below complex of free modules. For arbitrary DG modules over any graded associative DG algebras, the notion is defined by a different condition: cf. [33], where the next three propositions are established in general.

Proposition 1.3.1. If F is a semi-free DG module over a DG algebra A, then each diagram of morphisms of DG modules over A represented by solid arrows

[figure: U; ≃; β; F; γ; α; V]

with a surjective quasi-isomorphism β can be completed to a commutative diagram by a morphism γ, that is defined uniquely up to A–linear homotopy.

Remark. A degree d chain map F ⟶ V is nothing but a morphism F ⟶ Σ^{−d}V, so the proposition provides also a ‘unique lifting property’ for chain maps.

Proof. Note that F^{n} = ⨁_{|e_{λ}| ⩽ n} Ae_{λ} is a DG submodule of F over A, and F^{n} = 0 for n ≪ 0. By induction on n, we may assume that γ^{n}: F^{n} ⟶ U has been constructed, with α|_{F^{n}} = β∘γ^{n}.

For each λ ∈ Λ_{n+1}, we have ∂(α∂(e_{λ})) = α(∂^{2}(e_{λ})) = 0, so α∂(e_{λ}) is a cycle in V. Since β is a surjective quasi-isomorphism, there exists a cycle z′_{λ} ∈ U, such that β(z′_{λ}) = α∂(e_{λ}). Thus, z_{λ} = γ^{n}∂(e_{λ}) − z′_{λ} ∈ U satisfies

∂(z_{λ}) = γ^{n}∂^{2}(e_{λ}) − ∂(z′_{λ}) = 0  and  β(z_{λ}) = α∂(e_{λ}) − β(z′_{λ}) = 0 ,

that is, z_{λ} is a cycle in W = Ker β. The homology exact sequence of the short exact sequence of DG modules 0 ⟶ W ⟶ U ⟶ V ⟶ 0 shows that H(W) = 0, hence z_{λ} = ∂(y_{λ}) for some y_{λ} ∈ W. In view of our choices, the formula

γ^{n+1}(f + ∑_{λ∈Λ_{n+1}} a_{λ}e_{λ}) = γ^{n}(f) + ∑_{λ∈Λ_{n+1}} a_{λ}y_{λ} for f ∈ F^{n}

defines a morphism of DG modules γ^{n+1} : F^{n+1} ⟶ U, with γ^{n+1}|_{F^{n}} = γ^{n}, and completes the inductive construction. As F = ⋃_{n∈ℤ} F^{n}, setting γ(f) = γ^{n}(f) whenever f ∈ F^{n}, we get a morphism γ: F ⟶ U with α = βγ.

If γ′: F ⟶ U is a morphism with α = βγ′, then β(γ −γ′) = 0, hence there exists a morphism δ: F ⟶ W such that γ − γ′ = ιδ, where ι: W ⊆ U is the inclusion. Again, we assume by induction that a homotopy σ^{n}: F^{n} ⟶ W between δ|_{F^{n}} and 0 is available: δ|_{F^{n}} = ∂σ^{n} + σ^{n}∂. As

∂(δ(e_{λ}) − σ^{n}∂(e_{λ})) = δ∂(e_{λ}) − (∂σ^{n})(∂(e_{λ}))

= (δ − ∂σ^{n})(∂(e_{λ})) = (σ^{n}∂)(∂(e_{λ})) = 0

and H(W) = 0, there is a w_{λ} ∈ W such that ∂(w_{λ}) = δ(e_{λ}) − σ^{n}∂(e_{λ}). Now

σ^{n+1}(f + ∑_{λ∈Λ_{n+1}} a_{λ}e_{λ}) = σ^{n}(f) + ∑_{λ∈Λ_{n+1}}(−1)^{|a_{λ}|}a_{λ}w_{λ}

is a degree 1 homomorphism σ^{n+1} : F^{n+1} ⟶ W, with δ|_{F^{n+1}} = ∂σ^{n+1} + σ^{n+1}∂, and σ^{n+1}|_{F^{n}} = σ^{n}. In the limit, we get a homotopy σ: F ⟶ W from δ to 0, and then σ′ = ισ: F ⟶ U is a homotopy from γ to γ′. □

Proposition 1.3.2. If F is a semi-free DG module, then each quasi-isomorphism β: U ⟶ V of DG modules over A induces quasi-isomorphisms

Hom_{A} (F, β) : Hom_{A} (F, U) ⟶ Hom_{A} (F, V) ; F ⊗_{A} β: F ⊗_{A} U ⟶ F ⊗_{A} V .

Proof. To prove that Hom_{A} (F, β) is a quasi-isomorphism, we show the exactness of its mapping cone, which is isomorphic to Hom_{A} (F, C(β)) . Thus, we want to show that each chain map F ⟶ C(β) is homotopic to 0. Such a chain map is a lifting of F ⟶ 0 over the quasi-isomorphism C(β) ⟶ 0. Since 0: F ⟶ C(β) is another such lifting, they are homotopic by Proposition 1.3.1.

To prove that β ⊗_{A} F is a quasi-isomorphism, we use the exact sequences 0 ⟶ F^{n} ⟶ F^{n+1} ⟶ \overline{F}^{n+1} ⟶ 0 of DG modules over A, involving the submodules F^{n} from the preceding proof. The sequences split over A^{♮}, and so induce commutative diagrams with exact rows

0 ⟶ U ⊗_{A} F^{n} ⟶ U ⊗_{A} F^{n+1} ⟶ U ⊗_{A} \overline{F}^{n+1} ⟶ 0
β⊗_{A}F^{n}↓  ^{β⊗_{A}F^{n+1}}↓  _{β⊗_{A}\overline{F}^{n+1}}↓
0 ⟶ V ⊗_{A} F^{n} ⟶ V ⊗_{A} F^{n+1} ⟶ V ⊗_{A} \overline{F}^{n+1} ⟶ 0 .

By induction on n, we may assume that β ⊗_{A} F^{n} is a quasi-isomorphism. As

\overline{F}^{n+1} ≅ ⨁_{λ∈Λ_{n+1}} Ae_{λ} with ∂(e_{λ}) = 0 for all λ ∈ Λ_{n+1} ,

the map β ⊗_{A} \overline{F}^{n+1} is a quasi-isomorphism. By the Five-Lemma, β ⊗_{A} F^{n+1} is one as well, hence so is β ⊗_{A} F = β ⊗ (inj lim_{n} F^{n}) = inj lim_{n}(β ⊗_{A} F^{n}). □

Proposition 1.3.3. Let U be a DG module over a DG algebra A. Each quasi-iso-morphism γ: F ⟶ G of semi-free modules induces quasi-isomorphisms

Hom_{A} (γ, U) : Hom_{A} (G, U) ⟶ Hom_{A} (F, U) ; γ ⊗_{A} U: F ⊗_{A} U ⟶ G ⊗_{A} U.

Proof. The mapping cone C = C(γ) is exact. It is semi-free, so by the preceding proposition the quasi-isomorphism C ⟶ 0 induces a quasi-isomorphism Hom_{A} (C, C) ⟶ 0. Thus, there is a homotopy σ from id^{C} to 0^{C}. It is easily verified that Hom_{A} (σ, U) and σ ⊗_{A} U are null-homotopies on Hom_{A} (C, U) and C ⊗_{A} U, so these complexes are exact. They are isomorphic, respectively, to Σ^{−1} C(Hom_{A} (γ, U) ) and C(γ ⊗_{A} U), which are therefore exact. We conclude that Hom_{A} (γ, U) and γ ⊗_{A} U are quasi-isomorphisms. □

The preceding results have interesting applications even for complexes over a ring. A first illustration occurs in the proof of Proposition 1.1.2. Another one is in the following proof of the classical Künneth Theorem.

Proposition 1.3.4. If G is a bounded below complex of free R–modules, such that F = H(G) is free, then the Künneth map

κ^{GU}: H(G) ⊗_{R} H(U) ⟶ H(G ⊗_{R} U) ,
κ^{GU}(cls(g) ⊗ cls(u)) = cls(g ⊗ u) ,

is an isomorphism for each complex of R–modules U.

Proof. Set F = H(G). The composition of an R– linear splitting of the surjection Z(G) ⟶ F with the injection Z(G) ⟶ G is a quasi-isomorphism γ: F ⟶ G of semifree DG modules over R. By the last proposition, so is γ ⊗_{R} U: F ⊗_{R} U ⟶ G⊗_{R} U. The Künneth map being natural, it suffices to show that κ^{F U} is bijective. As ∂^{F} = 0 and each F_{n} is free, this is clear. □

2. Multiplicative structures on resolutions

Is it possible to ‘enrich’ resolutions over a commutative ring Q, by endowing them with DG module or DG algebra structures? The rather complete—if at first puzzling—answer comes in three parts:

• For residue rings of Q, algebra structures are carried by essentially all resolutions of length ≤ 3; for finite Q–modules, DG module structures exist on all resolutions of length ≤ 2.

• Beyond these bounds, not all resolutions support such structures.

• There always exist resolutions, that do carry the desired structure. In this chapter we present in detail the results available on all three counts. Most developments in the rest of the notes are built on resolutions that comply with the Fourth Commandment^{10}.

2.1. DG algebra resolutions. Let Q be a commutative ring and let R be a Q–algebra. A DG algebra resolution of R over Q consists of a (commutative) DG algebra A, such that A_{i} is a projective Q–module for each i, and a quasiisomorphism ϵ^{A}: A ⟶ R of DG algebras over Q.

The next example is the grandfather of all DG algebra resolutions.

Example 2.1.0. If R = Q/(f) for a Q–regular sequence f, then the Koszul complex on f is a DG algebra resolution of R over Q.

‘Short’ projective resolutions often carry DG algebra structures.

Example 2.1.1. If R = Q/I has a resolution A of length 1 of the form

0 ⟶ F_{1} ⟶ Q ⟶ 0

then the only product that makes it a graded algebra over Q is defined by the condition F_{1} · F_{1} = 0; it clearly makes A into a DG algebra.

If ϕ is a matrix, then for J, K ⊂ ℕ we denote ϕ_{J}^{K} the submatrix obtained by deleting the rows with indices from J and the columns with indices from K.

Example 2.1.2. If R = Q/I has a free resolution of length 2, then by the Hilbert-Burch Theorem there exist a non-zero-divisor a and an r × (r − 1) matrix ϕ, such that a free resolution of R over Q is given by the complex

A: 0 ⟶ ⨁_{k=1}^{r−1} Qf_{k} ⟶^{∂_{2}} ⨁_{j=1}^{r} Qe_{j} ⟶^{∂_{1}} Q ⟶ 0

∂_{2} = ϕ ∂_{1} = a( det(ϕ_{1}), …, (−1)^{j−1} det(ϕ_{j}), …, (−1)^{r−1} det(ϕ_{r}))

Herzog [86] shows that there exists a unique DG algebra structure on A, namely:

e_{j} · e_{k} = −e_{k} · e_{j} = −a ∑_{ℓ=1}^{r−1}(−1)^{j+k+ℓ} det(ϕ_{jk}^{ℓ})f_{ℓ}  for j < k;  e_{j} · e_{j} = 0 .

Example 2.1.3. An ideal I in a local ring (Q, 𝔫) is Gorenstein if R = Q/I has pd_{Q} R = p < ∞, Ext_{Q}^{n} (R, Q) = 0 for n ≠ p, and Ext_{Q}^{p} (R, Q) ≅ R; thus, when Q is regular, I is Gorenstein if and only if R is a Gorenstein ring.

10Resolve in kind!

If I is Gorenstein, pd_{Q} R = 3, and I is minimally generated by r elements, then J. Watanabe [157] proves that the number r is odd, and Buchsbaum-Eisenbud [47] show that there exists an alternating r × r matrix ϕ with elements in 𝔫, such that a minimal free resolution A of R over Q has the form

A: 0 ⟶ Qg ⟶^{∂_{3}} ⨁_{k=1}^{r} Qf_{k} ⟶^{∂_{2}} ⨁_{j=1}^{r} Qe_{j} ⟶^{∂_{1}} Q ⟶ 0

∂_{2} = ϕ ∂_{1} = ( pf(ϕ_{1}^{1}), …, (−1)^{j−1} pf(ϕ_{j}^{j}), …, (−1)^{r−1} pf(ϕ_{r}^{r})) = ∂_{3}^{∗}

where pf(α) is the Pfaffian of α. A DG algebra structure on A is given by

e_{j} · e_{k} = −e_{k} · e_{j} = ∑_{ℓ=1}^{r}(−1)^{j+k+ℓ}ρ_{jkℓ} pf(ϕ_{jkℓ}^{jkℓ})f_{ℓ} for j < k;

e_{j} · e_{j} = 0 ; e_{j} · f_{k} = f_{k} · e_{j} = δ_{jk}g ,

where ρ_{jkℓ} is equal to −1 if j < ℓ < k, and to 1 otherwise, cf. [21].

More generally, Buchsbaum and Eisenbud [47] prove that DG algebra structures always exist in projective dimension ≤ 3:

Proposition 2.1.4. If A is a projective resolution of a Q–module R, such that A_{0} = Q and A_{n} = 0 for n ≥ 4, then A has a structure of DG algebra.

Proof. For the construction, consider the complex S^{2}(A), that starts as

… ⟶ (A_{1} ⊗ A_{3}) ⊕ S^{2}(A_{2}) ⊕ A_{4} ⟶^{δ_{4}} (A_{1} ⊗ A_{2}) ⊕ A_{3}

⟶^{δ_{3}} (⋀^{2}A_{1}) ⊕ A_{2} ⟶^{δ_{2}} A_{1} ⟶^{δ_{1}} Q ⟶ 0

with differentials defined by the condition δ_{n}|_{A_{n}} = ∂_{n} and the formulas

δ_{2}(a ∧ b) = ∂_{1}(a)b − ∂_{1}(b)a;  δ_{3}(a ⊗ b) = −a ∧ ∂_{2}(b) + ∂_{1}(a)b;
δ_{4}(a ⊗ b) = ∂_{1}(a)b − a ∗ ∂_{3}(b) ;  δ_{4}(a ∗ b) = ∂_{2}(a) ⊗ b + ∂_{2}(b) ⊗ a ,

where ∗ denotes the product in the symmetric algebra. The complex S^{2}(A) is projective and naturally augmented to R, so by the Lifting Theorem there is a morphism μ: S^{2}(A) ⟶ A that extends the identity map of R.

Define a product on A (temporarily denoted ·) by composing the canonical projection A ⊗ A ⟶ S^{2}(A) with μ. With the unit 1 ∈ Q = A_{0}, one has all the properties required from a DG algebra except, possibly, associativity. Because A_{n} = 0 for n ≥ 4, this may be an issue only for a product of three elements a, b, c, of degree 1. For them we have

∂_{3}((a · b) · c) = ∂_{2}(a · b) · c + (a · b)∂_{1}(c)
= (∂_{1}(a)b) · c − (∂_{1}(b)a) · c + (a · b)∂_{1}(c)
= ∂_{1}(a)(b · c) − ∂_{1}(b)(a · c) + ∂_{1}(c)(a · b) .

A similar computation of ∂_{3}(a · (b · c)) yields the same result.

As ∂_{3} is injective, we conclude that (a · b) · c = a · (b · c). □

Next we describe two existence results in projective dimension 4.

Example 2.1.5. If Q is local, pd_{Q} Q/I = 4, and I is Gorenstein, then Kustin and Miller [101] prove that the minimal free resolution of R over Q has a DG algebra structure if Q ∋ \frac{1}{2}, a restriction removed later by Kustin [97].

Example 2.1.6. If I is a grade 4 perfect ideal generated by 5 elements in a local ring Q containing \frac{1}{2}, then Kustin [98], building on work of Palmer [127], constructs a DG algebra structure on the minimal free resolution of Q/I.

The question naturally arises whether it is possible to put a DG algebra structure on each minimal resolution of a residue ring of a local ring. As far as the projective dimension is concerned, the list above turns out to be essentially complete: for (perfect) counterexamples in dimension 4, cf. Theorem 2.3.1.

On the positive side, each Q–algebra has some DG algebra resolution, obtained by a universal construction: Given a cycle z in a DG algebra A, we embed A into a DG algebra A′ by freely adjoining a variable y such that ∂y = z. In A′ the cycle z has been killed: it has become a boundary.

Construction 2.1.7. Exterior variable. When |z| is even, 𝕜[y] is the exterior algebra over 𝕜 of a free 𝕜–module on a generator y of degree |z| + 1; the differential of A[y]^{♮} = A^{♮} ⊗_{𝕜} 𝕜[y] is given by

∂(a_{0} + a_{1}y) = ∂(a_{0}) + ∂(a_{1})y + (−1)^{|a_{1}|}a_{1}z;

thus, when A is concentrated in degree zero, A[y] is the Koszul complex K(z; A).

Construction 2.1.8. Polynomial variable. When |z| is odd, 𝕜[y] is the polynomial ring over 𝕜 on a variable y of degree |z| + 1, A[y]^{♮} = A^{♮} ⊗_{𝕜} 𝕜[y], and

∂( ∑_{i} a_{i}y^{i}) = ∑_{i} ∂(a_{i})y^{i} + ∑_{i}(−1)^{|a_{i}|}ia_{i}zy^{i−1} .

In either case, ∂ is the unique differential on A[y]^{♮} that extends the differential on A, satisfies ∂(y) = z, and the Leibniz formula; we call y a variable over A, and often use the more complete notation, A[y | ∂(y) = z].

Let u = cls(z) ∈ H(A) be the class of z. The quotient complex A[y]/A is trivial in degrees n ≤ |z| and is equal to A_{0}y in degree n = |z| + 1, so the homology exact sequence shows that the inclusion A ↪ A[y] induces a morphism of graded algebras H(A)/u H(A) ⟶ H(A[y]) that is bijective in degrees ≤ |z|.

A semi-free extension of A is a DG algebra A′ obtained by repeated adjunction of free variables. If Y is the set of all variables adjoined in the process, then we write A[Y] for A′; we also set Y_{n} = {y ∈ Y | |y| = n} and Y_{⩽ n} = ⋃_{i=0}^{n} Y_{i}. Semi-free algebra extensions have a lifting property:

Proposition 2.1.9. If A[Y] is a semi-free extension of a DG algebra A, then each diagram of morphisms of DG algebras over A represented by solid arrows

[figure: B; ≃; β; A[Y]; γ; α; C]

with a surjective quasi-isomorphism β can be completed to a commutative diagram by a morphism γ, that is defined uniquely up to A–linear homotopy.

Proof. Set A^{i} = A[Y_{⩽ i}]. Starting with the structure map A ⟶ B, we assume that for some n ≥ −1 we have a morphism γ^{n}: A^{n} ⟶ B of DG algebras over A, with βγ^{n} = α|_{A^{n}}. Over A^{n}, the set {1} ∪ Y_{n+1} generates a semi-free submodule F^{n+1} of A[Y]. By Theorem 1.3.1, γ^{n} extends to a morphism δ^{n+1} : F^{n+1} ⟶ B of DG modules over A^{n}. The graded commutative algebra (A^{n+1})^{♮} is freely generated over (A^{n})^{♮} by Y_{n+1}, so δ^{n+1} extends uniquely to a homomorphisms of graded algebras γ^{n+1} : A^{n+1} ⟶ B. Since δ^{n+1} is a morphism of DG modules, γ^{n+1} is necessarily a morphism of DG algebras. In the limit, one gets a morphism of DG algebras γ: A[Y] ⟶ B, with the desired properties. □

A resolvent of Q–algebra R is a DG algebra resolution of R over Q, that is a semi-free DG algebra extension of Q.

Proposition 2.1.10. Each (surjective) homomorphism ψ: Q ⟶ R has a resolvent Q[Y] (with Y_{0} = ∅). When Q is noetherian and R is a finitely generated Q–algebra there exists a resolvent with Y_{n} finite for each n.

Proof. Factor ψ as an inclusion Q ↪ Q[Y_{0}] into a polynomial ring on a set Y_{0} of variables and a surjective morphism ψ′: Q[Y_{0}] ↠ R that maps Y_{0} to a set of generators of the Q–algebra R. The Koszul complex Q ↪ Q[Y_{0}][[Y_{⩽ 1}] on a set of generators of Ker ψ′ is a semi-free extension of Q, with H_{0}(Q[Y_{⩽ 1}]) ≅ R.

By induction on i, assume that consecutive adjunctions to Q[Y_{⩽ 1}] of sets Y_{j} of variables of degrees j = 2, …, n have produced an extension Q ↪ Q[Y_{⩽ n}] with H_{i}(Q[Y_{⩽ n}]) = 0 for 0 < i < n. Adjoin to Q[Y_{⩽ n}] a set Y_{n+1} of variables of degree n + 1 that kill a set of generators of the Q[Y_{0}]–module H_{n}(Q[Y_{⩽ n}]). As observed above, we then get H_{i}(Q[Y_{⩽ n+1}]) = 0 for 0 < i < n + 1.

Going over the induction procedure with a noetherian hypothesis in hand, it is easy to see that a finite set Y_{n} suffices at each step. □

2.2. DG module resolutions. There is nothing esoteric about DG module structures on resolutions of modules. In fact, they offer a particularly adequate framework for important commutative algebra information.

Remark 2.2.1. Let U be a free resolution of a Q–module M. If f ∈ (0 :_{Q} M), then both f id^{U} and 0^{U} induce the zero map on M, hence they are homotopic, say f id^{U} = ∂σ + σ∂. A homotopy σ such that σ^{2} = 0 exists if and only if U can be made a DG module over the Koszul complex A = Q[y | ∂(y) = f]: just set yu = σ(u), and note that the homotopy condition for σ translates precisely into the Leibniz rule fu = ∂(yu) + y∂(u) for the action of y.

In some cases one can prove the existence of a square-zero homotopy.

Proposition 2.2.2. If (Q, 𝔫, k) is a local ring, f ∈ 𝔫 ∖ 𝔫^{2}, and M is a Q–module such that fM = 0, then the minimal free resolution U of M over Q has a structure of semi-free DG module over the Koszul complex A = Q[y | ∂(y) = f].

Proof. Setting f_{j} = f id^{U_{j}}, we restate the desired assertion as follows: For each j there is a homomorphism σ_{j}: U_{j} ⟶ U_{j+1}, such that:

∂_{j+1}σ_{j} + σ_{j−1}∂_{j} = f_{j} ; σ_{j−1}(U_{j−1}) = Ker σ_{j} ;
σ_{j−1}(U_{j−1}) is a direct summand of U_{j} .

Indeed, by the preceding remark the first two conditions define on U a structure of DG module over A; the third one is then equivalent to an isomorphism of A^{♮}– modules U^{♮} ≅ A^{♮} ⊗_{Q} V, with V = U^{♮}/yU^{♮}.

The map σ_{j} = 0 has the desired properties when j < 0, so we assume by induction that σ_{j} has been constructed for j ≤ i, with i ≥ −1. Since

∂_{i+1}(f_{i+1} − σ_{i}∂_{i+1}) = f∂_{i+1} − ∂_{i+1}σ_{i}∂_{i+1} = f∂_{i+1} − f∂_{i+1} + σ_{i−1}∂_{i}∂_{i+1} = 0 and U is acyclic, there exists a map σ_{i+1} such that ∂_{i+2}σ_{i+1} = f_{i+1} − σ_{i}∂_{i+1}.

Furthermore, as σ_{i}σ_{i−1} = 0 by the induction hypothesis, we have

(f_{i+1} − σ_{i}∂_{i+1})σ_{i} = fσ_{i} − σ_{i}∂_{i+1}σ_{i} = fσ_{i} − fσ_{i} + σ_{i}σ_{i−1}∂_{i} = 0 .

Thus, we can arrange for σ_{i+1} to be zero on the direct summand Im σ_{i} of U_{i+1}.

Let V_{i+1} be a complementary direct summand of Im σ_{i} in U_{i+1}. For v ∈ V_{i+1} ∖ 𝔫V_{i+1}, we have ∂_{i+2}σ_{i+1}(v) = fv − σ_{i}∂_{i+1}(v). The two terms on the right lie in distinct direct summands, and fv ∉ 𝔫^{2}V_{i+1}, so ∂_{i+2}σ_{i+1}(v) ∉ 𝔫^{2}U_{i+1}, and thus σ_{i+1}(v) ∉ 𝔫U_{i+2}. This shows that σ_{i+1} ⊗_{Q} k: V_{i+1} ⊗_{Q} k ⟶ U_{i+2} ⊗_{Q} k is injective, so σ_{i+1} is a split injection, completing the induction step. □

The construction of σ above is taken from Shamash [143]; it is implicit in Nagata’s [124] description of the syzygies of M over Q/(f) in terms of those over Q, presented next; neither source uses DG module structures.

Theorem 2.2.3. Let (Q, 𝔫, k) be a local ring, let f ∈ 𝔫 ∖ 𝔫^{2} be Q–regular, and let M be a finite module over R = Q/(f). If U is a minimal free resolution of M over Q, then there exists a homotopy σ from id^{U} to 0^{U}, such that

U′: … ⟶ \frac{U_{n}}{fU_{n} + σ(U_{n−1})} ⟶ … ⟶ \frac{U_{1}}{fU_{1} + σ(U_{0})} ⟶ \frac{U_{0}}{fU_{0}} ⟶ 0  (∗)

is a minimal R–free resolution of M and rank_{Q} U_{n} = rank_{R} U′_{n} + rank_{R} U′_{n−1}.

Proof. Set A = Q[y | ∂(y) = f]. By the preceding proposition, U is a semi-free DG module over A = Q[y | ∂(y) = f]. Let σ be the homotopy given by left multiplication with y. By Proposition 1.3.2 the quasi-isomorphism A ⟶ R induces a quasiisomorphism U ⟶ U ⊗_{A} R = U′. The complex U/(f, y)U = U/(fU + σ(U)) = U′ is obviously minimal, so we are done. □

DG module structures are more affordable than DG algebra structures.

Remark 2.2.4. Let S = Q/J, and let B be a DG algebra resolution of S over Q. If A is the Koszul complex on a sequence f ⊂ J, then by Proposition 2.1.9 the canonical map A ⟶ Q/(f) ⟶ S lifts over the surjective quasi-isomorphism B ⟶ S to a morphism A ⟶ B of DG algebras over Q.

Thus, any DG algebra resolution of S over Q is a DG module over each Koszul complex K(f; Q). However, DG module structures may exist even when DG algebra structures do not: for an explicit example, cf. Srinivasan [146].

Short projective resolutions always carry DG module structures: Iyengar [92] notes that a modification of the argument for Proposition 2.1.4 yields

Proposition 2.2.5. Let R = Q/I, and let M be an R–module. If U is a projective resolution of M, and U_{n} = 0 for n ≥ 3, then U has a structure of DG module over each DG algebra resolution A of R over Q. □

On the other hand, not all minimal resolutions of length ≥ 3 support DG module structures over DG algebras A ≠ Q, cf. Theorem 2.3.1.

Let A be a DG algebra resolution of R over Q, and let M be an R–module. A DG module resolution of M over A is a quasi-isomorphism ϵ^{U}: U ⟶ M of DG modules over A, such that for each n the Q–module U_{n} is projective. To construct such resolutions in general, we describe a ‘linear’ adjunction process.

Construction 2.2.6. Adjunction of basis elements. Let V be a DG module over A, and Z = {z_{λ} ∈ V }_{λ∈Λ} be a set of cycles. For a linearly independent set Y = {y_{λ} : |y_{λ}| = |z_{λ}| + 1}_{λ∈Λ} over the graded algebra A^{♮} underlying A, set

∂(v + ∑_{λ∈Λ} a_{λ}y_{λ}) = ∂(v) + ∑_{λ∈Λ} ∂(a_{λ})y_{λ} + (−1)^{|a_{λ}|} ∑_{λ∈Λ} a_{λ}z_{λ} .

This is the unique differential on V ⨁_{λ∈Λ} Ay_{λ} which extends that of V, satisfies the Leibniz rule, and has ∂(y_{λ}) = z_{λ} for λ ∈ Λ.

Proposition 2.2.7. If A is a DG algebra resolution of R over Q and M is an R–module, then M has a semi-free resolution U over A.

Proof. Pick a surjective homomorphism F ⟶ M from a free Q–module, and extend it to a chain map of DG modules ϵ^{0} : U^{0} = A ⊗_{Q} F ⟶ M. Clearly, H_{0}(ϵ^{0}) is surjective. If Z^{0} is a set of cycles whose classes generate Ker H_{0}(ϵ^{0}), then let U^{1} be the semi-free extension of U^{0}, obtained by adjunction of a linearly independent set Y^{1} that kills Z^{0}. Extend ϵ^{0} to ϵ^{1} : U^{1} ⟶ M by ϵ^{1}(Y^{1}) = 0, and note that H_{0}(ϵ^{1}) is an isomorphism. Successively adjoining linearly independent sets Y^{n}, of elements of degree n = 2, 3, … that kill sets Z^{n−1} of cycles generating H_{n−1}(U^{n−1}), we get a semi-free DG module U = ⋃_{n} U^{n} over A, with a quasi-isomorphism ϵ^{U}: U ⟶ M. □

In an important case, the constructions are essentially finite.

Proposition 2.2.8. Let Q be a noetherian ring

If R is a finite Q–algebra and M is a finite R–module, then there exist a DG algebra resolution A of R over Q and a DG module resolution U of M over A, such that the Q–modules Coker(η^{A}: Q ⟶ A_{0}), A_{n}, and U_{n}, are finite projective for all n and are trivial for n > max{pd_{Q} R, pd_{Q} M}.

Proof. If r_{1}, …, r_{s} generate R as a Q–module, then each r_{j} is a root of a monic polynomial f_{j} ∈ Q[x_{j}], hence R is a residue of Q′ = Q[x_{1}, …, x_{s}]/(f_{1}, …, f_{s}), which is a free Q–module. Use Proposition 2.1.10 to pick a resolvent A′ = Q′[Y] such that each A′_{n} is a finite free module over Q′. Then use Construction 2.2.7 to get a semi-free resolution U′ of M over A′ with each U′_{n} a finite free Q′–module; in particular, A′_{n} and U′_{n} are finite free Q–modules.

If max{pd_{Q} R, pd_{Q} M} = m < ∞, then define a Q–submodule V of U′ by setting V_{< m} = 0, V_{m} = ∂(U′_{m+1}), and V_{> m} = U′_{> m}. It is easy to check that V = {V_{n}} is a DG A′–submodule with H(V) = 0, hence U = U′/V has H(U) = M. The assumption on pd_{Q} M implies that the Q–module U_{m} is projective, so U is a DG module resolution of M over A′. Similarly, one sees that J ⊆ A′ defined by J_{< m} = 0, J_{m} = ∂(A′_{m+1}), and J_{> m} = A′_{> m}, is a DG ideal of A, such that A = A′/J is a DG algebra resolution of R. Finally, the Leibniz formula shows that JU′ ⊆ V, so U is a DG module over A.

The fact that Coker η is projective can be checked locally; Nakayama’s Lemma then shows that Im η is a direct summand of the free Q–module A_{0}. □

2.3. Products versus minimality. Our goal is the following non-existence

Theorem 2.3.1. Let k be a field, and Q be the polynomial ring k[s_{1}, s_{2}, s_{3}, s_{4}] with the usual grading, or the power series ring k[[s_{1}, s_{2}, s_{3}, s_{4}]]. There exists no DG algebra structure on the minimal Q–free resolution U of the residue ring

S = Q/I where I = (s_{1}^{2} , s_{1}s_{2} , s_{2}s_{3} , s_{3}s_{4} , s_{4}^{2})

or on the minimal Q–free resolution U′ of the Cohen-Macaulay residue ring

S′ = Q/I′ where I′ = I + (s_{1}s_{3}^{6} , s_{2}^{7} , s_{2}^{6}s_{4} , s_{3}^{7}) .

If A is a DG algebra over Q, and U or U′ is a DG module over A, then A = Q.

Remark. To prove the theorem, we check by a direct computation the non-vanishing of certain obstructions introduced by Avramov [22], and described in Theorem 3.2.6 below. Both the examples and the computations simplify those appearing in [22], and were developed in conversations with S. Iyengar.

As in [22], the examples can be used to generate, in any local ring Q with depth Q = g ≥ 4 (respectively, ≥ 6) perfect ideals of prescribed grades between 4 and g (respectively, Gorenstein ideals of prescribed grades between 6 and g), whose minimal free resolution admits no DG algebra structure.

Gorenstein ideals of grade 5 with this property had been missing, until the paper of Srinivasan [147]. The last open question, whether the minimal resolution of each non-cyclic module of projective dimension 3 (recall Proposition 2.1.4 and Proposition 2.2.5) carries a structure of DG module over some DG algebra A ≠ Q, was answered by Iyengar [92] with perfect counter-examples.

Construction 2.3.2. Tor algebras. Let S ⟵ R ⟶ k be homomorphisms of rings. If D is a DG algebra resolution D of S over R that is a resolution of S by free R–modules, cf. Proposition 2.1.10, then Tor^{R} (S, k) = H(D ⊗_{R} k) inherits a structure of graded algebra. It can be computed also from a DG algebra resolution E of the second argument, or from resolvents of both arguments, due to the quasiisomorphisms of DG algebras

D ⊗_{R} k ⟵^{D⊗_{R}ϵ^{E}} D ⊗_{R} E ⟶^{ϵ^{D}⊗_{R}E} S ⊗_{R} E .

Varying in these isomorphisms one varies one of D or E, while keeping the other fixed, one sees that the algebra structure on Tor does not depend on the choice of a DG algebra resolution. It can even be computed^{11} from projective resolutions D′ and E′ with no multiplicative structure: the unique up to homotopy lifting of μ^{S}: S ⊗_{R} S ⟶ S to a morphism μ^{D′}: D′ ⊗_{R} D′ ⟶ D′, that conspires with the Künneth map of Proposition 1.3.4 to produce

H(D′ ⊗_{R} k) ⊗_{R} H(D′ ⊗_{R} k) ⟶^{κ} H ((D′ ⊗_{R} k) ⊗_{R} (D′ ⊗_{R} k))

≅ H ((D′ ⊗_{R} D′) ⊗_{R} (k ⊗_{R} k)) ⟶^{H(μ^{D′} ⊗_{R}μ^{k})} H(D′ ⊗_{R} k) .

As the multiplication μ^{D}: D ⊗_{R} D ⟶ D also is a comparison map, the unique isomorphism H(D′ ⊗_{R} k) ≅ H(D ⊗_{R} k) transforms products into each other.

There is a related structure in the case of R–modules.

11In fact, this is how they were originally introduced by Cartan and Eilenberg [51].

Construction 2.3.3. Tor modules. Let ψ: Q ⟶ R and R ⟶ k be homomorphisms of rings, and let M be an R–module.

Choose a DG algebra resolution ϵ^{A}: A ⟶ R over Q, by Proposition 2.1.10, and a semi-free resolution ϵ^{U}: U ⟶ M over A, by Proposition 2.2.7. As both A and U are free over Q, we see that Tor^{Q} (M, k) = H(U ⊗_{Q} k) is a module over the graded algebra Tor^{Q} (R, k) = H(A ⊗_{Q} k) from the preceding construction. Recycling the discussion there, we verify that this structure is unique, and natural with respect to the module arguments.

The constructions also have a less well known naturality with respect to the ring argument; it is the one that we need.

Construction 2.3.4. Naturality. If ψ: Q ⟶ R is a ring homomorphism, then k becomes a Q–algebra, so pick a DG algebra resolution C of k over Q. By Proposition 2.1.9, there is a morphism of DG algebras γ: C ⟶ E over the identity map of k, that is unique up to Q–linear homotopy. The induced map H(M ⊗_{ψ} γ): H(M ⊗_{Q} C) ⟶ H(M ⊗_{R} E) is linear over H(R ⊗_{Q} C), and does not depend on the choice of γ. Thus, there is a natural homomorphism Tor^{ψ} (M, k) : Tor^{Q} (M, k) ⟶ Tor^{R} (M, k) of Tor^{Q} (R, k) –modules.

If U′ ⟶ M ⟵ V′ and C′ ⟶ k ⟵ E′ are arbitrary free resolutions, respectively over Q and over R, and β′: U′ ⟶ V′ and γ′: C′ ⟶ E′ are morphisms inducing id^{M} and id^{k}, then Tor^{ψ} (M, k) = H(β′ ⊗_{ψ} k) = H(M ⊗_{ψ} γ′).

Remark 2.3.5. Let (Q, 𝔫, k) be a local (or graded) ring, let f ∈ 𝔫 be a (homogeneous) regular element, let ψ: Q ⟶ Q/(f) = R be the natural projection, and let A be the Koszul complex Q[y | ∂(y) = f].

For a DG module resolution U of M over A, set \overline{U}⟨x⟩ = ⨁_{i ⩾ 0} Rx^{(i)} ⊗_{Q} \overline{U}, with |x^{(i)}| = 2i and ∂(x^{(i)} ⊗ u) = x^{(i−1)} ⊗ yu + x^{(i)} ⊗ ∂(u). In Example 3.1.2, we show that \overline{U}⟨x⟩ is a resolution of M over R. By Theorem 3.2.6 and Remark 3.2.7, if U is minimal, then Ker(Tor_{n}^{ψ} (M, k) ) = cls(y) Tor_{n−1}^{Q} (M, k) for n ≥ 1.

Proof of Theorem 2.3.1. First we look at the residue ring S. By Remark 2.2.4, it suffices to prove that its minimal free resolution U, over Q = k[s_{1}, s_{2}, s_{3}, s_{4}] or over Q = k[[s_{1}, s_{2}, s_{3}, s_{4}]], has no DG module structure over the Koszul complex A = Q[y | ∂(y) = f], where f = s_{1}^{2} + s_{4}^{2}. By the preceding remark, this will follow from cls(y) Tor_{3}^{Q} (S, k) ⊈ Ker(Tor_{4}^{ψ} (S, k) ). The Tor’s involved do not change under completion, so we restrict to the graded polynomial ring.

The Koszul resolvent C = Q[y_{1}, y_{2}, y_{3}, y_{4} | ∂(y_{i}) = s_{i}] of k over Q is a DG algebra over A, via the map y ↦ s_{1}y_{1} + s_{4}y_{4}; by Remark 2.3.5, the complex \overline{C}⟨x⟩, is a resolution of k over R = Q/(f). For K = S ⊗_{Q} C, we have

Tor^{Q} (S, k) = H(U ⊗_{Q} k) ≅ H(K)  (∗)

as modules over Tor^{Q} (R, k) = k[y | ∂(y) = 0], cf. Construction 2.3.3. The isomorphisms takes Ker Tor^{ψ} (S, k) to Ker H(ι), where ι is the inclusion K ⊂ L = S ⊗_{R} \overline{C}⟨x⟩. So it suffices to exhibit an element

w ∈ Ker H_{4}(ι) ∖ cls(y) H_{3}(K) .  (†)

It is easy to check that the cycles

z_{1} =(s_{1}s_{4}) y_{1} ∧ y_{2} ∧ y_{3} ,  z_{2} =(s_{1}s_{4}) y_{1} ∧ y_{2} ∧ y_{4} ,
z_{3} =(s_{1}s_{4}) y_{1} ∧ y_{3} ∧ y_{4} ,  z_{4} =(s_{2}s_{4}) y_{1} ∧ y_{3} ∧ y_{4} ,

are linearly independent modulo S∂(y_{1} ∧ ⋯ ∧ y_{4}). MACAULAY [40] shows that the minimal graded resolution U of the S over Q has the form

0 ⟶ Q(−6) ⟶^{∂_{4}} Q(−5)^{4} ⟶^{∂_{3}} Q(−4)^{3} ⊕ Q(−3)^{4} ⟶^{∂_{2}} Q(−2)^{5} ⟶^{∂_{1}} Q ⟶ 0 (‡)

with differentials given by the matrices

∂_{1} = (s_{1}^{2} s_{1}s_{2} s_{2}s_{3} s_{3}s_{4} s_{4}^{2})

∂_{2} = (\begin{matrix}0 & 0 & 0 & − s_{2} & − s_{4}^{2} & 0 & − s_{3}s_{4} \\ 0 & 0 & − s_{3} & s_{1} & 0 & − s_{4}^{2} & 0 \\ 0 & − s_{4} & s_{1} & 0 & 0 & 0 & 0 \\ −s_{4} & s_{2} & 0 & 0 & 0 & 0 & s_{1}^{2} \\ s_{3} & 0 & 0 & 0 & s_{1}^{2} & s_{1}s_{2} & 0 \end{matrix})

∂_{3} = (\begin{matrix}0 & 0 & s_{1}s_{2} & s_{1}^{2} \\ s_{1}^{2} & 0 & s_{1}s_{4} & 0 \\ s_{1}s_{4} & 0 & s_{4}^{2} & 0 \\ s_{3}s_{4} & s_{4}^{2} & 0 & 0 \\ 0 & −s_{2} & 0 & −s_{3} \\ 0 & s_{1} & −s_{3} & 0 \\ −s_{2} & 0 & 0 & s_{4} \end{matrix}) ∂_{4} = (\begin{matrix}−s_{4} \\ s_{3} \\ s_{1} \\ −s_{2} \end{matrix})

From (∗) and (‡) we get rank_{k} H_{3}(K) = 4, so cls(z_{1}), …, cls(z_{4}) is a basis of H_{3}(K). As yz_{i} = (s_{1}y_{1} + s_{4}y_{4})z_{i} = 0 for i = 1, …, 4, we get cls(y) H_{3}(K) = 0. Now

z = (s_{1}s_{4}) y_{1} ∧ y_{2} ∧ y_{3} ∧ y_{4} ∈ K_{4}

is a non-zero cycle, and so not a boundary in K, but becomes one in L:

z = ∂(s_{4} y_{2} ∧ y_{3} ∧ y_{4} y + s_{2} y_{3} x^{(2)}) ∈ L_{4} .

We have proved that w = cls(z) satisfies (†).

Turning to S′ = S/(s_{1}s_{3}^{6}, s_{2}^{7}, s_{2}^{6}s_{4}, s_{3}^{7}), consider the commutative square

K ⟶^{ι} L
π↓  ↓_{π′}
K′  S′ ⊗_{S} K ⟶^{ι′} S′ ⊗_{S} L  L′

of morphisms of complexes, with ι′ = S′ ⊗_{S} ι. For w′ = H(π)(w) it yields

H(ι′)(w′) = H(π′) H(ι)(w) = 0 .

Assume w′ ∈ cls(y) H_{3}(K′), set 𝔪 = (t_{1}, …, t_{4}), and consider the subcomplex

J: 0 ⟶ 𝔪^{3}K_{4} ⟶ 𝔪^{4}K_{3} ⟶ 𝔪^{5}K_{2} ⟶ 𝔪^{6}K_{1} ⟶ 𝔪^{7}K_{0} ⟶ 0

of K. Since (∗) and (‡) show that the non-zero homology of K is concentrated in internal degrees ≤ 6, cf. Remark 1.2.10, we conclude that H(J) = 0, so the projection ξ: K ⟶ K/J is a quasi-isomorphism. On the other hand, it is clear that J is a DG ideal of K, such that Ker π = 𝔪^{7}K ⊂ J. Thus, ξ = ρπ, where ρ: K′ ⟶ K/J is the canonical map. By the surjectivity of H(ξ), we have

H(ξ)(w) = H(ρ)(w′) ⊆ cls(y) H(ξ)( H_{3}(K)) = H(ξ)(cls(y) H_{3}(K)) .

The injectivity of H(ξ) implies w ∈ cls(y) H_{3}(K), violating (†).

Thus, we have found w′ ∈ Ker H_{4}(ι′)∖cls(y) H_{3}(K′). As above, this implies that U′ carries no structure of DG module over A. □

3. Change of rings

Fix a homomorphism of rings ψ: Q ⟶ R, and an R–module M.

We consider various aspects of the problem: How can homological information on R and M over Q be used to study the module M over R?

3.1. Universal resolutions. A recent result of Iyengar [92] addresses this problem on the level of resolutions^{12}.

Theorem 3.1.1. Let ϵ^{A}: A ⟶ R be a DG algebra resolution over Q with structure map η^{A}: Q ⟶ A, and let ϵ^{U}: U ⟶ M be a DG module resolution of M over A. With \overline{A} = (R ⊗_{Q} Coker η^{A}) and \overline{U} = (R ⊗_{Q} U), set

F_{n}(A, U) = ⨁_{p+i_{1}+···+i_{p}+j=n} \overline{A}_{i_{1}} ⊗_{R} ⋯ ⊗_{R} \overline{A}_{i_{p}} ⊗_{R} \overline{U}_{j} ;

∂′(\overline{a}_{1} ⊗ ⋯ ⊗ \overline{a}_{p} ⊗ \overline{u}) = ∑_{r=1}^{p}(−1)^{r+i_{1}+···+i_{r−1}} \overline{a}_{1} ⊗ ⋯ ⊗ ∂(\overline{a}_{r}) ⊗ ⋯ ⊗ \overline{a}_{p} ⊗ u
+ (−1)^{p+i_{1}+···+i_{p}} \overline{a}_{1} ⊗ ⋯ ⊗ \overline{a}_{p} ⊗ \overline{∂(u)} ;
∂′′(\overline{a}_{1} ⊗ ⋯ ⊗ \overline{a}_{p} ⊗ \overline{u}) = ∑_{r=1}^{p−1}(−1)^{r+i_{1}+···+i_{r}} \overline{a}_{1} ⊗ ⋯ ⊗ \overline{a_{r}a}_{r+1} ⊗ ⋯ ⊗ \overline{a}_{p} ⊗ u
+ (−1)^{p+i_{1}+···+i_{p−1}} \overline{a}_{1} ⊗ ⋯ ⊗ \overline{a}_{p−1} ⊗ \overline{a_{p}u} .

The homomorphisms ∂ = ∂′ + ∂′′: F_{n}(A, U) ⟶ F_{n−1}(A, U) make F(A, U) into a complex of R–modules. If the Q–modules Coker η_{i}^{A} and U_{i} are free for all i, then it is a free resolution of M over R.

When Q is a field, A = A_{0}, and U is an A–module, this is the well known standard resolution. The First Commandment^{13} points the way to generalizations: this is the philosophy of the proof presented at the end of this section.

Example 3.1.2. Let R = Q/(f) for a non-zero-divisor f. If U has a homotopy ∂σ + σ∂ = f id^{U} with σ^{2} = 0, then by Remark 2.2.1 it is a DG module over A = Q[y | ∂(y) = f]. As \overline{A}_{1} = R\overline{y} and \overline{A}_{i} = 0 for i ≠ 1, all but the last summands in the expressions for ∂′ and ∂′′ vanish, so F(A, U) takes the form

… ⟶ ⨁_{i} Rx^{(i)} ⊗_{Q} U_{n−2i} ⟶^{∂} ⨁_{i} Rx^{(i)} ⊗_{Q} U_{n−1−2i} ⟶ …

where x^{(i)} = \overline{y} ⊗ ⋯ ⊗ \overline{y} (i copies), and ∂(x^{(i)} ⊗ u) = x^{(i−1)} ⊗ σ(u) + x^{(i)} ⊗ ∂(u).

In the setup of the example, a resolution of M over R can be constructed even if no square-zero homotopy is available: this is the contents of the next theorem, due to Shamash [143]; the proof we present is from [25].

12In view of Proposition 2.2.8, if Q is noetherian and finite projective Q–modules are free, then a resolution of M over R is finitistically determined by matrix data over Q, namely, the multiplication tables of the algebra A and the module U, and the differentials in these finite complexes. With the help of computer algebra systems, such as MACAULAY [40], these data can be effectively gathered, at least when Q is a polynomial ring over a (small) field.

13Resolve!

Theorem 3.1.3. If R = Q/(f) for a non-zero-divisor f, M is an R–module, and U is a resolution of M by free Q–modules, then there exists a family of Q–linear homomorphisms σ = (σ^{[j]} ∈ Hom_{Q} (U, U)_{2j−1})_{j ⩾ 0}, such that

σ^{[0]} = ∂;  σ^{[0]}σ^{[1]} + σ^{[1]}σ^{[0]} = f id^{U} ;  ∑_{j=0}^{n} σ^{[j]}σ^{[n−j]} = 0 for n ≥ 2 .

If {x^{(i)} : |x^{(i)}| = 2i}_{i ⩾ 0} is a linearly independent set over R, then

… ⟶ ⨁_{i=0}^{n} Rx^{(i)} ⊗_{Q} U_{n−2i} ⟶^{∂} ⨁_{i=0}^{n−1} Rx^{(i)} ⊗_{Q} U_{n−1−2i} ⟶ …

∂(x^{(i)} ⊗ u) = ∑_{j=0}^{i} x^{(i−j)} ⊗ σ^{[j]}(u)

is a free resolution G(σ, U) of M over R.

Remark. Clearly, σ = σ^{[1]} is a homotopy between f id^{U} and 0^{U}. If σ^{2} = 0, then one can take σ^{[n]} = 0 for n ≥ 2, and both proposition and example yield the same resolution. In general, rewriting the condition for n = 2 in the form ∂σ^{[2]} + σ^{[2]}∂ = −σ^{2}, we see that σ^{[2]} is a homotopy which corrects the failure of σ^{2} to be actually 0. A similar interpretation applies to all σ^{[n]} with n ≥ 3, so σ is a family of higher homotopies between f id^{U} and 0^{U}.

Proof. Note that σ^{[0]} is determined, let σ^{[1]} be any homotopy such that f id^{U} = ∂σ^{[1]} + σ^{[1]}∂, and assume by induction that maps σ^{[j]} with the desired properties have been defined when 1 ≤ j < n for some n ≥ 2. Setting τ^{[1]} = f id^{U} and τ^{[j]} = − ∑_{h=1}^{j−1} σ^{[h]}σ^{[j−h]} for j ≥ 2, we have ∂σ^{[j]} = τ^{[j]} − σ^{[j]}∂, whence

∂σ^{[j]}σ^{[n−j]} = τ^{[j]}σ^{[n−j]} − σ^{[j]}τ^{[n−j]} + σ^{[j]}σ^{[n−j]}∂ for j = 1, …, n − 1 .

Summing up these equalities, we are left with ∂τ^{[n]} = τ^{[n]}∂, so τ^{[n]} is a cycle of degree 2n − 2 in the complex Hom_{Q} (U, U) . By Proposition 1.3.2, U ⟶^{≃} M induces a quasi-isomorphism Hom_{Q} (U, U) ⟶ Hom_{Q} (U, M) ; the latter complex is zero in positive degrees, hence τ^{[n]} is a boundary. Thus, there is a homomorphism σ^{[n]} : U ⟶ U of degree 2n − 1, such that τ^{[n]} = ∂σ^{[n]} + σ^{[n]}∂. This finishes the inductive construction of the family σ.

A direct computation shows that ∂^{2} = 0. Set G = G(σ, U), and note that there is an exact sequence 0 ⟶ R ⊗_{Q} U ⟶ G ⟶ Σ^{2}G ⟶ 0 of complexes of free R–modules. As H_{i}(R ⊗_{Q} U) ≅ Tor_{i}^{Q} (R, M) = 0 for i ≠ 0, 1, it yields

M ≅ H_{0}(R ⊗_{Q} U) ≅ H_{0}(G) , H_{n+2}(G) = H_{n}(Σ^{2}G) ≅ H_{n}(G) for n ≥ 1 ,

and an exact sequence

0 ⟶ H_{2}(G) ⟶ H_{2}(Σ^{2}G) ⟶^{ð} H_{1}(R ⊗_{Q} U) ⟶ H_{1}(G) ⟶ 0 .

Acyclicity of G will follow by induction on n, once we prove that ð is bijective.

If z ∈ (Σ^{2}G)_{2} = Rx ⊗_{Q} U_{0} is a cycle, then ð(cls(z)) is the class of ∂(x ⊗ z) = 1 ⊗ σ^{[1]}(z) ∈ R ⊗_{Q} U_{1}. To show that H_{0}(R ⊗_{Q} σ^{[1]}): H_{0}(R ⊗_{Q} U) ⟶ H_{1}(R ⊗_{Q} U) is bijective, note that σ^{[1]} is a homotopy between f id^{U} and 0^{U}, so we may replace U by any Q–free resolution V of M, and show that for some homotopy σ between f id^{V} and 0^{V} the map H_{0}(R ⊗σ) is bijective. Take V to be a semi-free resolution of M over A = Q[y | ∂(y) = f], and σ to be left multiplication by y, so that H_{0}(R ⊗ σ) is the action of 1 ⊗ y ∈ H_{1}(R ⊗_{Q} A) = Tor_{1}^{Q} (R, R) on H_{0}(R ⊗_{Q} V) = Tor_{0}^{Q} (R, M) . By Construction 2.3.3, it can be computed from the resolution A of R over Q, as the multiplication of H(A ⊗_{Q} M) = R[y] ⊗_{R} M with −(y ⊗ 1) ∈ H(A ⊗_{Q} R) = R[y]; this is obviously bijective. □

We now turn to the proof of Theorem 3.1.1. It uses a nice tool popular with algebraic topologists, cf. [50], [115], but neglected by commutative algebraists.

Construction 3.1.4. Bar construction. Consider a DG module U over a DG algebra A (as always, defined over 𝕜), and set Ã = Coker(η^{A}: 𝕜 ⟶ A). Let

S_{p}^{𝕜}(A, U) = A ⊗_{𝕜} \underbrace{Ã ⊗_{𝕜} ⋯ ⊗_{𝕜} Ã}_{p times} ⊗_{𝕜} U , for p ≥ 0 ,

be the DG module, with action of A on the leftmost factor and tensor product differential ∂^{p}, and set S_{p}^{𝕜}(A, U) = 0 for p < 0. The expression on the right in

δ^{p}(a ⊗ ã_{1} ⊗ ⋯ ⊗ ã_{p} ⊗ u) =(aa_{1}) ⊗ ã_{2} ⊗ ⋯ ⊗ ã_{p} ⊗ u
+ ∑_{i=1}^{p−1}(−1)^{i−1}a ⊗ ã_{1} ⊗ ⋯ ⊗ (ã_{i}a_{i+1}) ⊗ ⋯ ⊗ ã_{p} ⊗ u
(−1)^{p}a ⊗ ã_{1} ⊗ ⋯ ⊗ ã_{p−1} ⊗ (a_{p}u)

is easily seen to be well-defined. Another easy verification yields

δ^{p+1}δ^{p} = 0  and  ∂^{p−1}δ^{p} = δ^{p}∂^{p}  for all p .

Thus, (S^{𝕜}(A, U), δ) is a complex of DG modules^{14}, called the standard complex. It comes equipped with 𝕜-linear maps

π′: S_{0}^{𝕜}(A, U) ⟶ U , a ⊗ u ↦ au;
ι′: U ⟶ S_{0}^{𝕜}(A, U) , u ↦ 1 ⊗ u;
σ^{p}: S_{p}^{𝕜}(A, U) ⟶ S_{p+1}^{𝕜}(A, U) ,
a ⊗ ã_{1} ⊗ ⋯ ⊗ ã_{p} ⊗ u ↦ 1 ⊗ ã ⊗ ã_{1} ⊗ ⋯ ⊗ ã_{p} ⊗ u ,

that (are seen by another direct computation to) satisfy the relations

δ^{1}σ^{0} = id^{S𝕜_{0}(A,U)} −ι′π′;
δ^{p+1}σ^{p} + σ^{p−1}δ^{p} = id^{S𝕜_{p}(A,U)}  for p ≥ 1 .

In particular, when A and U have trivial differentials, H(S^{𝕜}(A, U)) ≅ U. If, furthermore, the 𝕜–modules Ã_{i} and U_{i} are free for all i, then this free resolution is known as the standard resolution of U over A.

Returning to the DG context, we reorganize S^{k}(A, U) into a DG module over A, by the process familiar ‘totaling’ procedure. The resulting DG module, with the action of A defined in Section 1.3, is the (normalized) bar construction:

B^{𝕜}(A, U) = ⨁_{p=0}^{∞} Σ^{p}(S_{p}^{𝕜}(A, U)) with ∂ = ∂′ + ∂′′

14Of course, a complex in the category of DG modules is a sequence of morphisms of DG modules δ^{p}: C^{p} ⟶ C^{p−1}, such that δ^{p−1}δ^{p} = 0.

where ∂′ denotes the differential of the DG module Σ^{p}S_{p}^{𝕜}(A, U), and ∂′ is the degree −1 map induced by the boundary δ of the complex of DG modules S^{k}(A, U); the equality ∂^{2} = results from the relations

∂′∂′ = 0 ,  ∂′′∂′′ = 0 ,  ∂′∂′′ + ∂′′∂′ = 0 ,

of which the first two are clear, and the last one is due to the difference by a factor (−1)^{p} of the differentials of S_{p}^{𝕜}(A, U) and Σ^{p}S_{p}^{𝕜}(A, U). Furthermore, the maps π′, ι′, σ^{p}, total to maps

π: B^{𝕜}(A, U) ⟶ U ,  ι: U ⟶ B^{𝕜}(A, U) ,  σ: B^{𝕜}(A, U) ⟶ B^{𝕜}(A, U) .

Clearly, π is a morphism of DG modules over A, ι is a morphism of complexes over 𝕜, σ is a degree 1 morphism of complexes over 𝕜, and they are related by

πι = 1_{U}  and  ∂σ + σ∂ = id^{B𝕜(A,U)} −ιπ .

Thus, H(π) and H(ι) are inverse isomorphisms, so π is a quasi-isomorphism. The canonical isomorphism of DG modules over the DG algebra A,

Σ^{p}(S_{p}^{𝕜}(A, U)) ≅ A ⊗_{k} \underbrace{Σ(Ã) ⊗_{𝕜} ⋯ ⊗_{𝕜} Σ(Ã)}_{p times} ⊗_{𝕜} U ,

expresses the degree n component of the bar construction as

B_{n}(A, U) = ⨁_{h+p+i_{1}+···+i_{p}+j=n} A_{h} ⊗_{𝕜} Ã_{i_{1}} ⊗_{𝕜} ⋯ ⊗_{𝕜} Ã_{i_{p}} ⊗_{𝕜} U_{j} .

The signs arising from the application of the shift, cf. Section 1.3, then yield the following expressions for the two parts of the differential:

∂′(a ⊗ ã_{1} ⊗ … ⊗ ã_{p} ⊗ u) = ∂(a) ⊗ ã_{1} ⊗ ⋯ ⊗ ã_{p} ⊗ u

+ ∑_{r=1}^{p}(−1)^{r+h+i_{1}+···+i_{r−1}}a ⊗ ã_{1} ⊗ ⋯ ⊗ ∂(ã_{r}) ⊗ ⋯ ⊗ ã_{p} ⊗ u

+ (−1)^{p+h+i_{1}+···+i_{p}}a ⊗ ã_{1} ⊗ ⋯ ⊗ ã_{p} ⊗ ∂(u)

∂′′(a ⊗ ã_{1} ⊗ … ⊗ ã_{p} ⊗ u) = (−1)^{h}(aa_{1}) ⊗ ã_{2} ⊗ ⋯ ⊗ ã_{p} ⊗ u

+ ∑_{r=1}^{p−1}(−1)^{r+h+i_{1}+···+i_{r}}a ⊗ ã_{1} ⊗ ⋯ ⊗ ã_{r}a_{r+1} ⊗ ⋯ ⊗ ã_{p} ⊗ u

+ (−1)^{p+h+i_{1}+···+i_{p−1}}a ⊗ ã_{1} ⊗ ⋯ ⊗ ã_{p−1} ⊗ (a_{p}u) .

This finishes our description of the bar construction.

Proof of Theorem 3.1.1. We apply Construction 3.1.4 to the DG algebra A and its DG module U, considered as complexes over the base ring 𝕜 = Q. Thus, we get quasi-isomorphisms

B^{Q}(A, U) ⟶^{π} U ⟶^{ϵ^{U}} M .

On the other hand, as the DG module B^{Q}(A, U) is semi-free over A, so

B^{Q}(A, U) = B^{Q}(A, U) ⊗_{A} A ⟶^{B^{Q}(A,U)⊗_{A}ϵ^{A}} B^{Q}(A, U) ⊗_{A} R

is a quasi-isomorphism by Proposition 1.3.2. Viewed as a complex of R–modules, B^{Q}(A, U) ⊗_{A} R is precisely the complex F(A, U) described in the statement of the theorem, so F(A, U) is a free resolution of M over R. □

3.2. Spectral sequences. Various spectral sequences relate (co)homological invariants of M over R and over Q. Those presented below are first quadrant, that is, have ^{r}E_{p,q} = 0 when p < 0 or q < 0, and of homological type of , meaning that their differentials follow the pattern ^{r}d_{p,q}: ^{r}E_{p,q} ⟶ ^{r}E_{p−r,q+r−1}.

For starters, here is a classical Cartan-Eilenberg spectral sequence [51].

Proposition 3.2.1. For each Q–module N there exists a spectral sequence

2E_{p,q} = Tor_{p}^{R} (M, Tor_{q}^{Q} (R, N)) ⟹ Tor_{p+q}^{Q} (M, N) .

Proof. Let V ⟶ M be a free resolution over R, and W ⟶ N be a free resolution over Q. By Proposition 1.3.2, the induced map V ⊗_{Q} W ⟶ M ⊗_{Q} W is a quasiisomorphism, hence H(V ⊗_{Q} W) ≅ Tor^{Q} (M, N) . As V ⊗_{Q} W ≅ V ⊗_{R} (R ⊗_{Q} W), the filtration (V_{⩽ p}) ⊗_{R} (R ⊗_{Q} W) yields a spectral sequence

2E_{p,q} = H_{p}(V ⊗_{R} H_{q}(R ⊗_{Q} W)) ⟹ Tor_{p+q}^{Q} (M, N)

where H_{q}(R ⊗_{Q} W) = Tor_{q}^{Q} (R, N) and H_{p}(V ⊗_{R} L) = Tor_{p}^{R} (M, L) . □

In simple cases, this spectral sequence degenerates to an exact sequence. This may be used to prove the next result, but we take a direct approach.

Proposition 3.2.2. If f is a non-zero-divisor on Q, and N is a module over R = Q/(f), then there is a long exact sequence

… ⟶ Tor_{n−1}^{R} (M, N) ⟶ Tor_{n}^{Q} (M, N) ⟶^{Tor_{n}^{ψ}(M,N)} Tor_{n}^{R} (M, N)
⟶^{ϑ_{n}} Tor_{n−2}^{R} (M, N) ⟶ Tor_{n−1}^{Q} (M, N) ⟶ ….

Proof. In the description of F(A, U) given in example 3.1.2, define a morphism ι: U ⟶ F(A, U) with ι(u) = x^{(0)} ⊗ u, and note that Coker ι ≅ Σ^{2} F(A, U). Thus, we have a short exact sequence of complexes of free R–modules

0 ⟶ R ⊗_{Q} U ⟶^{R⊗ι} F(A, U) ⟶^{ϑ} Σ^{2} F(A, U) ⟶ 0 .

Tensoring it with N over R, and writing down the homology exact sequence of the resulting short exact sequence of complexes, we get what we want. □

The next spectral sequence is introduced by Lescot [107].

Proposition 3.2.3. If ψ is surjective and k is a residue field of R, then there is a spectral sequence

2E_{p,q} = Tor_{p}^{Q} (k, M) ⊗_{k} Tor_{q}^{R} (k, k) ⟹ ( Tor^{Q} (k, k) ⊗_{k} Tor^{R} (M, k) )_{p+q} .

Proof. Choose free resolutions: U ⟶ k over Q; V ⟶ M and W ⟶ k over R. As U ⊗_{Q} V is a bounded below complex of free R–modules, Proposition 1.3.2 yields the first isomorphisms below; the third ones comes from Proposition 1.3.4:

H(U ⊗_{Q} V ⊗_{R} W) ≅ H((U ⊗_{Q} V) ⊗_{R} k) ≅ H((U ⊗_{Q} k) ⊗_{k} (V ⊗_{R} k))

≅ H(U ⊗_{Q} k) ⊗_{k} H(V ⊗_{R} k) = Tor^{Q} (k, k) ⊗_{k} Tor^{R} (M, k) .

Thus, the spectral sequence of the filtration (U ⊗_{Q} V) ⊗_{R} (W_{⩽ p}) has

2E_{p,q} = H_{p}(H_{q}(U ⊗_{Q} V) ⊗_{R} W) ⟹ ( Tor^{Q} (k, k) ⊗_{k} Tor^{R} (M, k) )_{p+q} .

Since U is a bounded below complex of free Q–modules, U ⊗_{Q} V ⟶ U ⊗_{Q} M is a quasi-isomorphism, so ^{2}E_{p,q} ≅ H_{p}(Tor_{q}^{Q} (k, M) ⊗_{R} W), and that module is equal to Tor_{p}^{R} (Tor_{q}^{Q} (k, M) , k) ≅ Tor_{p}^{Q} (k, M) ⊗_{k} Tor_{q}^{R} (k, k) . □

For us, the preceding sequences have the drawback of going in the ‘wrong’ direction: they require input of data over the ring of interest, R. Next we describe a sequence where the roles of Q and R are reversed. It belongs to the family of Eilenberg-Moore spectral sequences, cf. [122].

By Construction 2.3.3, Tor^{Q} (M, k) is a module over the graded algebra Tor^{Q} (R, k) . A homogeneous free resolution of the former over the latter provides an ‘approximation’ of a resolution of M over R: this is the contents of the following special case of a result of Avramov [22].

Proposition 3.2.4. When k is a residue field of R, there is a spectral sequence

2E_{p,q} = Tor_{p}^{Tor^{Q}(R,k)} (Tor^{Q} (M, k) , k)_{q} ⟹ Tor_{p+q}^{R} (M, k) .

Proof. In the notation of Construction 3.1.4, set B = A ⊗_{Q} k and V = U ⊗_{Q} k. The filtration ⨁_{i ⩽ p} (B̃^{⊗i} ⊗ V ) of B^{k}(B, V) ⊗_{B} k yields a spectral sequence

0E_{p,q} = (S_{p}^{k}(B, V) ⊗_{B} k)_{q} ⟹ H_{p+q}(B^{k}(B, V) ⊗_{B} k)

with ^{0}d_{p,q} equal to the tensor product differential. By Künneth,

1E_{p,q} = H_{q}(S_{p}^{k}(B, V) ⊗_{B} k)_{q} ≅ ( S_{p}^{k}(H(B), H(V)) ⊗_{H(B)} k)_{q} , ^{1}d_{p,q} = δ_{q}^{[p]} ⊗_{R} k .

As k is a field, S^{k}(H(B), H(V)) is a resolution of H(V) over H(B), so

2E_{p,q} = Tor_{p}^{H(B)} (H(V), k)_{q} .

Since H(B) = Tor^{Q} (R, k) and H(V) = Tor^{Q} (M, k) , the second page of the spectral sequence has the desired form. The isomorphisms

B^{k}(B, V) ⊗_{B} k ≅ ( B^{Q}(A, U) ⊗_{A} R) ⊗_{R} k = F(A, U) ⊗_{R} k

and Theorem 3.1.1 identify its abutment as Tor^{R} (M, k) . □

Corollary 3.2.5. If ψ is surjective, then for each n ≥ 0 there is an inclusion

∑_{i=1}^{n} Tor_{i}^{Q} (R, k) · Tor_{n−i}^{Q} (M, k) ⊆ Ker Tor_{n}^{ψ} (M, k) .

Proof. The natural map U ⊗_{Q} k ⟶ F(A, U), that is, V ⟶ B^{k}(B, V) ⊗_{B} k, identifies V with F^{1} ⊆ B^{k}(B, V) ⊗_{B} k. Thus, the map Tor_{n}^{ψ} (M, k) that it induces in homology factors through

ν_{n}: Tor_{n}^{Q} (M, k) = H_{n}(V) ⟶ ^{2}E_{0,n} = \frac{Tor_{n}^{Q} (M, k)}{∑_{i=1}^{n} Tor_{i}^{Q} (R, k) · Tor_{n−i}^{Q} (M, k)} .

We get Ker ν_{n} ⊆ Ker Tor_{n}^{ψ} (M, k) , which is the desired inclusion. □

The next theorem shows that the vector spaces

o_{n}^{ψ}(M) = \frac{Ker Tor_{n}^{ψ} (M, k)}{∑_{i=1}^{n} Tor_{i}^{Q} (R, k) · Tor_{n−i}^{Q} (M, k)}

are obstructions to DG module structures; they were found in [22].

Theorem 3.2.6. Let (Q, 𝔫, k) be a local ring, let ψ: Q ⟶ R be a surjective homomorphism of rings, and let M be a finite R–module.

If the minimal free resolution A of R over Q has a structure of DG algebra, and the minimal free resolution U of M over Q admits a structure of DG module over A, then o_{n}^{ψ}(M) = 0 for all n.

Proof. Under the hypotheses of the theorem, ∂′ ⊗_{Q} k = 0 in the bar construction B^{k}(B, V) ⊗_{B} k of Proposition 3.2.4, so the only non-zero differential in the spectral sequence constructed there acts on the first page. Thus, the sequence stops on the second page, yielding Ker Tor_{n}^{ψ} (M, k) = Ker ν_{n}. □

Remark 3.2.7. Let f be a Q–regular sequence. Computing the Tor algebra for R = Q/(f) with the help of the Koszul complex A = K(f; Q) we get

Tor^{Q} (R, k) = H(A ⊗_{R} k) = A ⊗_{R} k = ⋀(A_{1} ⊗_{R} k) = ⋀ Tor_{1}^{Q} (R, k)

and hence ∑_{i=1}^{n} Tor_{i}^{Q} (R, k) · Tor_{n−i}^{Q} (M, k) = Tor_{1}^{Q} (R, k) · Tor_{n−1}^{Q} (M, k) .

3.3. Upper bounds. In this section ψ: Q ⟶ R is a finite homomorphism of local rings that induces the identity on their common residue field k, and M is a finite R–module. We relate the Betti numbers of M over R and Q.

Often, such relations are expressed in terms of the formal power series

P_{M}^{R}(t) = ∑_{n=0}^{∞} β_{n}^{R}(M)t^{n} ∈ ℤ[[t]] ,

known as the Poincaré series of M over R, and the corresponding series over Q. Results then take the form of coefficientwise inequalities (denoted ≼ and ≽) of formal power series; equalities are significant.

Spectral sequence generate inequalities, by an elementary observation:

Remark 3.3.1. In a spectral sequence of vector spaces ^{r}E_{p,q} ⟹ E, r ≥ a, the space ^{r+1}E_{p,q} is a subquotient of ^{r}E_{p,q} for r ≥ a, and the spaces ^{∞}E_{p,q} are the subfactors of a filtration of E_{p+q}. Thus, there are (in)equalities

dim_{k} E_{n} = ∑_{p+q=n} rank_{k} ^{∞}E_{p,q} ≤ ∑_{p+q=n} rank_{k} ^{r}E_{p,q} ≤ ∑_{p+q=n} rank_{k} ^{a}E_{p,q} .

Multiplying the n’th one by t^{n}, and summing in ℤ[[t]], we get inequalities

∑_{n ⩾ 0} dim_{k} E_{n}t^{n} ≼ ∑_{n} ( ∑_{p+q=n} rank_{k} ^{r}E_{p,q} )t^{n} for r ≥ a .

The next result was initially deduced by Serre from the sequence in Proposition 3.2.1. It is more expedient to get it from that in Proposition 3.2.4.

Proposition 3.3.2. There is an inequality P_{M}^{R}(t) ≼ \frac{P_{M}^{Q}(t)}{1 − t( P_{R}^{Q}(t) − 1)} .

Proof. The spectral sequence of Proposition 3.2.4 has

∑_{n} ( ∑_{p+q=n} rank_{k} ^{1}E_{p,q} )t^{n} = ∑_{p} ( ∑_{q} rank_{k} ^{1}E_{p,q} t^{q})t^{p}

= ∑_{p} (( P_{R}^{Q}(t) − 1)^{p} P_{M}^{Q}(t))t^{p}

= P_{M}^{Q}(t) ∑_{p} ( P_{R}^{Q}(t) − 1)^{p}t^{p} = \frac{P_{M}^{Q}(t)}{1 − t( P_{R}^{Q}(t) − 1)}

so the desired inequality follows from the preceding remark. □

Remark. If equality holds with M = k, then ψ is called a Golod homomorphism. These maps, introduced by Levin [109], are studied in detail in [110], [24]; they are used in many computations of Poincaré series. The ‘absolute case’, when Q is a regular local ring, is the subject of Chapter 5.

Directly from the spectral sequence in Proposition 3.2.3, we read off

Proposition 3.3.3. There is an inequality P_{M}^{R}(t) P_{k}^{Q}(t) ≼ P_{M}^{Q}(t) P_{k}^{R}(t) . □

Remark. If equality holds, then the module M is said to be inert by ψ: these modules are introduced and studied by Lescot [107].

In special cases, universal bounds can be sharpened.

Recall [46] that a finite Q–module N has rank if for each prime ideal 𝔮 ∈ Ass Q the Q_{𝔮}–module N_{𝔮} is free, and its rank does not depend on 𝔮. The common rank of these free modules is called the Q–rank of N, and denoted rank_{Q} N; we write rank_{Q} N ≥ 0 to indicate that the rank of N is defined.

Proposition 3.3.4. If f is Q–regular and R = Q/(f), then

0 ≼ ∑_{n=1}^{∞} rank_{Q} Syz_{n}^{Q} (M) t^{n−1} = \frac{P_{M}^{Q}(t)}{(1 + t)} ≼ P_{M}^{R}(t) ≼ \frac{P_{M}^{Q}(t)}{(1 − t^{2})} .

Proof. Let U be a minimal resolution of M over Q. For each 𝔮 ∈ Ass Q we have f ∉ 𝔮. Thus, M_{𝔮} = 0, and so for each n there is an exact sequence

0 ⟶ Syz_{n+1}^{Q} (M)_{𝔮} ⟶ (U_{n})_{𝔮} ⟶ … ⟶ (U_{0})_{𝔮} ⟶ 0 .

It follows that Syz_{n+1}^{Q} (M)_{𝔮} is free of rank ∑_{i ⩾ 0}(−1)^{i}β_{n−i}^{Q}(M): this establishes the equality, and the first inequality.

For the second inequality, apply the exact sequence of Proposition 3.2.2, to get β_{n}^{Q}(M) ≤ β_{n−1}^{R}(M) + β_{n}^{R}(M) for all n.

The third inequality results from counting the ranks of the free modules in the resolution of M over R given by Theorem 3.1.3. □

There are useful sufficient conditions for equalities. One is essentially contained in Nagata [124]; the other, from Shamash [143], is given a new proof.

Proposition 3.3.5. Let f be a Q–regular element, and R = Q/(f).

(1) If f ∉ 𝔫^{2}, then P_{M}^{R}(t) = P_{M}^{Q}(t)/(1 + t) .

(2) If f ∈ 𝔫(0:_{Q} M), then P_{M}^{R}(t) = P_{M}^{Q}(t)/(1 − t^{2}) .

Proof. (1) This is just the last assertion of Theorem 2.2.3.

(2) Let s_{1}, …, s_{e} be a minimal set of generators of 𝔫, and write f = ∑_{j=1}^{e} a_{j}s_{j} with a_{j} ∈ (0:_{Q} M). In a DG algebra resolution V ⟶ k over Q, pick y_{1}, …, y_{e} ∈ V_{1} such that ∂(y_{j}) = s_{j}. As ∂( ∑_{j=1}^{e} a_{j}y_{j}) = f, Example 3.1.2 yields a resolution G of k over R, with G_{n} = ⨁_{i=0}^{n} Rx^{(i)} ⊗_{Q} V_{n−2i} and

∂(x^{(i)} ⊗_{Q} v) = ∑_{j=1}^{e} a_{j}x^{(i−1)} ⊗_{Q} y_{j}v + x^{(i)} ⊗_{Q} ∂(v) .

For b ∈ M, the induced differential of M ⊗_{R} G then satisfies

∂(bx^{(i)} ⊗_{Q} v) = ∑_{j=1}^{e} a_{j}bx^{(i−1)} ⊗_{Q} y_{j}v + bx^{(i)} ⊗_{Q} ∂(v) = bx^{(i)} ⊗_{Q} ∂(v) ,

so M ⊗_{R} G ≅ ⨁_{i=0}^{∞} Σ^{2i}(M ⊗_{Q} V) as complexes of R–modules. This yields

Tor^{R} (M, k) = H(M ⊗_{R} G) = ⨁_{i=0}^{∞} Σ^{2i} Tor^{Q} (M, k) .

The desired equality of Poincaré series is now obvious. □

Remark. The resolution G(σ, U) of Theorem 3.1.3 has

∑ rank_{R} G_{n}t^{n} = ( ∑ rank_{Q} U_{i}t^{i})/(1 − t^{2}) .

Thus, if U is a minimal resolution of M over Q and f ∈ 𝔫(0 :_{Q} M), then by (2) G(σ, U) is a minimal R–free resolution of M. Another case of minimality is given by Construction 5.1.2, which shows that if pd_{Q} M = 1, then Syz_{n}^{R} (M) has a minimal resolution of that form for each n ≥ 1. Quite the opposite happens ‘in general’: it is proved in [32] that if R is a complete intersection and the Betti numbers of M are not bounded, then Syz_{n}^{R} (M) has such a minimal resolution for at most one value of n.

4. Growth of resolutions

The gap between regularity and singularity widens to a chasm in homological local algebra: Minimal resolutions are always finite over a regular local ring, and (very) rarely over a singular one. A bridge^{15} is provided by the Cohen Structure Theorem: the completion of each local ring is a residue of a regular ring, so by change of rings techniques homological invariants over the singular ring may be approached from those—essentially finite—over the regular one.

To describe and compare resolutions of modules over a singular local ring, we go beyond the primitive dichotomy of finite versus infinite projective dimension, and analyze infinite sequences of integers, such as ranks of matrices, or Betti numbers. For that purpose there in no better choice than to follow the time-tested approach of calculus, and compare sizes of resolutions to the functions we^{16} understand best: polynomials and exponentials.

4.1. Regular presentations. Let I be an ideal in a noetherian ring R. Recall that the minimal number of generators ν_{R}(I), the height, and the depth of I are always related by inequalities, due to Rees and to Krull:

depth_{R}(I, R) ≤ height I ≤ ν_{R}(I) .

For the rest of this section, (R, 𝔪, k) is a local ring; ν_{R}(𝔪) is then known as its embedding dimension, denoted edim R, and the inequalities read

depth R ≤ dim R ≤ edim R .

Discrepancies between these numbers provide measures of irregularity: • cmd R = dim R − depth R is the Cohen-Macaulay defect of R;

• codim R = edim R − dim R is the codimension^{17} of R;

• codepth R = edim R − depth R is the codepth^{17} of R.

We use the vanishing of codepth R to define^{18} the regularity of R. Thus, if R is regular, then each minimal generating set of 𝔪 is a regular sequence.

Two cornerstone results of commutative ring theory determine the role of regular rings in the study of free resolutions.

The Auslander-Buchsbaum-Serre Theorem describes them homologically.

Theorem 4.1.1. The following conditions are equivalent.

(i) R is regular.

(ii) pd_{R} M < ∞ for each finite R–module M .

(iii) pd_{R} k < ∞.

15Warning: crossing may take an infinite time.

16Algebraists.

17Because ‘codimension’ has been used to denote depth, and ‘codepth’ to denote Cohen-Macaulay defect, the notions described here are sometimes qualified by ‘embedding’; it would be too cumbersome to stick to that terminology, and to devise new notation.

18This clearly implies the usual definition, in terms of the vanishing of codim R. Conversely, if R is regular, then the associated graded ring S = ⨁_{n} 𝔪^{n}/𝔪^{n+1} is the quotient of a polynomial ring in e = edim R variables. Since S and R have equal Hilbert-Samuel functions, dim S = dim R = e, so S is the polynomial ring on the classes in S_{1} of a minimal set of generators t of 𝔪. In particular, these classes form an S–regular sequence, and then a standard argument shows that t is an R– regular sequence.

Proof. (i) ⟹ (iii). By Example 1.1.1, the Koszul complex on a minimal set of generators for 𝔪 is a free resolution of k.

(iii) ⟹ (ii). As Tor_{n}^{R} (M, k) = 0 for n ≫ 0, apply Proposition 1.2.2.

(ii) ⟹ (i). We prove that codepth R = 0 by induction on d = depth R. If d = 0, then k is free by Proposition 1.2.7.1, hence 𝔪 = 0. If d > 0, then a standard prime avoidance argument yields a regular element g ∈ 𝔪 ∖ 𝔪^{2}. Set R′ = R/(g), and note that codepth R = codepth R′. As pd_{R′} k < ∞ by Theorem 2.2.3, we have codepth R′ = 0 by the induction hypothesis. □

Corollary 4.1.2. If R is regular, then so is R_{𝔭} for each 𝔭 ∈ Spec R.

Proof. By the theorem, R/𝔭 has a finite R–free resolution F; then F_{𝔭} is a finite R_{𝔭}–free resolution of k(𝔭) = R_{𝔭}/𝔭R_{𝔭}, so R_{𝔭} is regular by the theorem. □

The Cohen Structure Theorem establishes the dominating position of regular rings. A regular presentation of R is an isomorphism R ≅ Q/I, where Q is a regular local ring. Many local rings (for example, all those arising in classical algebraic or analytic geometry), come equipped with such a presentation. By Cohen’s theorem, every complete local ring has a regular presentation.

Here is how the two theorems above apply to the study of resolutions.

Since the 𝔪-adic completion R̂ is a faithfully flat R–module, a complex of R– modules F is a (minimal) free resolution of M over R if and only if F̂ = R̂ ⊗_{R} F is a (minimal) free resolution of M̂ = R̂ ⊗_{R} M over R̂; in particular, P_{M}^{R}(t) = PR̂ ; M̂(t). The point is: as M̂ iand R̂ have finite free resolutions over the ring Q, all change of rings results apply with finite entry data.

If (Q, 𝔫, k) is a regular local ring, R = Q/I, and f ∈ I ∖ 𝔫^{2}, then Q/(f) is also regular, and maps onto R. Iterating, one sees that if R has some regular presentation, then it has a minimal one, with edim R = edim Q.

Minimal presentations often fade into the background, because some of their invariants can be computed directly over the ring R, from Koszul complexes on minimal sets of generators t of 𝔪. Different choices of t lead to isomorphic complexes, so when we do not need to make an explicit choice of generators, we write K^{R} instead of K(t; R), and set K^{M} = K^{R} ⊗_{R} M.

Lemma 4.1.3. If K^{R} is a Koszul complex on a minimal set of generators of 𝔪, and R̂ = Q′/I′ (respectively, R = Q/I) is a minimal regular presentation, then

(1) H(K^{M}) ≅ H(KM̂) ≅ Tor^{Q′}(M̂, k) ( ≅ Tor^{Q} (M, k) ).

(2) sup{i | H_{i}(K^{M}) ≠ 0} = sup{i | H_{i}(KM̂) ≠ 0} = pd_{Q′} M̂ ( = pd_{Q} M).

(3) H_{1}(K^{R}) ≅ H_{1}(KR̂) ≅ I′ ⊗_{Q′} k ( ≅ I ⊗_{Q} k).

Proof. The fact that R̂ is faithfully flat over R, and 𝔪R̂ is its maximal ideal yields the relations on both ends, so we argue for those in the middle. By Example 1.1.1, K^{Q′} is a minimal free resolution of k over Q′: this yields (1), and then (2) follows from Proposition 1.2.2. For (3), use the long exact sequence of Tor^{Q′} (−, k) applied to the exact sequence 0 ⟶ I′ ⟶ Q′ ⟶ R̂ ⟶ 0. □

In view of the lemma, Proposition 3.3.2 translates into:

Proposition 4.1.4. For each finite R–module M there is an inequality

P_{M}^{R}(t) ≼ \frac{∑_{i=0}^{edim R−depth M} rank_{k} H_{i}(K^{M})t^{i}}{1 − ∑_{j=1}^{codepth R} rank_{k} H_{j}(K^{R})t^{j+1}} .  □

Corollary 4.1.5. There is an α ∈ ℝ, such that β_{n}^{R}(M) ≤ α^{n} for n ≥ 1. □

A local ring homomorphism φ: R ⟶ R′, such that 𝔪R′ is the maximal ideal of R′, may be lifted—in more than one way—to a morphism of DG algebras K^{φ}: K^{R} ⟶ K^{R′} (we use a ‘functorial’ notation, because in the cases treated below the choice of a specific lifting will be of no consequence, while it helps to distinguish K^{φ} from K^{R} ⊗_{R} φ: K^{R} ⟶ K^{R} ⊗_{R} R′).

The following easily proved statements have unexpectedly strong consequences. The second is the key to Serre’s original proof that pd_{R} k characterizes regularity in [141]. The third, also due to Serre, is important in the study of multiplicities, cf. [142], [17]; its proof below is from Eagon and Fraser [54].

Lemma 4.1.6. The complexes K^{R} and K^{M} have the following properties.

(1) If g ∈ 𝔪 ∖ 𝔪^{2} is R–regular, then the homomorphism φ: R ⟶ R/(g) induces a surjective quasi-isomorphism K^{φ}: K^{R} ⟶ K^{R/(g)}.

(2) If a ∈ K^{R} satisfies ∂(a) ∈ 𝔪^{2}K^{R}, then a ∈ 𝔪K^{R}.

(3) For each finite R–module M there is an integer s such that the complex

C^{i}: 0 ⟶ 𝔪^{i−e}K_{e}^{M} ⟶ … ⟶ 𝔪^{i−1}K_{1}^{M} ⟶ 𝔪^{i}K_{0}^{M} ⟶ 0  (∗)

is exact for i ≥ s; for each i, C^{i} is a DG submodule of K^{M} over K^{R}.

Proof. (1) Choose y_{1}, …, y_{e} ∈ K_{1}^{R}, such that {∂(y_{i}) = t_{i} | j = 1, …, e} is a minimal generating set for 𝔪. We may assume that g = t_{e}, and set D = R[y | ∂(y) = g]. As K^{R} is a semi-free DG module over D, by Proposition 1.3.2 the quasiisomorphism D ⟶ R/(g) induces a quasi-isomorphism K^{R} ⟶ R/(g) ⊗_{D} K^{R} = K^{R/(g)}.

(2) The statement may be rephrased as follows: the map K_{n}^{R}/𝔪K_{n}^{R} ⟶ 𝔪K_{n−1}^{R}/𝔪^{2}K_{n−1}^{R} induced by the differential ∂_{n} of K^{R} is injective for all n ≥ 1. This is a direct consequence of the formula for the Koszul differential, and the minimality of the generating set t_{1}, …, t_{e}.

(3) As C^{1} = Ker(K^{R} ⟶ k) ⊂ K^{R} is a DG ideal, C^{i} = (C^{1})^{i}K^{M} is a DG submodule of K^{M}. For each 1 ≤ n ≤ e and i ≫ 0, we have equalities

Z_{n}(C^{i}) = Z_{n}(K^{M}) ∩ 𝔪^{i−n}K_{n}^{M} = 𝔪(Z_{n}(K^{M}) ∩ 𝔪^{i−n−1}K_{n}^{M})

(the first by definition, the second by Artin-Rees). Increasing i, we may assume they hold simultaneously for all n. Thus, each z ∈ Z_{n}(C^{i}) can be written as z = ∑_{j=1}^{e} t_{j}v_{j} with v_{j} ∈ 𝔪^{i−n−1}K_{n}^{M}, so z = ∂y for y = ∑_{j=1}^{e} y_{j}v_{j} ∈ C_{n+1}^{i}. □

Next we present a result of [21], which shows that the minimal resolution of each R–module M is part of that of ‘most’ of its residue modules. It is best stated in terms of a property of minimal complexes.

Remark 4.1.7. Let F be a minimal complex of free R–modules, with F_{n} = 0 for n < 0. If ϵ: F ⟶ N is a morphism to a finite R–module N, let α: F ⟶ G be a lifting of ϵ to a minimal resolution G of N. Any two liftings are homotopic, so the homomorphism k ⊗_{R} α = H(k ⊗_{R} α) depends only on ϵ.

We say that ϵ is essential if k ⊗_{R} α_{n} is injective for each n. In that case, each α_{n} is a split injection, hence α maps F isomorphically onto a subcomplex of G, that splits off as a graded R–module; the entire resolution of N is obtained then from α(F) by adjunction of basis elements, as in Construction 2.2.6.

When pd_{R} M is finite the next result is a straightforward application of the Artin-Rees Lemma. Replacing resolutions over R by resolutions over K^{R}, we make Artin-Rees work simultaneously in infinitely many dimensions.

Theorem 4.1.8. Let F be a minimal free resolution of a finite R–module M. There exists an integer s ≥ 1, such that for each submodule M′ ⊆ 𝔪^{s}M the augmentation ϵ: F ⟶ M′′ = M/M′ is essential. In particular,

P_{M′′}^{R}(t) = P_{M}^{R}(t) + t P_{M′}^{R}(t) .

Proof. Choose s as in Lemma 4.1.6.3, so that C^{s} ⊆ K^{M} is an exact DG submodule. The projection ρ: K^{M} ⟶ K^{M}/C^{s} is then a quasi-isomorphism of DG modules over K^{R}. Let U be a semi-free resolution of k over K^{R}, let M′ be a submodule in 𝔪^{s}M, let π: M ⟶ M′′ be the canonical map. The composition

U ⊗_{K^{R}} (K^{R} ⊗_{R} M) ⟶^{π′} U ⊗_{K^{R}} (K^{R} ⊗_{R} M′′) ⟶ U ⊗_{K^{R}} (K^{M}/C^{s})

of morphisms of DG modules, where π′ = U ⊗_{K^{R}} (K^{R} ⊗_{R} π), is equal to U ⊗_{K^{R}} ρ, and so is a quasi-isomorphism by Proposition 1.3.2. It follows that H(π′) is injective. As U is a free resolution of k over R and U ⊗_{K^{R}} (K^{R} ⊗_{R} π) = U ⊗_{R} π: U ⊗_{R} M ⟶ U ⊗_{R} M′′, we see that H(π′) = Tor^{R} (k, π) . If F′′ is a minimal free resolution of M′′, then k ⊗_{R} π′: k ⊗_{R} F ⟶ k ⊗_{R} F′′ is another avatar of the same map, so it is injective, as desired. For the equality of Poincaré series, apply Tor^{R} (k, −) to 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0. □

In a precise sense, the residue field has the ‘largest’ resolution.

Corollary 4.1.9. For each finite R–module M there exists an integer ℓ ≥ 1, such that P_{M}^{R}(t) ≼ ℓ P_{k}^{R}(t) .

Proof. By the theorem, P_{M}^{R}(t) ≼ P_{M/𝔪^{s}M}^{R}(t) for some s, so we may assume that length_{R} M < ∞. The obvious induction on length, using the exact sequence of Tor^{R} (−, k) then establishes the inequality with ℓ = length_{R} M. □

For another manifestation of the ubiquity of P_{k}^{R}(t), cf. Theorem 6.3.6.

4.2. Complexity and curvature. In this section we introduce and begin to study measures for the asymptotic size of resolutions.

On the polynomial scale, the complexity of M over R is defined by

cx_{R} M = inf \{d ∈ ℕ | \begin{matrix}there exists a real number β such that \\ β_{n}^{R}(M) ≤ βn^{d−1} for n ≫ 0 \end{matrix}\};

this is an adaptation from [25], [26], of a concept originally introduced by Alperin and Evens [2] to study modular representations of finite groups; clearly, one gets the same concept by requiring inequalities for n ≫ 0.

Example 4.2.1. (1) If R′ = k[s_{1}]/(s_{1}^{2}), and t_{1} ∈ R′ is the image of s_{1}, then

F′: … ⟶ R′ ⟶^{t_{1}} R′ ⟶^{t_{1}} … R′ ⟶^{t_{1}} R′ ⟶^{t_{1}} R′ ⟶ 0 .  (∗)

is a minimal free resolution of k = R′/(t_{1}). Thus, β_{n}^{R′}(k) = 1 for n ≥ 0, and cx_{R′} k = 1. Each finite module M over the principal ideal ring k[t_{1}] is a finite direct sum of copies of k and of R′, hence β_{n}^{R′}(M) = β_{1}^{R′}(M) for n ≥ 1.

(2) Set R = k[s_{1}, s_{2}]/(s_{1}^{2}, s_{2}^{2}) = k[t_{1}, t_{2}]. As R is a free module over its subring R′, the complex R ⊗_{R′} F′ is a minimal resolution of the cyclic module M = R/(t_{1}) over R. Thus, β_{n}^{R}(M) = 1 for n ≥ 0, and cx_{R} M = 1.

On the other hand, note that Syz_{1}^{R} (R/(t_{1}t_{2})) ≅ k, and let F′′ be the complex corresponding to F′ over R′′ = k[t_{2}]. By the Künneth Theorem 1.3.4, F′ ⊗_{k} F′′ is a resolution of k ⊗_{k} k = k over R = R′ ⊗_{k} R′′, and is obviously minimal. Thus, β_{n}^{R}(k) = n + 1 for n ≥ 0, and cx_{R} (R/(t_{1}t_{2})) = cx_{R} (R/(t_{1}, t_{2})) = 2.

In particular, over the ring R there exist modules of complexity 0, 1, and 2; in fact, these are all the possible values: cf. Proposition 4.2.5.4.

Complexity may itself be infinite.

Example 4.2.2. Let R = k[s_{1}, s_{2}]/(s_{1}^{2}, s_{1}s_{2}, s_{2}^{2}). The isomorphism 𝔪 ≅ k^{2} and the exact sequence 0 ⟶ 𝔪 ⟶ R ⟶ k ⟶ 0 show that β_{n+1}^{R}(k) = 2β_{n}^{R}(k) for n ≥ 1, hence β_{n}^{R}(k) = 2^{n} for n ≥ 0, and cx_{R} k = ∞. If M is any finite R–module, then Syz_{1}^{R} (M) ⊆ 𝔪F_{0} is isomorphic to a finite direct sum of copies of k, so cx_{R} M = ∞ unless M is free.

For modules of infinite complexity, the exponential scale is used in [29] to introduce a notion of curvature^{19} by the formula

curv_{R} M = lim sup_{n→∞} \sqrt[n]{β_{n}^{R}(M)} .

Thus, in the last example, either M is free or curv_{R} M = 2.

Some relations between these asymptotic invariants follow directly from the definitions, except for (5), which is a consequence of Corollary 4.1.5:

Remark 4.2.3. For a finite R–module M the following hold:

(1) pd_{R} M < ∞ ⟺ cx_{R} M = 0 ⟺ curv_{R} M = 0 .

(2) pd_{R} M = ∞ ⟺ cx_{R} M ≥ 1 ⟺ curv_{R} M ≥ 1 .

(3) cx_{R} M ≤ 1 ⟺ M has bounded Betti numbers.

(4) cx_{R} M < ∞ ⟹ curv_{R} M ≤ 1 .

(5) curv_{R} M < ∞.

With respect to change of modules, properties of complexity and curvature mirror well known properties of projective dimension.

Proposition 4.2.4. When M is a finite R–module the following hold. (1) cx_{R} M ≤ cx_{R} k and curv_{R} M ≤ curv_{R} k .

(2) For each n there are equalities

cx_{R} M = cx_{R} Syz_{n}^{R} (M)  and  curv_{R} M = curv_{R} Syz_{n}^{R} (M) .

19So called because it is the inverse of the radius of convergence of P_{M}^{R} (t).

(3) If M′ and M′′ are R–modules, then

cx_{R}(M′ ⊕ M′′) = max {cx_{R} M′, cx_{R} M′′};
curv_{R}(M′ ⊕ M′′) = max {curv_{R} M′, curv_{R} M′′} .

(4) If 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 is an exact sequence of R–modules, then

cx_{R} M ≤ cx_{R} M′ + cx_{R} M′′ and curv_{R} M ≤ curv_{R} M′ + curv_{R} M′′ .

(5) If a sequence g ⊂ R is regular on R and on M, then

cx_{R}(M/(g)M) = cx_{R} M  and  curv_{R}(M/(g)M) = curv_{R} M .

(6) If N is a finite R–module such that Tor_{n}^{R} (M, N) = 0 for n > 0, then

max{cx_{R} M, cx_{R} N} ≤ cx_{R}(M ⊗_{R} N) ≤ cx_{R} M + cx_{R} N ;
curv_{R}(M ⊗_{R} N) = max{curv_{R} M, curv_{R} N} .

Proof. (1) comes from Proposition 4.1.9; (2), (3), and (4) are clear.

(5) By Example 1.1.1, Tor_{n}^{R} (M, R/(g)) = H_{n}(K(g; M)), and the latter module vanishes for n > 0. Thus, the desired equalities follow from (6).

(6) Let F and G be minimal free resolutions of M and N, respectively; as H_{n}(F ⊗_{R} G) = Tor_{n}^{R} (M, N) = 0 for n ≥ 1, the complex F ⊗_{R} G is a free resolution of M ⊗_{R} N. It is obviously minimal, so

max{β_{n}^{R}(M), β_{n}^{R}(N)} ≤ ∑_{p+q=n} β_{p}^{R}(M)β_{q}^{R}(N) = β_{n}^{R}(M ⊗_{R} N) .

The inequalities for complexities follow. To get the equality for curvatures, rewrite the relations above in terms of Poincaré series:

max{P_{M}^{R}(t), P_{N}^{R}(t)} ≼ P_{M}^{R}(t) P_{N}^{R}(t) = P_{M⊗_{R}N}^{R}(t) .

A product converges in the smaller of the circles of convergence of its factors, so the equality of power series yields curv_{R}(M ⊗_{R} N) ≤ max{curv_{R} M, curv_{R} N}; the converse inequality comes from the inequality of power series. □

The finiteness of projective dimension of a module is notoriously unstable under change of rings. Complexity and curvature fare better:

Proposition 4.2.5. Let M be a finite R–module, let φ: R ⟶ R′ be a homomorphism of local rings, and set M′ = R′ ⊗_{R} M.

(1) For each prime ideal 𝔭 in R there are inequalities

cx_{R_{𝔭}} M_{𝔭} ≤ cx_{R} M  and  curv_{R_{𝔭}} M_{𝔭} ≤ curv_{R} M .

(2) If φ is a flat local homomorphism, then

cx_{R′} M′ = cx_{R} M  and  curv_{R′} M′ = curv_{R} M .

(3) If Tor_{n}^{R} (R′, M) = 0 for n ≫ 0, then

cx_{R′} M′ ≤ cx_{R} M  and  curv_{R′} M′ ≤ curv_{R} M .

(4) If g is an R–regular sequence of length r and R′ = R/(g), then

cx_{R} M ≤ cx_{R′} M′ ≤ cx_{R} M + r;
curv_{R′} M′ = curv_{R} M when pd_{R} M = ∞ .

Proof. Both (1) and (2) are immediate. (3) results from Proposition 1.2.3. (4) follows from the inequalities of Poincaré series in Proposition 3.3.4. □

Finally, we note a result specific to Cohen-Macaulay rings; it fails in general, as shown by the ring R = k[[s_{1}, s_{2}]]/(s_{1}^{2}, s_{1}s_{2}) of multiplicity 1.

Proposition 4.2.6. If R is Cohen-Macaulay, then curv_{R} M ≤ mult R−1; equality holds when R has minimal multiplicity (defined in Example 15.2.8)

This follows from a lemma, that sharpens the result of Ramras [135].

Lemma 4.2.7. If R is Cohen-Macaulay, mult R = l, and type R = s, then

(l − 1)β_{n}^{R}(M) ≥ β_{n+1}^{R}(M) ≥ \frac{s}{l − s} β_{n}^{R}(M) for n > depth R − depth M .

Proof. By Remark 1.2.9, we may assume that R is artinian, with length R = l, and length(0:_{R} 𝔪) = s. In a minimal resolution F of M, Syz_{n+1}^{R} (M) ⊆ 𝔪F_{n}, so

(l − 1)β_{n}^{R}(M) = length 𝔪F_{n} ≥ length Syz_{n+1}^{R} (M) ≥ β_{n+1}^{R}(M) for n ≥ 1 .

As ∂((0:_{R} 𝔪)F_{i}) ⊆ (0:_{R} 𝔪)𝔪F_{i−1} = 0, we have (0 :_{R} 𝔪)F_{i} ⊆ Syz_{i+1}^{R} (M), and hence length Syz_{i+1}^{R} (M) ≥ sβ_{i}^{R}(M). The exact sequence

0 ⟶ Syz_{n+2}^{R} (M) ⟶ F_{n+1} ⟶ Syz_{n+1}^{R} (M) ⟶ 0

now yields lβ_{n+1}^{R}(M) = length F_{n+1} ≥ sβ_{n+1}^{R}(M) + sβ_{n}^{R}(M). □

4.3. Growth problems. As of today, a distinctive feature of the state of knowledge of infinite free resolutions is that tantalizing questions on the behavior of basic invariants can be stated in very simple terms. We present four groups of interrelated problems, that set benchmarks for many results in the text. Some are (variants) of problems discussed in more detail in [27], others are new. They could be viewed in the broader context of growth of algebraic structures, to which the survey of Ufnarovskij [153] is a good introduction.

For the rest of this section, (R, 𝔪, k) is a local ring, and M is a finite R–module; to avoid distracting special cases, we assume that pd_{R} M = ∞.

All problems discussed below have positive answers for modules over Golod rings and over complete intersections. These two cases, for which exhaustive information is available, are treated in detail in later chapters.

Growth. The easily proved inequality over Cohen-Macaulay rings suggests

Problem 4.3.1. Is lim sup_{n⟶∞} \frac{β_{n+1}^{R}(M)}{β_{n}^{R}(M)} always finite?

There is no problem when the Betti numbers of M are bounded, but little is known on when such modules occur. The only general construction that I am aware of, presented in Example 5.1.3, yields modules with periodic of period 2 minimal resolutions. Such a periodicity implies constant Betti numbers, but not conversely, cf. Remark 5.1.4. On the other hand, there exist rings over which all infinite Betti sequences are unbounded, cf. Theorem 5.3.3.

The simplest pattern of bounded Betti numbers is highlighted in

Problem 4.3.2. Is a bounded sequence {β_{n}^{R}(M)} eventually constant?

Problem 4.3.2, proposed in [134], is subsumed in the next one, from [23]:

Problem 4.3.3. Is the sequence {β_{n}^{R}(M)} eventually non-decreasing?

An early class of artinian examples is given by Gover and Ramras [75]. Many more can be found in the papers discussed after Problem 4.3.9. Complexity. Complexity is a measure for infinite projective dimensions, in the polynomial range of the growth spectrum. It is very interesting to know whether it satisfies an analogue of the Auslander-Buchsbaum result: depth R is an upper bound for all finite projective dimensions. More precisely, we propose

Problem 4.3.4. Does cx_{R} M < ∞ imply cx_{R} M ≤ codepth R?

Over a complete intersection the inequality holds for all modules, and all values between 0 and codepth R occur. More generally, for modules of finite CIdimension^{20} the problem has a positive solution, that comes with a bonus: if R is Cohen-Macaulay (respectively, Gorenstein), but not a complete intersection, then cx_{R} M ≤ codepth R − 2 (respectively, ≤ codepth R − 3).

Finite complexity only imposes an upper bound on the Betti numbers, prescribing no asymptote. As in calculus, when {b_{n}} and {c_{n}} are sequences of positive real numbers, we write b_{n} ∼ c_{n} to denote lim_{n} b_{n}/c_{n} = 1.

Problem 4.3.5. If cx_{R} M = d < ∞, is then β_{n}^{R}(M) ∼ αn^{d−1} for some α ∈ ℝ?

Would Betti sequences turn up that have subpolynomial but not asymptotically polynomial growth, then complexity should be refined by the number lim sup_{n} ln (β_{n}^{R}(M))/ln(n), modeled on^{21} Gelfand-Kirillov dimension, cf. [71].

Curvature. By d’Alembert’s convergence criterion, curv_{R} M is contained between lim inf_{n} β_{n+1}^{R}(M)/β_{n}^{R}(M) and lim sup_{n} β_{n+1}^{R}(M)/β_{n}^{R}(M). To quantify the observation that ‘at infinity’ Betti numbers display uniform behavior, we risk

Problem 4.3.6. Is curv_{R} M = lim_{n⟶∞} \frac{β_{n+1}^{R}(M)}{β_{n}^{R}(M)} ?

A positive answer would solve Problem 4.3.2, and also Problem 4.3.3 when curv_{R} M > 1. If curv_{R} M = 1, then the Betti sequence of M grows subexponentially; none is known to grow superpolynomially, so we propose

Problem 4.3.7. Does curv_{R} M = 1 imply cx_{R} M < ∞ ?

For k the answer is positive, cf. Theorem 8.2.1. The dichotomy implied by a general positive answer would be all the more remarkable for the fact, that intermediate growth does occur in most (even finitely presented!) algebraic systems: associative algebras, Lie algebras, and groups, cf. [153].

Remark 4.2.3 and Proposition 4.2.4 show that the curvatures of all R–modules (of infinite projective dimension) lie on [1, curv_{R} k], but their actual distribution is a mystery. Obviously, the first question to ask is:

Problem 4.3.8. Is the set {curv_{R} M | M a finite R–module} finite?

We also propose an exponential version of Problem 4.3.5:

Problem 4.3.9. If curv_{R} M = β > 1, is then β_{n}^{R}(M) ∼ αβ^{n} sor some α ∈ ℝ?

20That class, introduced by Avramov, Gasharov, and Peeva [32], includes the modules of finite virtual projective dimension of [25], and hence all modules over a complete intersection.

21This is a GK-dimension: that of Ext_{R} (M, k) over Ext_{R} (k, k) , cf. Chapter 10.

This is proved by Sun [150] over generalized Golod rings, cf. Theorem 10.3.3.2. Intermediate results are known in many other cases. Gasharov and Peeva [70], [130], prove that if R is Cohen-Macaulay of multiplicity ≤ 8, then there is a real number γ > 1, such that β_{n+1}^{R}(M) > γβ_{n}^{R}(M) for n ≫ 0. This property is called ‘termwise exponential growth’ by Fan [63], who extends some methods of [70] to handle artinian rings of ‘large’ embedding dimension. For a Cohen presentation R̂ = Q/I Choi [52], [53] shows that c = rank_{K} (𝔫(I : 𝔫)/𝔫I) is an invariant of R, and that β_{n+2}^{R}(M) > cβ_{n}^{R}(M) for n ≥ 1.

A weaker property, called ‘strongly^{22} exponential growth’ in [26], is established when 𝔪^{3} = 0 by Lescot [106]: for each M there is a real number γ > 1, such that β_{n}^{R}(M) ≥ γ^{n} for n ≥ 0. This special case is significant, as Anick and Gulliksen [13] prove that each P_{k}^{R}(t) is rationally related to P_{k}^{S}(t) for some artinian k–algebra (S, 𝔫, k) with 𝔫^{3} = 0.

Rationality. The study of infinite resolutions over local rings was triggered by a question, variously and appropriately linked to the names of Kaplansky, Kostrikin, Serre, and Shafarevich: Does P_{k}^{R}(t) represent a rational function?

Gulliksen [79] raised the stakes, proving that a positive answer for all (R, 𝔪, k) would imply that P_{M}^{R}(t) is rational for all R and all M. Anick [10], [11] answered the original question, with a graded artinian local k–algebra^{23}, such that P_{k}^{R}(t) is transcendental, but the following is widely open:

Problem 4.3.10. Over which R do all modules have rational Poincaré series?

Anick’s construction, as reworked by Löfwall and Roos [114], is described in detail by Roos [138], Babenko [39], and Ufnarovskij [153]; these surveys, and that of Anick [12], also describe a finite CW complex, whose loop space homology has a transcendental Poincaré series. Bøgvad [44] uses the artinian examples to produces a Gorenstein ring with irrational P_{k}^{R}(t). Fröberg, Gulliksen and Löfwall [69] show that the rationality of the Poincaré series of the residue fields is not preserved in flat families of local rings.

Jacobsson [93] provides the shortest path to irrational Poincaré series. He also shows that the rationality of P_{k}^{R}(t) does not imply that of all P_{M}^{R}(t), hence:

Problem 4.3.11. Do all rational P_{M}^{R}(t) over R have a common denominator?

A result of Levin [111], cf. Corollary 6.3.7, shows that for the last two problems it suffices to consider modules of finite length. Positive solutions are obtained over generalized Golod rings in [28], cf. Theorem 10.3.3.1.

Besides the aesthetic of the formula in ‘closed form’ that it embodies, a rational expression for a Poincaré series has practical applications. First, it provides a recurrent relation for Betti numbers that can be useful in constructing a minimal resolution. Second, it allows for efficient estimates of the asymptotic behavior of Betti sequences; for instance, rationality implies a positive solution to Problem 4.3.7, and yields some information on Problem 4.3.2: a bounded Betti sequence is eventually periodic, cf. e.g. [26].

22The terminological discrepancy reflects the growth of expectations, over a decade.

23Of rank 13.

5. Modules over Golod rings

In this chapter (R, 𝔪, k) is a local ring.

Golod [74] characterized those rings R over which the resolution of k has the fastest growth allowed by Proposition 4.1.4, that is, which have

P_{k}^{R}(t) = \frac{(1 + t)^{edim R}}{1 − ∑_{j=1}^{codepth R} rank_{k} H_{j}(K^{R})t^{j+1}} .  (5.0.1)

They are now known as Golod rings. We present some highlights of the abundant information available on resolutions of modules over them. It neatly splits into two pieces, corresponding to codepth R ≤ 1 and codepth R ≥ 2.

5.1. Hypersurfaces. A local ring with codepth R ≤ 1 is called a hypersurface.

To account for the name, consider a minimal presentation R̂ ≅ Q/I. Lemma 4.1.3.2 yields pd_{Q} Q/I ≤ 1, so I is principal, say I = (f); in particular, regular rings are hypersurfaces. If R is a hypersurface, then P_{k}^{R}(t) = (1 + t)^{edim R} by Example 1.1.1 when it is regular, and P_{k}^{R}(t) = (1 + t)^{edim R}/(1 − t^{2}) by Proposition 3.3.5.2 when it is singular. Thus, a hypersurface is a Golod ring.

Resolutions over hypersurface rings have a nice periodicity, discovered by Eisenbud [57]. In particular, cx_{R} M ≤ 1 for each R–module M. Remark 8.1.1.3 contains a strong converse: if cx_{R} k ≤ 1, then R is a hypersurface.

Theorem 5.1.1. If M is a finite module over a hypersurface ring R, then β_{n+1}^{R}(M) = β_{n}^{R}(M) for n > m = depth R − depth M. The minimal free resolution of M becomes periodic of period 2 after at most m + 1 steps. It is periodic if and only if M is maximal Cohen-Macaulay without free direct summand.

Proof. By the uniqueness of the minimal free resolution F, to say that it is periodic of period 2 after s steps amounts to saying that the syzygy modules Syz_{s}^{R} (M) and Syz_{s+2}^{R} (M) are isomorphic. We may test the isomorphism of these finite modules after tensoring by the faithfully flat R–module R̂. Since R̂⊗_{R}Syz_{n}^{R} (M) ≅ SyzR̂ ; n(M̂), we may assume that the ring R is complete, and hence of the form Q/(f) for some regular local ring Q.

For m = depth R −depth M +1, Proposition 1.2.8 and Corollary 1.2.5 show that Syz_{m}^{R} (M) is maximal Cohen-Macaulay without free direct summand; this establishes the ‘only if’ part of the last assertion. As Q is regular, and depth Syz_{m}^{R} (M) = depth R = depth Q − 1, we have pd_{Q} Syz_{m}^{R} (M) = 1, so the other assertions follow from the next construction. □

Construction 5.1.2. Periodic resolutions. Let (Q, 𝔫, k) be a local ring, let f ∈ 𝔫 be a regular element, and let M be a finite module over R = Q/(f), with pd_{Q} M = 1. By Proposition 1.2.7, its minimal resolution U over Q is of the form 0 ⟶ U_{1} ⟶ U_{0} ⟶ 0 with U_{1} ≅ U_{0} ≅ Q^{b}. Since fM = 0, the homothety f id_{U_{0}} lifts to a homomorphism σ: U_{0} ⟶ U_{1}.

Considered as a degree 1 map of complexes σ: U ⟶ U, it is a homotopy between f id^{U} and 0. If δ: U_{1} ⟶ U_{0} is the differential of U, this means that δσ = f id^{U_{0}} and σδ = f id^{U_{1}}. Thinking of σ and δ as b × b matrices, we get matrix equalities δσ = fI_{b} = σδ, called by Eisenbud [57] a matrix factorization of f. Clearly, this defines an infinite complex of free R–modules

F (δ, σ): … ⟶^{R⊗_{Q}δ} R ⊗_{Q} U_{0} ⟶^{R⊗_{Q}σ} R ⊗_{Q} U_{1} ⟶^{R⊗_{Q}δ} R ⊗_{Q} U_{0} ⟶ ⋯ (∗)

If 1 ⊗ u_{1} ∈ Ker(R ⊗ δ) for u_{1} ∈ U_{1}, then δ(u_{1}) = fu_{0} = δσ(u_{0}) with u_{0} ∈ U_{0}. As δ is injective, u_{1} = σ(u_{0}), and thus Ker(R ⊗ δ) ⊆ Im(R ⊗ σ). By a symmetric argument Ker(R ⊗ σ) ⊆ Im(R ⊗ δ), so F (δ, σ) resolves Coker(R ⊗_{Q} δ) = M.

When M has a free direct summand, its minimal resolution is not periodic, so F (δ, σ) is not minimal. Conversely, if F (δ, σ) is not minimal, then N = Im(R⊗σ) ⊈ 𝔪(R ⊗_{Q} U_{1}), so R^{b} has a basis element in N. Thus, N has a free direct summand; since N ≅ Coker(R ⊗ δ) ≅ M, so does M.

Iyengar [92] notes an alternative approach to the preceding construction:

Remark. The shortness of U forces σ^{2} = 0; reversing Remark 2.2.1, we define a DG module structure of U over the Koszul complex A = Q[y | ∂(y) = f], by setting yu = σ(u) for u ∈ U. Comparison of formulas then shows that F (δ, σ) coincides with the resolution F (A, U) of Example 3.1.2.

Modules with periodic resolutions exist over all non-linear hypersurface sections: the relevant example, due to Buchweitz, Greuel, and Schreyer [48], is adapted for the present purpose by Herzog, Ulrich, and Backelin [87].

Example 5.1.3. In the notation of Construction 5.1.2, assume that f ∈ 𝔫^{2}, so that f = ∑_{i=1}^{e} a_{i}s_{i}, with a_{i}, s_{i} ∈ 𝔪. Let K be the Koszul complex on s_{1}, …, s_{e}, let x_{1}, …, x_{e} be a basis of K_{1}, such that ∂(x_{i}) = s_{i} for i = 1, …, e. Left multiplication with ∑_{i=1}^{e} a_{i}x_{i} ∈ K_{1} yields a map τ such that ∂τ + τ∂ = f id_{K}, cf. 2.2.1. Setting U_{0} = ⨁_{i} K_{2i} and U_{1} = ⨁_{i} K_{2i+1}, one then sees that

δ = ∑_{i}(∂_{2i+1} + τ_{2i+1}): U_{1} ⟶ U_{0}  and  σ = ∑_{i}(∂_{2i} + τ_{2i}): U_{0} ⟶ U_{1}

is a matrix factorization of f over Q, so the preceding remark provides a periodic of period 2 minimal resolution of the R–module Coker(R ⊗_{Q} δ) = M.

Remark 5.1.4. A ring R satisfies the Eisenbud conjecture if each module of complexity 1 has a resolution that is eventually periodic of period 2. This is known for various R: complete intersections (Eisenbud, [57]); with codepth R ≤ 3 (Avramov, [26]); with codepth R ≤ 4 that are Gorenstein ([26]) or Cohen-Macaulay almost complete intersections (Kustin and Palmer [102]); Cohen-Macaulay of multiplicity ≤ 7 or Gorenstein of multiplicity ≤ 11 (Gasharov and Peeva, [70]); some ‘determinantal’ cases ([26]; Kustin [99], [100]).

On the other hand, Gasharov and Peeva [70] introduce a series of graded rings of embedding dimension 4 and multiplicity 8, setting

R = k[s_{1}, s_{2}, s_{3}, s_{4}]/(as_{1}s_{3} + s_{2}s_{3} , s_{1}s_{4} + s_{2}s_{4} , s_{3}s_{4} , s_{1}^{2} , s_{2}^{2} , s_{3}^{2} , s_{4}^{2})

for some non-zero element a in a field k. It is easy to check that

… ⟶ R^{2}(−n) ⟶^{∂_{n}} R^{2}(−n + 1) ⟶ … with ∂_{n} = (\begin{matrix}t_{1} & a^{n}t_{3} + t_{4} \\ 0 & t_{2} \end{matrix})

is a minimal exact complex of graded R–modules, hence M = Coker ∂_{1} has β_{n}^{R}(M) = 2 for all n ≥ 0. It is proved in [70] that this complex is periodic of period q if and only if q is the order of a in the multiplicative group of k, so Eisenbud’s conjecture fails when q > 2; similar examples over Gorenstein rings of embedding dimension 5 and multiplicity 12 are also given.

5.2. Golod rings. To investigate the rings satisfying (5.0.1), Golod [74] introduced in commutative algebra certain higher order homology operations, that have become an important tool for the construction and study of resolutions.

In this section, we use the shorthand notation \overline{a} = (−1)^{|a|+1}a.

Remark 5.2.1. Let A be a DG algebra, with H_{0}(A) ≅ k. We say that A admits a trivial Massey operation, if for some k–basis b = {h_{λ}}_{λ∈Λ} of H_{⩾ 1}(A) there exists a function μ: ⨆_{i=1}^{∞} b^{i} ⟶ A, such that

μ(h_{λ}) = z_{λ} ∈ Z(A) with cls(z_{λ}) = h_{λ};

∂μ(h_{λ_{1}}, …, h_{λ_{p}}) = ∑_{j=1}^{p−1} \overline{μ(h_{λ_{1}}, …, h_{λ_{j}})}μ(h_{λ_{j+1}}, …, h_{λ_{p}}) .

The fact that A admits a trivial Massey operation means that the algebra structure on H(A) is highly trivial. For example, z_{λ_{1}}z_{λ_{2}} = ±∂μ(h_{λ_{1}}, h_{λ_{2}}) implies that H_{⩾ 1}(A) · H_{⩾ 1}(A) = 0. Furthermore, as

|μ(h_{λ_{1}}, …, h_{λ_{p}})| = |h_{λ_{1}}| + ⋯ + |h_{λ_{p}}| + p − 1 ,

if H_{i}(A) = 0 for i ≫ 0, then there are only finitely many obstructions—known as Massey products—to the construction of a trivial Massey operation.

Theorem 5.2.2. If the Koszul complex K^{R} of a local ring (R, 𝔪, k) admits a trivial Massey operation μ, defined on a basis b = {h_{λ}}_{λ∈Λ} of H_{⩾ 1}(K^{R}), then

G_{n} = ⨁_{p+h+i_{1}+···+i_{p}=n} K_{h}^{R} ⊗_{R} V_{i_{1}} ⊗_{R} ⋯ ⊗_{R} V_{i_{p}} ,

where each V_{n} is an R–module with basis {v_{λ} : n = |v_{λ}| = |h_{λ}| + 1}_{λ∈Λ}, and

∂(a ⊗ v_{λ_{1}} ⊗ ⋯ ⊗ v_{λ_{p}}) = ∂(a) ⊗ v_{λ_{1}} ⊗ ⋯ ⊗ v_{λ_{p}}

+ (−1)^{|a|} ∑_{j=1}^{p} aμ(h_{λ_{1}}, …, h_{λ_{j}}) ⊗ v_{λ_{j+1}} ⊗ ⋯ ⊗ v_{λ_{p}}

is a minimal free resolution (G, ∂) of k. In particular, the ring R is Golod.

Remark 5.2.3. The theorem contains one direction of Golod’s result: in [74] he also shows that, conversely, the equality of Poincaré series implies that each basis of H(K^{R}) has a trivial Massey operation, cf. also [83]. The proof given below is taken from Levin [110], and uses an idea of Ghione and Gulliksen [72].

The condition for R to be Golod means that the differentials ^{r}d_{p,q} of the spectral sequence 3.2.4 vanish for r ≥ 1. Thus, the theorem is ‘explained’ by Guggenheim and May’s description [76] of the differentials of Eilenberg-Moore spectral sequences in terms of certain matric Massey products, introduced by May [118]; for proofs using that approach, cf. [18], [24].

Proof. The verification that ∂^{2} = 0 is direct, using the formulas in Remark 5.2.1. To see that G is exact, note that K = K^{R} is naturally a subcomplex of G, and G/K ≅ G ⊗_{R} V with ∂(g ⊗ v) = ∂(g) ⊗ v. The exact sequence

0 ⟶ K ⟶ G ⟶ G ⊗_{R} V ⟶ 0

of complexes of R–modules has a homology exact sequence

… ⟶ ⨁_{j} ( H_{n−j}(G) ⊗_{R} V_{j+1}) ⟶^{ð_{n+1}} H_{n}(K) ⟶ H_{n}(G)
⟶ ⨁_{j} ( H_{n−j}(G) ⊗_{R} V_{j}) ⟶^{ð_{n}} H_{n−1}(K) ⟶ …

It is clear that H_{0}(G) = k, and that ð_{n+1}(1 ⊗ v_{λ}) = h_{λ}. Thus, ð_{n+1} is surjective, and the homology sequence splits. In the exact sequence

0 ⟶ H_{1}(G) ⟶ H_{0}(K) ⊗_{R} V_{2} ⟶^{ð_{1}} H_{1}(K) ⟶ 0

we have H_{0}(K) ⊗_{R} V_{2} ≅ H_{1}(K), hence H_{1}(G) = 0. Working backwards from here, we see that H_{n}(G) = 0 for n ≥ 1.

We induce on p to prove that μ(h_{λ_{1}}, …, h_{λ_{p}}) ∈ 𝔪K^{R} for all sequences h_{λ_{1}}, …, h_{λ_{p}}. If p = 1, then μ(h_{λ}) ⊆ Z(K^{R}), and Z(K^{R}) ⊆ 𝔪K^{R} by Lemma 4.1.6.2. If p > 1, then the definition and the induction hypothesis imply ∂μ(h_{λ_{1}}, …, h_{λ_{p}}) ∈ 𝔪^{2}K^{R}, so μ(h_{λ_{1}}, …, h_{λ_{p}}) ∈ 𝔪K^{R}, again by loc. cit.

Thus, G is a minimal resolution of k. A computation identical with the one for the upper bound in the proof of Proposition 3.3.2 shows that ∑_{n} rank_{R} G_{n}t^{n} is given by the right hand side of (5.0.1), so R is Golod. □

The following conditions often suffice to recognize a Golod ring,

Proposition 5.2.4. A local ring R is Golod if some of the following hold: (1) H_{⩾ 1}(K^{R}) is generated by a set of cycles Z, such that Z^{2} = 0.

(2) R/(g) is Golod for a regular element g ∈ 𝔪 ∖ 𝔪^{2}.

(3) R ≅ Q/(f) for a Golod ring (Q, 𝔫, k) and a regular element f ∈ 𝔫 ∖ 𝔫^{2}.

(4) R̂ ≅ Q/I, where (Q, 𝔫, k) is regular, edim Q = edim R, and the minimal resolution A of R̂ over Q is a DG algebra with A_{⩾ 1} · A_{⩾ 1} ⊆ 𝔫A.

Proof. (1) Select a subset {z_{λ} ∈ Z}_{λ∈Λ} such that b = {cls(z_{λ})}_{λ∈Λ} is a basis of H_{⩾ 1}(K^{R}). A trivial Massey operation can then be defined by setting μ(h_{λ_{1}}, …, h_{λ_{j}}) = 0 for j ≥ 2, so R is Golod by Theorem 5.2.2.

(2) and (3). We note that: edim R = edim(R/(g)) − 1 (obvious); P_{k}^{R}(t) = (1 + t) P_{k}^{R/(g)}(t) (Proposition 3.3.5.1); rank_{k} H_{n}(K^{R}) = rank_{k} H_{n}(K^{R/(g)}) for each n (Lemma 4.1.6.1). Putting these equalities together, we see that R satisfies the defining equality (5.0.1) if and only if R/(g) does.

(4) The Golod conditions for R and R̂ are equivalent, so we may assume that Q/I is a regular presentation of R. By Theorem 5.2.2, it suffices to show that K^{R} admits a trivial Massey operation.

Choose a set of cycles {z_{λ} ∈ A⊗_{Q} K^{Q}}_{λ∈Λ}, such that b = {h_{λ} = cls(z_{λ})}_{λ∈Λ} is a basis of H_{⩾ 1}(A ⊗_{Q} K^{Q}). Set μ(h_{λ}) = z_{λ}, and assume by induction that a function μ: ⨆_{i=1}^{p−1} b^{i} ⟶ A ⊗_{Q} K^{Q} has been constructed for some p ≥ 2, and satisfies the conditions of Remark 5.2.1. The element

z_{λ_{1},...,λ_{p}} = ∑_{j=1}^{p−1} \overline{μ(h_{λ_{1}}, …, h_{λ_{j}})}μ(h_{λ_{j+1}}, …, h_{λ_{p}})

is then a cycle in A ⊗_{Q} K^{Q}. If ϵ: K^{Q} ⟶^{≃} k is the augmentation, then

(A ⊗_{Q} ϵ)(z_{λ_{1},...,λ_{p}}) ∈ (A ⊗_{Q} k)_{⩾ 1} · (A ⊗_{Q} k)_{⩾ 1} = 0 .

Since (A ⊗_{Q} ϵ) is a quasi-isomorphism, z_{λ_{1},...,λ_{p}} is a boundary, so extend μ to ⨆_{i=1}^{p} b^{i} by choosing μ(h_{λ_{1}}, …, h_{λ_{p}}) such that ∂(μ(h_{λ_{1}}, …, h_{λ_{p}})) = z_{λ_{1},...,λ_{p}} This completes the inductive construction of a trivial Massey operation on A ⊗_{Q} K^{Q}. The augmentation A ⟶ R induces a surjective quasi-isomorphism ϕ: A ⊗_{Q} K^{Q} ⟶ K^{R}, so to get a trivial Massey operation on the basis H(ϕ)(b) of H_{⩾ 1}(K^{R}), set μ′(ϕ(h_{λ_{1}}), …, ϕ(h_{λ_{p}})) = ϕ(μ(h_{λ_{1}}, …, h_{λ_{p}})). □

As a first application, we present a result of Shamash [143].

Proposition 5.2.5. If codim R ≤ 1, then R is Golod.

Proof. We may assume that R is a complete, and so has a minimal regular presentation R ≅ Q/I. As Q is a catenary domain, height I ≤ 1; since Q is factorial, there is an f ∈ 𝔫 such that I = fJ. In K^{R} = R ⊗_{Q} K^{Q} each cycle of degree ≥ 1 has the form 1 ⊗ z, where z ∈ K^{Q} satisfies ∂(z) = fv for some v ∈ K^{Q}. Also, f∂(v) = ∂(fv) = ∂^{2}(z) = 0, and so ∂(v) = 0. Choosing u ∈ K^{Q} such that ∂(u) = f, we get ∂(z − uv) = 0. As K_{1}^{Q} is acyclic, z − uv = ∂(w) for some w ∈ K^{Q}. Thus, cls(1 ⊗ z) = cls(1 ⊗ u) cls(1 ⊗ v), so H_{⩾ 1}(K^{R}) is generated by (1 ⊗ u)Z(K^{R}). As u^{2} = 0, case (1) of Proposition 5.2.4 applies. □

Remark. It is not clear how the Golod property of a local ring relates to other characteristics of its singularity. For one thing, it does not fit in the hierarchy

regular ⟹ complete intersection ⟹ Gorenstein ⟹ Cohen-Macaulay.

The only Golod rings that are Gorenstein are the hypersurfaces: compare Remark 5.2.3 with the fact that if R is Gorenstein, then H(K^{R}) has Poincaré duality. On the other hand, a Golod ring may or may not be Cohen-Macaulay: compare the preceding proposition with Example 5.2.8. Furthermore, the Golod condition is not stable under localization, as demonstrated by the next example.

Example 5.2.6. Let Q be the regular ring k[s_{1}, s_{2}, s_{3}]_{(s_{1},s_{2},s_{3})}. The ring R = Q/(s_{1}^{2}s_{3}, s_{2}^{2}s_{3}) is Golod by Proposition 5.2.5. On the other hand, its localization at 𝔭 = (s_{1}, s_{2}) is the ring ℓ[s_{1}, s_{2}]/(s_{1}^{2}, s_{2}^{2}), where ℓ = k(s_{3}). By example 4.2.1.2, P_{ℓ}^{R_{𝔭}}(t) = 1/(1 − t)^{2} = (1 + t)^{2}/(1 − 2t^{2} + t^{4}), so R_{𝔭} is not Golod.

Golod rings are defined by an extremal property; this might be why they appear frequently as solutions to extremal problems.

Example 5.2.7. Let Q be a graded polynomial ring generated by Q_{1} over k, and let S be a residue ring of Q, such that rank_{k} S_{1} = rank_{k} Q_{1} = e. A (now) famous theorem of Macaulay proves that there exists a lex-segment monomial ideal I, such that R = Q/I and S rank_{k} S_{n} = rank_{k} R_{n} for all n, and β_{1j}^{Q}(S) ≤ β_{1j}^{Q}(R) for the first graded Betti number, cf˙ Remark 1.2.10.

Bigatti [42] and Hulett [90] in characteristic zero, and Pardue [128] in general, extend Macaulay’s theorem to a coefficientwise inequality of Poincaré series in two variables

P_{S}^{Q}(t, u) = ∑_{n ,j} β_{nj}^{Q}(S)t^{n}u^{j} ≼ ∑_{n ,j} β_{nj}^{Q}(R)t^{n}u^{j} = P_{R}^{Q}(t, u) .

As I is a stable monomial ideal, P_{R}^{Q}(t, u) is known explicitly from the minimal free resolution A of R over Q, given by Eliahou and Kervaire [60].

Peeva [129] constructs a DG algebra structure on A, such that A_{i}A_{j} ⊆ 𝔫A_{i+j} when i ≥ 1 and j ≥ 1. By Proposition 5.2.4.4, this implies that R is Golod. The same conclusion is obtained by Aramova and Herzog [14], who verify that condition 5.2.4.1 holds. Thus, there are (in)equalities

P_{k}^{S}(t, u) ≼ \frac{(1 + tu)^{e}}{1 + t − t P_{S}^{Q}(t, u)} ≼ \frac{(1 + tu)^{e}}{1 + t − t P_{R}^{Q}(t, u)} = P_{k}^{R}(t, u) .

Example 5.2.8. Let R be a Cohen-Macaulay ring of dimension d.

As the residue field of R′ = R[t]_{𝔪[t]} is infinite, we can choose a regular sequence g = {g_{1}, …, g_{d}} such that the length of R′′ = R′/(g) is equal to mult R, the multiplicity of R, cf. Remark 1.2.9. Such a sequence is linearly independent modulo 𝔪^{2}, hence edim R′′ = edim R′ − d = codim R′ = codim R. Thus, 1 + edim R′′ = 1 + length(𝔪′′/𝔪′′^{2}) = length(R′′/𝔪′′^{2}) ≤ length R′′. This translates to an inequality codim R ≤ mult R − 1, noted by Abhyankar [1].

If equality holds, then R is said to have minimal multiplicity. Such a ring is Golod. Indeed, it suffices to prove that for R′. Proposition 5.2.4.2 reduces the problem to R′′. As Z(K^{R′′}) ⊆ 𝔪′′K^{R′′} by Lemma 4.1.6.2 and 𝔪′′^{2} = 0, Proposition 5.2.4.1 shows that R′′ is Golod. Another look at R′′ shows that mult R = type R+1, so for each M with pd_{R} M = ∞ Lemma 4.2.7 yields

P_{M}^{R}(t) = f(t) + \frac{at^{m+1}}{1 − (mult R − 1)t}

with a positive a ∈ ℤ, and f(t) ∈ ℤ[t] of degree m = dim R − depth M.

5.3. Golod modules. A Golod module is a finite R–module M, whose Poincaré series reaches the upper bound in the inequality of Proposition 4.1.4:

P_{M}^{R}(t) = \frac{∑_{i ⩾ 0} rank_{k} H_{i}(K^{M})t^{i}}{1 − ∑_{j ⩾ 1} rank_{k} H_{j}(K^{R})t^{j+1}} .  (5.3.1)

Thus, R is a Golod ring if and only if k is a Golod module. A result of Lescot [107] establishes a tight connection between Golod conditions on a ring and on its modules; part (1) is independently due to Levin [111].

Theorem 5.3.2. (1) If R has a Golod module M ≠ 0, then R is a Golod ring.

(2) If R is a Golod ring, and M is a finite R–module, then the module Syz_{n}^{R} (M) is Golod for n ≥ p = edim R − depth M.

Proof. (1) Referring successively to the definition, Proposition 3.3.3, and the inequality at the beginning of this section, we get a sequence of (in)equalities

\frac{∑_{i ⩾ 0} rank_{k} H_{i}(K^{M})t^{i}}{1 − ∑_{j ⩾ 1} rank_{k} H_{j}(K^{R})t^{j+1}} · P_{k}^{Q}(t) = P_{M}^{R}(t) · P_{k}^{Q}(t)

≼ P_{M}^{Q}(t) · P_{k}^{R}(t) ≼ P_{M}^{Q}(t) · \frac{∑_{i ⩾ 0} rank_{k} H_{i}(K^{R})t^{i}}{1 − ∑_{j ⩾ 1} rank_{k} H_{j}(K^{R})t^{j+1}} .

Lemma 4.1.3 shows that the expressions at the ends are equal, so equalities hold throughout. Cancelling P_{M}^{Q}(t) from both sides of the last equality, we see that P_{k}^{R}(t) satisfies the defining formula for Golod rings.

(2) Set M_{i} = Syz_{i}^{R} (M). Tensoring the short exact sequence of complexes from the proof of Theorem 5.2.2 with M_{i}, we get exact sequences

0 ⟶ K^{M_{i}} ⟶ M_{i} ⊗_{R} G ⟶ M_{i} ⊗_{R} (G ⊗_{R} V) ⟶ 0 .

Setting q = codepth R, note that

H_{n}(M_{i} ⊗_{R} G) = Tor_{n}^{R} (M_{i}, k) ;
H_{n} ((M_{i} ⊗_{R} G) ⊗_{R} V) = ⨁_{j=1}^{q} Tor_{n−j−1}^{R} (M_{i}, k) ⊗_{k} H_{j}(K^{R}) .

The short exact sequences produce commutative diagrams with exact rows

\xymatrix{Tor_{n+1}^{R} (M_{i}, k) \ar[r]^{ι_{n+1}^{i}} \ar[d]_{ð_{n}^{i}} & ⨁_{j=1}^{q} Tor_{n−j}^{R} (M_{i}, k) ⊗ H_{j}(K^{R}) \ar[r] \ar[d]^{⨁_{j}(ð_{j}^{i}⊗id)} & H_{n}(K^{M_{i}}) \\ _{TorR n (M_{i+1}, k)} \ar[r]^{ι_{n}^{i+1}} & ⨁_{j=1}^{p} Tor_{n−j−1}^{R} (M_{i+1}, k) ⊗ H_{j}(K^{R}) \ar[r] & H_{n−1}(K^{M_{i+1}})}

where the connecting homomorphisms ð_{n}^{i} come from the exact sequences

0 ⟶ M_{i+1} ⟶ R^{β_{i}} ⟶ M_{i} ⟶ 0 .

Note that H_{s}(K^{M}) = 0 for s > p, so ι_{s}^{0} is surjective for s > p + 1. Assume by induction that ι_{s}^{i} is surjective for s > p − i + 1. Since ð_{j}^{i} is surjective for all j, both vertical arrows are onto, so the diagram shows that ι_{s}^{i+1} is surjective for s > p − i. We conclude that ι_{s}^{p} is surjective for s > 1. Since the complex G ⊗_{R} V starts in degree 2, the map ι_{1}^{p} is surjective as well. Thus,

Tor_{n}^{R} (M_{p}, k) ≅ H_{n}(K^{M_{p}}) ⊕ ⨁_{j=1}^{q} Tor_{n−j−1}^{R} (M_{p}, k) ⊗_{k} H_{j}(K^{R}) for n ≥ 0

and hence

β_{n}^{R}(M_{p}) = rank_{k} H_{n}(K^{M}) + ∑_{j=1}^{q} β_{n−j−1}^{R}(M_{p}) rank_{k} H_{j}(K^{R}) for n ≥ 0 .

These numerical equalities add up to the defining equation (5.3.1) . □

Betti numbers of modules over Golod rings are well documented in the next theorem. It collects results from work of several authors: Ghione and Gulliksen [72] establish the rational expression in (1); the bound on its numerator, and a proof that β_{n+1}^{R}(M) > β_{n}^{R}(M) for n ≥ edim R, are due to Lescot [107]; the exponential growth result in (5) is established by Peeva [130], while the asymptotic formula in (3) is a consequence of the result of Sun [150].

Hypersurfaces are excluded from the statement of the theorem, as they have been dealt with in Section 5.1.

Theorem 5.3.3. Let R be a Golod ring with codepth R = q ≥ 2, and let M be a finite R–module with edim R − depth M = p. If pd_{R} M = ∞, then:

(1) There exist polynomials with positive integer coefficients, p(t) of degree p − 1 and q(t) of degree ≤ q, such that

P_{M}^{R}(t) = p(t) + \frac{q(t)}{1 − ∑_{j=1}^{q} rank_{k} H_{j}(K^{R})t^{j+1}} · t^{p} .

(2) cx_{R} M = ∞ and curv_{R} M = β > 1 .

(3) β_{n}^{R}(M) ∼ αβ^{n} for some real number α > 0 .

(4) lim_{n→∞} \frac{β_{n+1}^{R}(M)}{β_{n}^{R}(M)} = β.

(5) \frac{β_{n+1}^{R}(M)}{β_{n}^{R}(M)} ≥ min \{ \frac{β_{i+1}^{R}(M)}{β_{i}^{R}(M)} | p ≤ i < p + q\} = γ > 1 for n ≥ p.

Proof. The module M_{p} = Syz_{p}^{R} (M) is Golod by Theorem 5.3.2.2, and has

p = edim R − depth M_{p} = depth R − depth M_{p} + codepth R ≥ q + 2 .

Proposition 1.2.8 shows that its depth is equal to depth R. Replacing M by M_{p}, we see that it suffices to prove the theorem when M is a Golod module, and p = 0. For simplicity, we write β_{n} instead of β_{n}^{R}(M) .

(1) then holds by definition.

(5) Let R̂ = Q/I be a minimal Cohen presentation. With r_{i} = rank_{Q} Syz_{i}^{Q}(M̂) and s_{j} = rank_{Q} Syz_{j}^{Q}(R̂), we have

P_{M}^{R}(t) = \frac{∑_{i} rank_{k} H_{i}(K^{M})t^{i}}{1 − ∑_{j} rank_{k} H_{j}(K^{R})t^{j+1}} = \frac{PQ ; M̂(t)}{1 + t − t PQ ; R̂(t)} = \frac{∑_{i=1}^{p} r_{i}t^{i−1}}{1 − t ∑_{j=1}^{q} s_{j}t^{j−1}}

where the first equality holds because M is Golod, the second by Lemma 4.1.3.1, and the third by Proposition 3.3.5.2. This yields numerical relations

β_{n+1} = β_{n} + s_{1}β_{n−1} + ⋯ + s_{q−1}β_{n−q+1} + r_{n+1} for n ≥ 0 .

Since s_{1} ≥ 1 because R is not a hypersurface, we get β_{n+1} > β_{n} for n ≥ 0. Thus, min{β_{i+1}^{R}(M)/β_{i}^{R}(M) | 0 ≤ i < q} = γ > 1, so assume by induction that n ≥ q − 1 and β_{i+1}/β_{i} ≥ γ for i ≤ n. As r_{n+1} = 0 for n ≥ q − 1, we have

β_{n+1} = β_{n} + s_{1}β_{n−1} + ⋯ + s_{q−1}β_{n−q+1}
≥ γβ_{n−1} + s_{1}γβ_{n−2} + ⋯ + s_{q−1}γβ_{n−q} = γβ_{n} .

(2) is a direct consequence of (5).

(3) Let ρ be the radius of convergence of P_{M}^{R}(t). Since this is a series with positive coefficients, ρ is a root of the polynomial g(t) = 1 − ∑_{j=1}^{q} s_{j}t^{j}. Because g(0) = 1 > 0 and g(1) = 1 − ∑ rank_{Q} Syz_{j+1}^{Q} (R) < 0, we have ρ < 1. As g′(ρ) = − ∑_{j=1}^{q} js_{j}ρ^{j−1} < 0, the root ρ is simple. Let ξ_{1}, …, ξ_{m} be the remaining roots of g(t); assuming that one of them is equal to ζρ for some ζ ∈ ℂ with |ζ| = 1 and ζ ≠ 1, we get

1 = | ∑_{j=1}^{q} s_{j}(ζρ)^{j}| < ∑_{j=1}^{q} |s_{j}(ζρ)^{j}| = ∑_{j=1}^{q} s_{j}ρ^{j} = 1

which is absurd. Thus, ρ < |ξ_{i}| for i = 1, …, m. Set ξ_{0} = ρ, and write P_{M}^{R}(t) as a sum of prime fractions α_{ih}/(1 − ξ_{i}^{−1}t)^{h} with α_{ih} ∈ ℂ for i = 0, …, m. Expanding each one of them by the binomial formula, we see that

β_{n} = αρ^{−n} + ∑_{i=1}^{m} ∑_{h=1}^{n_{i}} α_{ih}\binom{h + n − 1}{h − 1}ξ_{i}^{−n} with α ≠ 0 .

Thus, α = lim_{n⟶∞} β_{n}ρ^{n} > 0; as ρ^{−1} = curv_{R} M = β, this is what we want. (4) is a direct consequence of (3). □

The next result is due to Scheja [139]; the proof is from [21].

Proposition 5.3.4. Let e = edim R and r = rank_{k} H_{1}(K^{R}). If codepth R = 2, then either R is Golod, and then P_{k}^{R}(t) = (1 + t)^{e−1}/(1 − t − (r − 1)t^{2}), or R is a complete intersection, and then P_{k}^{R}(t) = (1 + t)^{e−2}/(1 − 2t + t^{2}) .

Proof. Completing if necessary, we may assume that R ≅ Q/I, with (Q, 𝔫, k) a regular local ring, and I ⊆ 𝔫^{2}. The Auslander-Buchsbaum Equality yields pd_{Q} R = depth Q−depth R = codepth R = 2. By Example 2.1.2, the minimal free resolution A of R over Q is a DG algebra, such that A_{1}A_{1} ⊆ 𝔫A_{2}, unless I is generated by a regular sequence. In the latter case R is a complete intersection by definition; in the former, it is Golod by Proposition 5.2.4.4. The Poincaré series come from Proposition 3.3.5.2 and formula (5.0.1), respectively. □

We complement the discussion in Section 1 with Iyengar’s [92] construction of minimal resolutions over Golod rings of codepth 2; the very different case of complete intersection is treated by Avramov and Buchweitz [31].

Example 5.3.5. Let R = Q/I be a Golod ring with codepth R = 2. If edim R − depth M = p, then N = Syz_{p}^{R} (M) has pd_{Q} N = 2 by Proposition 1.2.8 and Lemma 1.2.6. By Proposition 2.2.5, the minimal free resolution U of N over Q is a DG module over the DG algebra A of Example 2.1.2. If F(A, U) = F is the resolution of N over R given by Theorem 3.1.1, then

∑_{n=0}^{∞} rank_{R} F_{n}t^{n} = \frac{P_{N}^{Q}(t)}{1 + t − t P_{R}^{Q}(t)} = P_{N}^{R}(t)

because the R–module N is Golod by Theorem 5.3.2.2. Thus, F is minimal.

6. Tate resolutions

A process of killing cycles of odd degree by adjunction of divided powers— rather than polynomial—variables, was systematically used by H. Cartan in his spectacular computation [50] of the homology of Eilenberg-MacLane spaces [56]. Their potential for commutative algebra was realized by Tate^{24} [151].

Section 1 presents most of Tate’s paper. Section 3 contains a major theorem of Gulliksen [77] and Schoeller [140]; it is proved essentially by Gulliksen’s arguments, but in a new framework motivated by Quillen’s construction of the cotangent complex in characteristic 0 and developed in Section 2.

6.1. Construction. Let A be a DG algebra, and let z ∈ A be a cycle.

Construction 6.1.1. Divided powers variable. The 𝕜–algebra 𝕜⟨x⟩ on a divided powers variable x of positive even degree is the free 𝕜–module with basis {x^{(i)} : |x^{(i)}| = i|x|}_{i ⩾ 0} and multiplication table

x^{(i)}x^{(j)} = \binom{i + j}{i}x^{(i+j)} for i, j ≥ 0 ;

it is customary to set x^{(1)} = x, x^{(0)} = 1, and x^{(i)} = 0 for i < 0.

Set A⟨x⟩^{♮} = A^{♮} ⊗_{𝕜} 𝕜⟨x⟩. If z ∈ A is cycle of positive odd degree, then

∂( ∑_{i} a_{i}x^{(i)}) = ∑_{i} ∂(a_{i})x^{(i)} + ∑_{i}(−1)^{|a_{i}|}a_{i}zx^{(i−1)}

is a differential on A⟨x⟩, that extends that of A, and satisfies the Leibniz rule.

For uniformity of notation, when |x| is odd A⟨x | ∂(x) = z⟩ stands for the algebra A[x | ∂(x) = z], described in Construction 2.1.7; when |x| = 0 we set A⟨x | ∂(x) = 0⟩ = A[x | ∂(x) = 0], cf. Construction 2.1.8.

Example 6.1.2. Let B be a graded commutative algebra. An element u ∈ B_{d} is regular on B if it is not invertible, and has the smallest possible annihilator: When d is even this means that (0 :_{B} u) = 0, the usual concept for commutative rings; when d is odd this means that (0:_{B} u) = Bu, because u^{2} = 0 implies uB ⊆ (0:_{B} u).

It is easy to see that u is regular if and only if π: B⟨x | ∂(x) = u⟩ ⟶ B/(u), π( ∑ b_{i}x^{(i)}) = b_{0} + (u), is a quasi-isomorphism. Indeed, this is classical if d is even. When d is odd, z = ∑ b_{i}x^{(i)} is a cycle precisely when ub_{i} = 0 for i > 0; if u is regular, then b_{i} = a_{i}u for each i, so z = b_{0} + ∂( ∑_{i}(−1)^{|a_{i}|}a_{i}x^{(i+1)}); else, there is a v ∉ (u) such that uv = 0, hence Ker H(π) ∋ cls(ux) ≠ 0.

A semi-free Γ -extension A⟨X⟩ of A is a DG algebra obtained by iterated (possibly, transfinite) sequence of adjunctions of the three types of variables described above; we say that the elements of X are Γ -variables over A.

Remark 6.1.3. Let A ↪ A⟨X⟩ be a semi-free Γ-extension, with ∂(x_{λ}) = z_{λ} ∈ A. Consider the semi-free extension A[Y | ∂(y_{λ}) = z_{λ}], and let α: A[Y] ⟶ A⟨X⟩ be the morphism of DG algebras defined by α(y_{λ}) = x_{λ} for each λ. A simple induction yields α(y_{λ}^{i}) = (i!)x_{λ}^{(i)}. Thus, if n! is invertible in A, then

24From the introduction of [151]: ‘Our “adjunction of variables” is a naïve approach to the exterior algebras and twisted polynomial rings familiar to topologists, and the ideas involved were clarified in my mind by conversations with John Moore.’

β (x_{λ_{1}}^{(i_{1})} ⋯ x_{λ_{q}}^{(i_{q})}) = \frac{y_{λ_{1}}^{i_{1}} ⋯ y_{λ_{q}}^{i_{q}}}{(i_{1})! ⋯ (i_{q})!} for i_{1} + ⋯ + i_{q} ≤ n

defines a morphism of complexes (A⟨X⟩)_{⩽ n} ⟶ (A[Y])_{⩽ n}, inverse to α_{⩽ n}. When A_{0} is a ℚ–algebra, α and β are inverse isomorphisms of DG algebras.

Let φ: R ⟶ S be a homomorphism of commutative rings. A factorization of φ through a semi-free Γ-extension R ↪ R⟨X⟩ followed by a surjective quasiisomorphism R⟨X⟩ ⟶ S is called a Tate resolution of S over R (or, of the R– algebra S). Revisiting the proof of Proposition 2.1.10, this time killing classes of odd degree by adjunction of divided powers variables, one gets

Proposition 6.1.4. Each R–algebra S has a Tate resolution. If R is noetherian and S is a finitely generated R–algebra (and a residue ring of R), then such a resolution exists with all X_{i} finite (and X_{0} = ∅). □

We take a close look at the effect that an adjunctions of Γ-variables has on homology. If ι: A ↪ A⟨X⟩ is a semi-free Γ-extension, and ∂(X) ⊆ A, then W = {cls(∂(x)) ∈ H(A) | x ∈ X} is contained in the kernel of the algebra homomorphism H(ι): H(A) ⟶ H(A⟨X⟩), hence there is an induced homomorphism of graded algebras \overline{ι}: H(A)/(W) H(A) ⟶ H(A⟨X⟩).

Let A ↪ A⟨x | ∂(x) = z⟩ be an extension with |z| = d and w = cls(z).

Remark 6.1.5. When d ≥ 0 is even, there is an exact sequence of chain maps

0 ⟶ A ⟶^{ι} A⟨x⟩ ⟶^{ϑ} A ⟶ 0  where  ϑ(a + xb) = b ,

has degree −d − 1, and the homology exact sequence is

… ⟶ H_{n−d}(A) ⟶^{w} H_{n}(A) ⟶^{H_{n}(ι)} H_{n}(A⟨x⟩) ⟶^{H_{n}(ϑ)} H_{n−d−1}(A) ⟶ ….

In a special case, it appears in most textbooks on commutative algebra: A = K(f; Q) is the Koszul complex on f and A⟨x⟩ = K(f, g; Q) is that on f ∪ {g}.

Remark 6.1.6. When d > 0 is odd, there is an exact sequence of chain maps

0 ⟶ A ⟶^{ι} A⟨x⟩ ⟶^{ϑ} A⟨x⟩ ⟶ 0  where  ϑ( ∑_{i} a_{i}x^{(i)}) = ∑_{i} a_{i}x^{(i−1)}

of degree −d − 1, and the homology exact sequence is

… ⟶ H_{n−d}(A⟨x⟩) ⟶^{ð_{n+1}} H_{n}(A) ⟶^{H_{n}(ι)} H_{n}(A⟨x⟩) ⟶^{H_{n}(ϑ)} H_{n−d−1}(A⟨x⟩) ⟶ …

Unlike the preceding case, multiplication by w does not appear as a map in this sequence. To analyze its impact, consider the spectral sequence of the filtration ∑_{i ⩽ p} Ax^{(i)}. Its module ^{2}E_{p,q} is the homology of the complex

H_{q−d(p+1)}(A) ⟶^{w} H_{q−dp}(A) ⟶^{w} H_{q−d(p−1)}(A) ,

so for all q there are equalities

2E_{0,q} = \frac{H_{q}(A)}{w H_{q−d}(A)}  and  ^{2}E_{p,q} = \frac{(0:_{H(A)} w)_{q−pd}}{w H_{q−(p+1)d}(A)} when p ≥ 1 .

Setting s = inf{n | (0 :_{H(A)} w)_{n} ≠ w H_{n−d}(A)}, we get ^{2}E_{p,q} = 0 for p ≥ 1 and q < dp + s. Since ^{r}d_{p,q} maps ^{r}E_{p,q} to ^{r}E_{p−r, q+r−1}, we have ^{2}E_{0,q} = ^{∞}E_{0,q} for q ≤ 2d + s and ^{2}E_{1,q} = ^{∞}E_{1,q} for q ≤ 3d + s. The spectral sequence converges to H(A⟨x⟩), so for n ≤ s the inclusion ι: A ↪ A⟨X⟩ induces isomorphisms

\overline{ι}_{n}: \frac{H_{n}(A)}{w H_{n−d}(A)} ≅ H_{n}(A⟨X⟩) for 0 ≤ n ≤ d + s ,

and short exact sequences

0 ⟶ \frac{H_{n}(A)}{w H_{n−d}(A)} ⟶^{\overline{ι}_{n}} H_{n}(A⟨x⟩) ⟶ \frac{(0:_{H(A)} w)_{n}}{w H_{n−d}(A)} ⟶ 0 for d + s + 1 ≤ n ≤ 2d + s .

We extend to the garded commutative setup a basic tool of a commutative algebraist’s trade: A sequence u = u_{1}, …, u_{i}, ⋯ ⊆ B is regular on B if (u) ≠ B and the image of u_{i} is regular on B/(u_{1}, …, u_{i−1})B for each i ≥ 1. The following result is from the source [151]. When A = A_{0}, the extension A⟨X⟩ is an usual Koszul complex and condition (ii) means that H_{n}(A⟨X⟩) = 0 for n > 0, so we have an extension of the classical characterization of regular sequences. On the other hand, when ∂^{A} = 0, condition (iii) extends the homological description of regular elements in Example 6.1.2.

Proposition 6.1.7. Let A be a DG algebra, let w = w_{1}, …, w_{j}, … be a sequence of classes in H(A), let z be a sequence of cycles such that cls(z_{j}) = w_{j} for each i, and let ι: A ↪ A⟨X | ∂(X) = z⟩ be a semi-free Γ -extension.

Implications (i) ⟹ (ii) ⟹ (iii) then hold among the conditions:

(i) The sequence w is regular in H(A).

(ii) The canonical map \overline{ι}: \frac{H(A)}{(w) H(A)} ⟶ H(A⟨X⟩) is an isomorphism.

(iii) The canonical map H(ι): H(A) ⟶ H(A⟨X⟩) is surjective.

The conditions are equivalent if w = w_{1}, or |w_{j}| > 0 for all j, or each H_{i}(A) is noetherian over S = H_{0}(A) and all w_{j} of degree zero are in the Jacobson radical of S. In these cases, any permutation of a regular sequence is itself regular.

Proof. Homology commutes with direct limits, so we may assume that the sequence z is finite, say z = z_{1}, …, z_{i}, and induce on i. For i = 1, set z = z_{1}, w = w_{1}, and d = |w|. When d is even Remark 6.1.5 readily yields (iii) ⟹ (i) ⟹ (ii), so assume that d is odd.

(i) ⟹ (ii). If w is regular on H(A), then ^{2}E_{p,q} = 0 for p > 0 in the spectral sequence in Remark 6.1.6, hence

H_{q}(A)/w H_{q−d}(A) = ^{2}E_{0,q} = ^{∞}E_{0,q} = H_{q}(A⟨X⟩) .

(iii) ⟹ (i). If w is not regular, then by Remark 6.1.6 the sequence

H_{s+d+1}(A) ⟶^{H(ι)} H_{s+d+1}(A⟨x⟩) ⟶ \frac{(0:_{H(A)} w)_{s}}{w H_{s−d}(A)} ⟶ 0

is exact, with non-trivial quotient. This contradicts the surjectivity of \overline{ι}.

Let i > 1, assume that the proposition has been proved for sequences of length i−1, set X′ = x_{1}, …, x_{i−1}, X = X′ ∪{x_{i}}, and consider the semi-free Γ-extensions ι′: A ⟶ A⟨X′⟩ and A⟨X′⟩ ⟶ A⟨X′⟩⟨x_{i}⟩ = A⟨X⟩.

(i) ⟹ (ii). If w = w_{1}, …, w_{i} is regular on H(A), then so is the sequence w′ = w_{1}, …, w_{i−1}, hence H(A)/(w′) H(A) ≅ H(A⟨X′⟩) by the induction hypothesis. As w_{i} is regular on H(A)/(w′) H(A), the basis of the induction yields an isomorphism H(A)/(w) H(A) ≅ H(A⟨X⟩).

(iii) ⟹ (i). Since the homomorphism H(ι) factors as

H(A) ⟶^{α} H(A⟨X′⟩) ⟶^{β} \frac{H(A⟨X′⟩)}{w_{i} H(A⟨X′⟩)} ⟶^{γ} H(A⟨X⟩) ,

where α = H(ι′), we see that γ is onto. The basis of our induction shows that w_{i} is regular on H(A⟨X′⟩), and that γ is bijective. Thus, βα is onto, so

H_{n}(A⟨X′⟩) = α(H_{n}(A)) + w_{i} H_{n−d}(A⟨X′⟩) for n ∈ ℤ and d = |w_{i}| .

When d > 0, induction on n shows that α is surjective; when d = 0, the same conclusion comes from Nakayama’s Lemma, which applies since H_{n}(A⟨X′⟩) is a finite H_{0}(A⟨X′⟩)–module. The surjectivity of α and the induction hypothesis imply that w′ is regular on H(A); thus, w is regular on H(A). □

Theorem 6.1.8. Let Q be a ring, let B be a DG algebra resolution of S = Q/(s_{1}, …, s_{e}), and choose x_{1}, …, x_{e} ∈ B_{1} such that ∂(x_{i}) = s_{i} for i = 1, …, e. For f = f_{1}, …, f_{r} ∈ Q with f_{j} = ∑_{i=1}^{e} a_{ij}s_{i} for j = 1, …, r, set z_{j} = ∑_{i=1}^{e} \overline{a}_{ij}x_{i}, where overlines denote images in R = Q/(f).

If f is Q–regular, then C = \overline{B}⟨x_{e+1}, …, x_{e+r} | ∂(x_{e+j}) = z_{j}⟩ resolves S over R.

Proof. With A = Q⟨Y | ∂(y_{j}) = f_{j}⟩, we have quasi-isomorphisms of DG algebras α: A ⟶ R and β: B ⟶ S, and hence induced quasi-isomorphisms

\overline{B} = B ⊗_{Q} R ⟵^{B⊗_{Q}α} B ⊗_{Q} A ⟶^{β⊗_{Q}A} S ⊗_{Q} A = S⟨y_{1}, …, y_{r} | ∂(y_{j}) = 0⟩ .

As z_{j} = (B ⊗_{Q} α)( ∑_{i=1}^{e} a_{ij}x_{i} − y_{j}) and (β ⊗_{Q} A)( ∑_{i=1}^{e} a_{ij}x_{i} − y_{j}) = −y_{j}, we see that H_{1}(\overline{B}) is a free module on w_{1}, …, w_{r}, where w_{j} = cls(z_{j}), and H(\overline{B}) is the exterior algebra ⋀ H_{1}(\overline{B}). Thus, the sequence w_{1}, …, w_{r} is regular on H(\overline{B}), so H(C) = H(\overline{B})/(w_{1}, …, w_{r}) = S by the preceding proposition. □

For B = Q⟨x_{1}, …, x_{e} | ∂(x_{i}) = s_{i}⟩ and t_{i} = \overline{s}_{i} ∈ R, the theorem yields

Corollary 6.1.9. If both f and s are Q–regular sequences, then the DG algebra

C = R⟨x_{1}, …, x_{e+r} | ∂(x_{i}) = t_{i} for i = 1, …, e; ∂(x_{e+j}) = z_{j} for j = 1, …, r⟩

resolves S over R. □

Remark. A result of Blanco, Majadas, and Rodicio [43] rounds off this circle of ideas: H_{n}(R⟨X_{1}, X_{2}⟩) = 0 for n ≥ 1 if and only if {cls(∂(x)) | x ∈ X_{2}} is a basis of H = H_{1}(R⟨X_{1}⟩) over S = H_{0}(R⟨X_{1}⟩), and the canonical map of graded algebras ⋀_{S} H ⟶ H(R⟨X_{1}⟩) is bijective.

In characteristic 0, the theorem holds with R⟨X⟩ replaced by R[X], cf. Remark 6.1.3; in general, the use of divided powers is essential:

Example 6.1.10. Let Q = k[[s]], where k is a field of characteristic p > 0, set R = Q/(s^{m+1}) for some m ≥ 1, and let t be the image of s.

For each i ≥ 0, the DG algebra G = R[y_{1}, y_{2} | ∂(y_{1}) = t; ∂(y_{2}) = t^{m}y_{1}] has G_{2i} = y_{2}^{i} and G_{2i+1} = Ry_{1}y_{2}^{i}, with ∂(y_{2}^{i}) = it^{m}y_{1}y^{i−1} and ∂(y_{1}y_{2}^{i}) = ty_{2}^{i}. Thus, H_{2ip}(G) ≅ H_{2ip−1}(G) ≅ k for i ≥ 0. If R[Y] is a resolvent of k, then it contains a subalgebra isomorphic to G, and so cannot be minimal.

6.2. Derivations. Throughout this section, A ↪ A⟨X⟩ is a semi-free Γ-extension. First, we describe a convenient basis.

Remark 6.2.1. The following conventions are in force: x^{(i)} = 0 and x^{(0)} = 1 for all x ∈ X and all i < 0; when |x| is odd, x^{(i)} is defined only for i ≤ 1, and x^{(1)} = x; when |x| = 0, x^{(i)} stands for x^{i}.

Order X = {x_{λ}}_{λ∈Λ}, first by x_{λ} < x_{μ} if |x_{λ}| < |x_{μ}|, then by well-ordering each X_{n}. For every sequence of Γ-variables x_{μ} < ⋯ < x_{ν}, and every sequence of integers i_{μ} ≥ 1, …, i_{ν} ≥ 1, the product x_{μ}^{(i_{μ})} ⋯ x_{ν}^{(i_{ν})} is called a normal Γ -monomial on X; its degree is i_{μ}|x_{μ}| + ⋯ + i_{ν}|x_{ν}|; by convention, 1 is a normal monomial. The normal monomials form the standard basis of A⟨X⟩^{♮} over A^{♮}.

To contain a proliferation of signs, we use the canonical bimodule structure carried by each DG module V over a garded commutative DG algebra B. Namely, B operates on V on the right by vb = (−1)^{|v||b|}bv for all v ∈ V and b ∈ B. This operation is right associative (that is, v(bb′) = (vb)b′), distributive, unitary, and commutes with the original action: (bv)b′ = b(vb′).

Remark 6.2.2. Let U be a module over A⟨X⟩^{♮}.

A map of 𝕜–modules ϑ: A⟨X⟩ ⟶ U, such that

ϑ(a) = 0  for all a ∈ A;
ϑ(bb′) = ϑ(b)b′ + (−1)^{|b||ϑ|}bϑ(b′)  for all b, b′ ∈ A⟨X⟩;
ϑ(x^{(i)}) = ϑ(x)x^{(i−1)}  for all x ∈ X_{even} and all i ∈ ℕ ,

is called an A–linear Γ -derivation; it is a homomorphism of A^{♮}–modules. It is easy to see by induction that

ϑ(x^{(i)}) = \begin{cases}ix^{(i−1)} & if |x| = 0 ; \\ x^{(i−1)} & if |x| ≥ 1 , \end{cases}

and

ϑ (x_{λ_{1}}^{(i_{λ_{1}} )} ⋯ x_{λ_{q}}^{(i_{λq})}) = ∑_{j=1}^{q}(−1)^{s_{j−1}}x_{λ_{1}}^{(i_{λ_{1}} )} ⋯ ϑ (x_{λ_{j}}^{(i_{λ_{j}})}) ⋯ x_{λ_{q}}^{(i_{λq})}

where s_{j} = |ϑ|(i_{λ_{1}}|x_{λ_{1}}| + ⋯ + i_{λ_{j}}|x_{λ_{j}}|).

In particular, ϑ is determined by its value on X. Conversely, each homogeneous map X ⟶ U extends to a (necessarily unique) A–linear Γ-derivation ϑ: A⟨X⟩ ⟶ U. Indeed, define the action of ϑ on the standard basis by the formulas above, and extend it by A–linearity; it suffices to check the Leibniz rule on products of normal monomials, and this is straightforward.

Let Der_{A}^{γ} (A⟨X⟩, U) be the set of all A–linear Γ-derivations from A⟨X⟩ to U. It is easy to see that Der_{A}^{γ} (A⟨X⟩, U) is a submodule of Hom_{A} (A⟨X⟩, U) , for the operation of A⟨X⟩^{♮} on the target: (bα)(b′) = bα(b′).

If U is a DG module over A⟨X⟩, and ϑ is an A–linear Γ-derivation, then so is ∂ϑ− (−1)^{|ϑ|}ϑ∂, hence Hom_{A} (A⟨X⟩, U) contains Der_{A}^{γ} (A⟨X⟩, U) as a DG submodule over A⟨X⟩; we call it the DG module of A–linear Γ -derivations. If ϑ is a cycle in Der_{A}^{γ} (A⟨X⟩, U), that is, if ∂ϑ = (−1)^{|ϑ|}ϑ∂, then we say that ϑ is a chain Γ - derivation. Clearly, if β: U ⟶ V is a homomorphism of DG modules over A⟨X⟩, then ϑ ↦ β∘ϑ is a natural homomorphism

Der_{A}^{γ} (A⟨X⟩, β) : Der_{A}^{γ} (A⟨X⟩, U) ⟶ Der_{A}^{γ} (A⟨X⟩, V) ,

of DG modules over A⟨X⟩, which is a chain map if β is one.

Over a commutative ring, the derivation functor is representable by module of Kähler differentials. The next proposition establishes the representability of the functor of Γ-derivations of semi-free Γ-extensions.

Proposition 6.2.3. There exist a semi-free DG module Diff_{A}^{γ} A⟨X⟩ over A⟨X⟩ and a degree zero chain Γ -derivation d: A⟨X⟩ ⟶ Diff_{A}^{γ} A⟨X⟩ such that

(1) ( Diff_{A}^{γ} A⟨X⟩ )^{♮} has a basis dX = {dx : |dx| = |x|}_{x∈X} over A⟨X⟩^{♮}.

(2) d(x) = dx for all x ∈ X.

(3) ∂(b(dx)) = ∂(b)(dx) + (−1)^{|b|}b d(∂(x)) for all b ∈ A⟨X⟩.

(4) The map β ↦ β∘d is a natural in U isomorphism

Hom_{A⟨X⟩} (Diff_{A}^{γ} A⟨X⟩, U) ⟶ Der_{A}^{γ} (A⟨X⟩, U)

of DG modules over A⟨X⟩, with inverse given by

ϑ̃( ∑_{x∈X} a_{x}dx) = ∑_{x∈X}(−1)^{|ϑ||a_{x}|}a_{x}ϑ(x) .

Remark. We call Diff_{A}^{γ} A⟨X⟩ the DG module of Γ -differentials of A⟨X⟩ over A, and d the universal chain Γ -derivation of A⟨X⟩ over A.

Proof. Let D be a module with basis dX = {dx : |dx| = |x|}_{x∈X} over A⟨X⟩^{♮}. By Remark 6.2.2, there is a unique degree zero Γ-derivation d: A⟨X⟩ ⟶ D, such that d(x) = dx for all x ∈ X. A short computation shows that ∂∘d − d∘∂: A⟨X⟩ ⟶ D is an A–linear Γ-derivation. It is trivial on X, hence ∂∘d = d∘∂. In particular, ∂^{2}(dx)) = d(∂^{2}(x)) = 0 for all x ∈ X, so ∂^{2} = 0.

The DG module Diff_{A}^{γ} A⟨X⟩ = (D, ∂) has the first three properties by construction. The last one is verified by inspection. □

In combination with Proposition 1.3.2, the preceding result yields:

Corollary 6.2.4. If U ⟶ V is a (surjective) quasi-isomorphism of DG modules over A⟨X⟩, then so is Der_{A}^{γ} (A⟨X⟩, U) ⟶ Der_{A}^{γ} (A⟨X⟩, V). □

Construction 6.2.5. Indecomposables. Let J denote the kernel of the morphism A ⟶ S = H_{0}(A⟨X⟩), and let X^{(⩾ 2)} be the set of normal Γ-monomials x_{μ}^{(i_{μ})} ⋯ x_{ν}^{(i_{ν})} that are decomposable, that is, satisfy i_{μ} + ⋯ + i_{ν} ≥ 2. It is clear that A + JX + AX^{(⩾ 2)} is a DG submodule of A⟨X⟩ over A, hence the projection π: A⟨X⟩ ⟶ A⟨X⟩/(A + JX + AX^{(⩾ 2)}) defines a complex of free S–modules

Ind_{A}^{γ} A⟨X⟩: … ⟶ SX_{n+1} ⟶^{δ_{n+1}} SX_{n} ⟶ ….

We call it the complex of Γ -indecomposables of the extension A ↪ A⟨X⟩. It is used to construct DG Γ-derivations, by means of the next lemma.

Lemma 6.2.6. Let V be a complex of S–modules, let U a DG module over A with U_{i} = 0 for i < 0, and let β: U ⟶ V a surjective quasi-isomorphism.

For each chain map ξ: Ind_{A}^{γ} A⟨X⟩ ⟶ V of degree n there exists a degree n chain Γ -derivation ϑ: A⟨X⟩ ⟶ U such that βϑ = ξπ; any two such derivations are homotopic by a homotopy that is itself an A–linear Γ -derivation.

Furthermore, for each family {u_{x} ∈ U_{0} | β(u_{x}) = ξ(x) for x ∈ X_{n}} ⊆ U there is a chain Γ -derivation ϑ, satisfying ϑ(x) = u_{x} for all x ∈ X_{n}.

Proof. The canonical projection A⟨X⟩ ⟶ Ind_{A}^{γ} A⟨X⟩ is an A–linear chain Γderivation, so Proposition 6.2.3.4 yields a morphism D ⟶ Ind_{A}^{γ} A⟨X⟩ of DG modules over A⟨X⟩, that maps dx to x for each x ∈ X. It induces a morphism of complexes of free S–modules S ⊗_{A⟨X⟩} D ⟶ Ind_{A}^{γ} A⟨X⟩ that is bijective on the bases, and hence is an isomorphism. Thus, we have isomorphisms

Hom_{A⟨X⟩} (D, V) ≅ Hom_{S} (S⊗_{A⟨X⟩}D, V ) ≅ Hom_{S} (Ind_{A}^{γ} A⟨X⟩, V)

On the other hand, for the semi-free module D = Diff_{A}^{γ} A⟨X⟩ over A⟨X⟩, Corollary 6.2.4 gives the surjective quasi-isomorphism below

Der_{A}^{γ} (A⟨X⟩, U) ⟶^{≃} Der_{A}^{γ} (A⟨X⟩, V) ≅ Hom_{A⟨X⟩} (D, V)

while the isomorphism comes from Proposition 6.2.3.4.

Concatenating these two sequences of morphisms, we get a surjective quasi-so-morphism α: Der_{A}^{γ} (A⟨X⟩, U) ⟶ Hom_{S} (Ind_{A}^{γ} A⟨X⟩, V) , so we can choose a cycle ϑ ∈ Der_{A}^{γ} (A⟨X⟩, U) with α(ϑ) = ξ: this is the desired chain Γ-derivation.ϑ(x) = u_{x} for all x ∈ X_{n}. Any two choices differ by the boundary of some υ ∈ Der_{A}^{γ} (A⟨X⟩, U), that is, of a Γ-derivation. Finally, observe that the ϑ_{i} = 0 for i > n, and the choice of ϑ_{n} is only subject to the condition β_{0}ϑ_{n} = ξ_{0}π_{n}, so ϑ_{n}(x) = u_{x} for x ∈ X_{n} is a possible choice. □

In these notes, applications of the lemma go through the following

Proposition 6.2.7. Assume that A⟨X⟩ ⟶ S is a quasi-isomorphism.

If x ∈ X_{n} ⊂ Ind_{A}^{γ} A⟨X⟩ is such that \overline{x} = x ∈ X_{n} +δ_{n+1} is a free direct summand of Coker δ_{n+1}, then there is an A-linear chain Γ -derivation ϑ: A⟨X⟩ ⟶ A⟨X⟩ of degree −n, with ϑ(x) = 1 and ϑ(X_{n}{x}) = 0.

Proof. Since Sx ∩ Im δ_{n+1} = 0, the homomorphism of S–modules ξ_{n}: SX_{n} ⟶ S defined by ξ_{n}(x) = 1 and ξ_{n}(X_{n} ∖ {x}) = 0 extends to a morphisms of complexes Ind_{A}^{γ} A⟨X⟩ ⟶ S; apply the lemma with U = A⟨X⟩ and V = S. □

6.3. Acyclic closures. The notion is introduced by Gulliksen in [83], where the main results below may be found. Our approach is somewhat different, as it is based on techniques from the preceding section.

Construction 6.3.1. Acyclic closures. Let A be a DG algebra, such that A_{0} is a local ring (R, 𝔪, k), and each R–module H_{n}(A) is finitely generated, let A ⟶ S be a surjective augmentation, and set J = Ker(A ⟶ S).

Successively adjoining finite packages of Γ-variables in degrees 1, 2, 3, etc., one arrives at a semi-free Γ-extension A ↪ A⟨X⟩, such that H_{0}(A⟨X⟩) = S, and H_{n}(A⟨X⟩) = 0 for n ≠ 0 (recall the argument for Proposition 2.1.10).

The Third Commandment imposes the following decisions: (1) X = X_{⩾ 1} ; (2) ∂(X_{1}) minimally generates J_{0} mod ∂_{1}(A_{1}) ; (3) {cls(∂(x)) | x ∈ X_{n+1}} minimally generates H_{n}(A⟨X_{⩽ n}⟩) for n ≥ 1 . Extensions obtained in that way are called acyclic closures of S over A.

The set X_{n} is finite for each n, so we number the Γ-variables X by the natural numbers, in such a way that |x_{i}| ≤ |x_{j}| for i < j. The standard basis of Remark 6.2.1 is then indexed by infinite sequences I = (i_{1}, …, i_{j}, …), such that i_{j} is a non-negative integer, i_{j} ≤ 1 when |x_{j}| is odd, and i_{j} = 0 for j > q = q(I); we call such an I an indexing sequence, and set x^{(I)} = x_{1}^{(i_{1})} ⋯ x_{q}^{(i_{q})}.

We set |I| = ∑_{j=0}^{∞} i_{j}|x_{j}|. For an indexing sequence H = (h_{1}, …, h_{j}, …), we set I > H if |I| > |H|; when |I| = |H|, we set I > H if there is an ℓ ≥ 0, such that i_{ℓ} > h_{ℓ}, and i_{j} = h_{j} for j > ℓ. We now have a linear order on all indexing sequences, and we linearly order the basis accordingly. Since |x^{(I)}| = |I|, it refines the order on the variables, and is just (an extension of) the usual degree-lexicographic order.

To recognize an acyclic closure when we see one, we prove:

Lemma 6.3.2. A semi-free Γ -extension A ↪ A⟨X⟩ is an acyclic closure of S if and only if X = X_{⩾ 1}, H_{0}(A⟨X⟩) = S, and the complex of free S–modules Ind_{A}^{γ} A⟨X⟩ of Construction 6.2.5 is minimal.

Proof. As X_{0} = ∅, the complex Ind_{A}^{γ} A⟨X⟩ is trivial in degrees ≤ 0.

Assume that A⟨X⟩ is an acyclic closure. For x′ ∈ X_{n+1}, write z = ∂(x′) as ∑_{x} a_{x}x + w with x ∈ X_{n}, a_{x} ∈ R, and w ∈ A⟨X_{< n}⟩. If n = 1, then ∂(z) = 0 means ∑_{x} a_{x}∂(x) ∈ ∂_{1}(A_{1}), so a_{x} ∈ 𝔪 by (2). If n ≥ 2, then ∑_{x} a_{x} cls(∂(x)) = 0, so a_{x} ∈ 𝔪 by (3); thus, Ind_{A}^{γ} A⟨X⟩ is minimal.

Assume that Ind_{A}^{γ} A⟨X⟩ is minimal. If (2) or (3) fails, then there are x_{i_{1}}, …, x_{i_{s}} ∈ X_{n+1}; a_{i_{2}}, …, a_{i_{s}} ∈ R; y ∈ A⟨X_{⩽ n}⟩, such that x_{i_{1}} − a_{i_{2}}x_{i_{2}} − ⋯ − a_{i_{s}}x_{i_{s}}−y is a cycle. In A⟨X⟩, it is equal to ∂(u+∑_{v∈X_{n+1}} c_{v}v+w), with u ∈ RX_{n+2}, c_{v} ∈ R, and w ∈ A⟨X_{⩽ n}⟩. As ∂(c_{v}v) = ∂(c_{v})v − c_{v}∂(v) ∈ JX_{n+1} + AX^{(⩾ 2)} and ∂(w) ∈ AX^{(⩾ 2)}, where X^{(⩾ 2)} is the set of decomposable Γ-monomials from Construction 6.2.5, we get ∂(u) = x_{i_{1}} − a_{2}x_{i_{2}} − ⋯ − a_{s}x_{i_{s}} ∉ 𝔪 Ind_{A}^{γ} A⟨X⟩. This contradicts the minimality of Ind_{A}^{γ} A⟨X⟩. □

We are now ready to prove a key technical fact.

Lemma 6.3.3. If A⟨X⟩ is an acyclic closure of k over A, then there exist A–linear chain Γ -derivations ϑ_{i}: A⟨X⟩ ⟶ A⟨X⟩ for i ≥ 1, such that:

(1)  ϑ_{i}(x_{h}) = \begin{cases}0 & for |x_{h}| ≤ |x_{i}| and h ≠ i; \\ 1 & for h = i . \end{cases}

(2) Each ϑ_{i} is unique up to an A–linear Γ -derivation homotopy.

(3) When I is an indexing sequence, q is such that i_{j} = 0 for j > q, ϑ^{I} = ϑ_{q}^{i_{q}} ⋯ ϑ_{1}^{i_{1}}, and H is an indexing sequence, then

ϑ^{I}(x^{(H)}) = \begin{cases}0 & for H < I; \\ 1 & for H = I . \end{cases}

Remark. In the composition ϑ^{I}, the indices of ϑ_{i_{j}} appear in decreasing order.

Proof. By Lemma 6.3.2, Ind_{A}^{γ} A⟨X⟩ is a complex of k–vector spaces with trivial differential, so derivations ϑ_{i} satisfying (1) and (2) are provided by Proposition 6.2.7. As the ϑ_{i} are Γ-derivations, (3) follows by induction on ∑_{j} i_{j}. □

The next result is due to Gulliksen [83].

Theorem 6.3.4. Let A be a DG algebra, such that A_{0} is a local ring (R, 𝔪, k), and each R–module H_{n}(A) is finitely generated. If A⟨X⟩ is an acyclic closure of k over A, then ∂(A⟨X⟩) ⊆ JA⟨X⟩, where J_{0} = 𝔪, and J_{n} = A_{n} for n > 0.

Proof. Take an arbitrary b ∈ A⟨X⟩, and write its boundary in the standard basis: ∂(b) = ∑_{H} a_{H}x^{(H)}. We have to prove that if |H| = |b|−1, then a_{H} ∈ 𝔪. Assuming the contrary, we can find an indexing sequence I with a_{I} ∉ 𝔪 and a_{H} ∈ 𝔪 for H > I. Using the preceding lemma, we get

±∂_{1}(ϑ^{I}(b)) = ϑ^{I}(∂(b)) = a_{I} + ∑_{H>I} a_{H}ϑ^{I}(x^{(H)}) ≡ a_{I} mod 𝔪A⟨X⟩ .

This is a contradiction, because ∂_{1}(A⟨X⟩_{1}) = 𝔪. □

The important special case when A = A_{0} is proved independently by Gulliksen [77] (using derivations) and Schoeller [140] (using Hopf algebras):

Theorem 6.3.5. If (R, 𝔪, k) is a local ring and R⟨X⟩ is an acyclic closure of the R–algebra k, then R⟨X⟩ is a minimal resolution of the R–module k. □

As a first application we prove a result of Levin [111], where H_{N}^{R}(t) denotes the Hilbert series ∑_{n=0}^{∞} rank_{k}(𝔪^{n}N/𝔪^{n+1}N)t^{n} of a finite R–module N.

Theorem 6.3.6. For each finite R–module M there is an integer s such that

P_{𝔪^{i}M}^{R}(t) = H_{𝔪^{i}M}^{R}(−t) P_{k}^{R}(t) for each i ≥ s .

Proof. By Theorem 6.3.5, there is a minimal resolution U = R⟨X⟩ of k over R that is a semi-free DG module over the Koszul complex K = R⟨X_{1}⟩. Choose s as in Lemma 4.1.6.3, so that for i ≥ s the complexes

C^{i}: 0 ⟶ 𝔪^{i−e}M ⊗_{R} K_{e} ⟶ … ⟶ 𝔪^{i−1}M ⊗_{R} K_{1} ⟶ 𝔪^{i}M ⊗_{R} K_{0} ⟶ 0

are exact (here e = edim R). Fix such an i, set N = 𝔪^{i}M, and for each p ≥ 0 set F^{p} = ⨁_{|e_{λ}| ⩽ p}(𝔪N ⊗_{R} K)e_{λ}.

Take z ∈ Z_{n}(𝔪N ⊗_{R} U) ∩ F^{p}. When p = 0 we have

Z_{n}(𝔪N ⊗_{R} K) = Z_{n}(𝔪^{i+1}M ⊗_{R} K) = ∂(𝔪^{i}M ⊗_{R} K_{n+1}) = ∂(N ⊗_{R} K_{n+1})

with the second equality due to the exactness of C^{i+1+n}. When p > 0, assume by induction that Z(𝔪N ⊗_{R} U) ∩ F^{p−1} ⊆ ∂(N ⊗_{R} U), and write z = ∑_{λ∈Λ_{p}} a_{λ}e_{λ} + v with v ∈ F^{p−1}. and {e_{λ}}_{λ∈Λ} a K^{♮}–basis of U^{♮}. Now

0 = ∂(z) = ∑_{λ∈Λ_{p}} ∂(a_{λ})e_{λ} ± ∑_{λ∈Λ_{p}} a_{λ}∂(e_{λ}) + ∂(v)

implies ∂(a_{λ}) = 0 for λ ∈ Λ_{p}, hence a_{λ} = ∂(b_{λ}) with b_{λ} ∈ N ⊗_{R} K, and

z = ∑_{λ∈Λ_{p}} ∂(b_{λ})e_{λ} + v = ∂( ∑_{λ∈Λ_{p}} b_{λ}e_{λ}) ∓ ∑_{λ∈Λ_{p}} b_{λ}∂(e_{λ}) + v .

Since u = ∑_{λ∈Λ_{p}} b_{λ}∂(e_{λ}) ∈ N ⊗_{R} 𝔪U = 𝔪N ⊗_{R} U, we see that u + v lies in Z(𝔪N ⊗_{R} U) ∩ F^{p−1} ⊆ ∂(N ⊗_{R} U), and hence z ∈ ∂(N ⊗_{R} U).

We have Z(𝔪N ⊗_{R} U) ⊆ ∂(N ⊗_{R} U), so 𝔪N ⊗_{R} U ⊆ N ⊗_{R} U induces the zero map in homology, that is, Tor^{R} (𝔪^{i+1}M, k) ⟶ Tor^{R} (𝔪^{i}M, k) is trivial. Thus, for each n ∈ ℤ we get an exact sequence

0 ⟶ Tor_{n}^{R} (𝔪^{i}M, k) ⟶ Tor_{n}^{R} (𝔪^{i}M/𝔪^{i+1}M, k) ⟶ Tor_{n−1}^{R} (𝔪^{i+1}M, k) ⟶ 0

of k–vector spaces. They yield an equality of Poincaré series

P_{𝔪^{j}M}^{R}(t) + t P_{𝔪^{j+1}M}^{R}(t) = rank_{k}(𝔪^{j}M/𝔪^{j+1}M) P_{k}^{R}(t)  (∗_{j})

for each j ≥ i. Multiplying (∗_{j}) by (−t)^{j−i}, and summing the resulting equalities in ℤ[[t]] over j ≥ i, we obtain P_{𝔪^{i}M}^{R}(t) = H_{𝔪^{i}M}^{R}(−t) P_{k}^{R}(t), as desired. □

The^{25} reader will note that the argument above may be used to yield a new proof of Theorem 4.1.8. By that result, P_{M}^{R}(t) = P_{M/𝔪^{i}M}^{R}(t) − t P_{𝔪^{i}M}^{R}(t); by Hilbert theory we know that H_{𝔪^{i}M}^{R}(t)(1 − t)^{dim M} ∈ ℤ[t] (for each i), hence

Corollary 6.3.7. If all R–modules of finite length have rational Poincaré series, then P_{M}^{R}(t) is rational for all R–modules M. □

A second application, from Gulliksen [78], treats partial acyclic closures.

Proposition 6.3.8. If e = edim R, then H_{⩾ 1}(R⟨X_{⩽ n}⟩)^{e+1} = 0 for each n ≥ 1.

Proof. By Theorem 6.3.5, Z_{i}(R⟨X⟩) = (∂(R⟨X⟩))_{i} ⊆ 𝔪(R⟨X⟩)_{i} for i ≥ 1, so

Z_{i}(R⟨X_{⩽ n}⟩) = Z(R⟨X⟩) ∩ R⟨X_{⩽ n}⟩_{i} ⊆ 𝔪(R⟨X⟩) ∩ R⟨X_{⩽ n}⟩_{i} = 𝔪(R⟨X_{⩽ n}⟩_{i}) .

Thus, every cycle z ∈ Z_{i}(R⟨X_{⩽ n}⟩) can be written in the form

z = ∑_{j=1}^{e} t_{j}v_{j} = ∑_{j=1}^{e} ∂(x_{j})v_{j} = ∑_{j=1}^{e} x_{j}∂(v_{j}) + ∂( ∑_{j=1}^{e} x_{j}v_{j}) ,

so each element of H_{⩾ 1}(R⟨X_{⩽ n}⟩) is represented by a cycle in X_{1}R⟨X_{⩽ n}⟩. As (X_{1}R⟨X_{⩽ n}⟩)^{e+1} = (X_{1})^{e+1}R⟨X_{⩽ n}⟩ = 0, we get H_{⩾ 1}(R⟨X_{⩽ n}⟩)^{e+1} = 0. □

Remark 6.3.9. To study ‘uniqueness’ of acyclic closures, one needs the category of DG algebras with divided powers: these are DG algebras, whose elements of positive even degree are equipped with a family of operations {a ↦ a^{(i)}}_{i ⩾ 0} that satisfy (among other things) the conditions imposed on the Γ-variables in Construction 6.1.1; they are also known as DG Γ -algebras^{26}, due to the use by Eilenberg-MacLane [56] and Cartan [50] of γ_{i}(a) to denote a^{(i)}.

It is proved in [83] that R⟨X⟩ has a unique structure of DG Γ-algebra, that extends the natural divided powers of the Γ-variables in X, and if R⟨X′⟩ is an acyclic closure of S, then R⟨X⟩ ≅ R⟨X′⟩ as DG Γ -algebras over R.

25Attentive.

26This accounts for the Γ’s appearing from Section 6.1 onward.

7. Deviations of a local ring

In this chapter (R, 𝔪, k) is a local ring.

We describe a sequence of homological invariants of the R–module k, that are ‘logarithmically’ related to its Betti numbers. They are introduced formalistically in Section 1, and shown to measure the deviation of R from being regular or a complete intersection in Section 3. In Section 2 we develop tools for their study, that reduce some problems over the singular ring R to problems over a regular ring, by means of minimal models for R.

7.1. Deviations and Betti numbers. We need an elementary observation.

Remark 7.1.1. For each formal power series P (t) = 1 + ∑_{j=1}^{∞} b_{j}t^{j} with b_{j} ∈ ℤ, there exist uniquely defined e_{n} ∈ ℤ, such that

P (t) = \frac{∏_{i=1}^{∞}(1 + t^{2i−1})^{e_{2i−1}}}{∏_{i=1}^{∞}(1 − t^{2i})^{e_{2i}}}

where the product converges in the (t)-adic topology of the ring ℤ[[t]].

Indeed, let p_{j}(t) = (1 − (−t)^{j})^{(−1)^{j+1}}. Setting P_{0}(t) = 1, assume by induction that P_{n−1}(t) = ∏_{h=1}^{n−1} p_{h}(t)^{e_{h}} satisfies P (t) ≡ P_{n−1}(t) (mod t^{n}) with uniquely defined e_{h}. If P (t) − P_{n−1}(t) ≡ e_{n}t^{n} (mod t^{n+1}), then set P_{n}(t) = P_{n−1}(t) · p_{n}(t)^{e_{n}}. The binomial expansion of p_{n}(t)^{e_{n}} shows that P (t) ≡ P_{n}(t) (mod t^{n+1}), and that e_{n} is the only integer with that property.

The exponent e_{n} defined by the product decomposition of the Poincaré series P_{k}^{R}(t) is denoted^{27} ε_{n}(R) and called the n’th deviation of R (for reasons to be clarified in Section 3); we set ε_{n}(R) = 0 for n ≤ 0. Here are the first few relations between Betti numbers β_{n} = β_{n}^{R}(k) and deviations ε_{n} = ε_{n}(R):

\begin{aligned}β_{1} & = ε_{1} ; \\ β_{2} & = ε_{2} + \binom{ε_{1}}{2} ; \end{aligned}  \begin{aligned}β_{3} & = ε_{3} + ε_{2}ε_{1} + \binom{ε_{1}}{3} ; \\ β_{4} & = ε_{4} + ε_{3}ε_{1} + \binom{ε_{2}}{2} + ε_{2}\binom{ε_{1}}{2} + \binom{ε_{1}}{4} . \end{aligned}

Remark 7.1.2. The equality P_{k}^{R}(t) = PR̂ ; k(t) and the uniqueness of the product decomposition show that ε_{n}(R) = ε_{n}(R̂) for all n.

A first algebraic description of the deviations is given by

Theorem 7.1.3. If R⟨X⟩ is an acyclic closure of k over R, then

card X_{n} = ε_{n}(R) for n ∈ ℤ .

Proof. By the minimality of R⟨X⟩ established in Theorem 6.3.5, we have equalities Tor^{R} (k, k) = H(R⟨X⟩ ⊗_{R}k) = k⟨X⟩ = ⨂_{x∈X} k⟨x⟩. Furthermore,

∑_{n=0}^{∞} rank_{k}(k⟨x⟩_{n})t^{n} = \begin{cases}1 + t^{2i−1} & if x ∈ X_{2i−1} ; \\ 1/(1 − t^{2i}) & if x ∈ X_{2i} , \end{cases}

27This numbering is at odds with [83], where ε_{n} stands for ε_{n+1}(R).

by Constructions 2.1.7 and 6.1.1, so we get

P_{k}^{R}(t) = \frac{∏_{i=1}^{∞}(1 + t^{2i−1})^{card(X_{2i−1})}}{∏_{i=1}^{∞}(1 − t^{2i})^{card(X_{2i})}} .

The coefficient of t^{n} depends only on the first n factors, so the product converges in the t-adic topology, and the desired equalities follow from Remark 7.1.1. □

We record a couple of easy consequences.

Corollary 7.1.4. ε_{n}(R) ≥ 0 for all n ∈ ℤ. □

Corollary 7.1.5. If R̂ = Q/I is a minimal regular presentation, then ε_{1}(R) = ν_{Q}(𝔫) and ε_{2}(R) = ν_{Q}(I) .

Proof. By Remark 7.1.2, we may assume that R = Q/I. Condition (6.3.1.0) for acyclic closures then yields the first equality, and implies that R⟨X_{1}⟩ is the Koszul complex K^{R}. The second equality now comes from Lemma 4.1.3.3. □

A most important property of deviations is their behavior under change of rings: it is additive, as opposed to the multiplicative nature of Betti numbers. The logarithmic nature of the deviations will reappear in Theorem 7.4.2.

Proposition 7.1.6. If g ∈ R is regular, then

ε_{1}(R/(g)) = \begin{cases}ε_{1}(R) − 1 & if g ∉ 𝔪^{2} ; \\ ε_{1}(R) & if g ∈ 𝔪^{2} ; \end{cases}

ε_{2}(R/(g)) = \begin{cases}ε_{2}(R) & if g ∉ 𝔪^{2} ; \\ ε_{2}(R) + 1 & if g ∈ 𝔪^{2} ; \end{cases}
ε_{n}(R/(g)) = ε_{n}(R)  for n ≥ 3 .

Proof. For R′ = R/(g), Proposition 3.3.5 yields P_{k}^{R}(t) = (1 − t^{2}) P_{k}^{R′}(t) if g ∈ 𝔪^{2} and P_{k}^{R}(t) = (1 + t) P_{k}^{R′}(t) if g ∉ 𝔪^{2}; now apply Remark 7.1.1. □

7.2. Minimal models. In this section (Q, 𝔫, k) is a local ring.

A minimal DG algebra over Q is a semi-free extension Q ↪ Q[Y] such that Y = Y_{⩾ 1} and the differential ∂ is decomposable in the sense that

∂(Y_{1}) ⊆ QY_{0}  and  ∂(Y_{n}) ⊆ ∑_{i=1}^{n−1} QY_{i−1}Y_{n−i}  for n ≥ 2

where Y_{0} denotes a minimal set of generators of 𝔫; when ∂(Y_{1}) ⊆ 𝔫^{2}, the minimality condition can be rewritten in the handy format ∂(Q[Y]) ⊆ (Y)^{2}Q[Y].

Remark 7.2.1. Along with a minimal DG algebra Q[Y], we consider the residue DG algebras Q[Y]/(Y_{< n})Q[Y] = k[Y_{⩾ n}] for n ≥ 1. Their initial homology is easy to compute: for degree reasons, the decomposability of ∂ implies that ∂(k[Y_{⩾ n}]_{i}) = 0 when i ≤ 2n, hence

H_{i} (k[Y_{⩾ n}]) = \begin{cases}k & if i = 0 ; \\ 0 & if 0 < i < n; \\ kY_{i} & if n ≤ i < 2n . \end{cases}

Mimicking the proof of Lemma 6.3.2, we get a criterion for minimality:

Lemma 7.2.2. A semi-free extension Q ↪ Q[Y] with Y = Y_{⩾ 1} is minimal if and only if ∂(Y_{1}) minimally generates Ker (Q ⟶ H_{0}(Q[Y])) and ∂(Y_{n}) minimally generates H_{n−1}(Q[Y_{⩽ n−1}]) for n ≥ 2. □

Minimal DG algebras are ‘as unique as’ minimal complexes.

Lemma 7.2.3. Each quasi-isomorphism ϕ: Q[Y] ⟶ Q[Y′] of minimal DG algebras over Q is an isomorphism.

Proof. Consider the restrictions ϕ^{⩽ n}: Q[Y_{⩽ n}] ⟶ Q[Y′_{⩽ n}] and the morphisms ϕ^{> n} = (k ⊗_{Q[Y_{⩽ n}]} ϕ): k[Y_{> n}] ⟶ k[Y′_{> n}] induced by ϕ. By the preceding lemma, ∂(Y_{1}) and ∂(Y′_{1}) are minimal sets of generators of Ker(Q ⟶ H_{0}(Q[Y])), so by Nakayama ϕ_{1} : QY_{1} ⟶ QY′_{1} is an isomorphism of Q–modules, and hence ϕ^{⩽ 1} is an isomorphism of DG algebras over Q.

Assume by induction that ϕ^{⩽ n} is bijective for some n ≥ 1. By Proposition 1.3.3 then so is ϕ^{> n}, hence H_{n+1}(ϕ^{> n}) is an isomorphism. By Remark 7.2.1, this is simply ϕ_{n+1}^{> n}: kY_{n+1} ⟶ kY′_{n+1}, hence (k ⊗_{Q[Y_{⩽ n}]} ϕ^{⩽ n+1}): k[Y_{⩽ n+1}] ⟶ k[Y_{⩽ n+1}] is bijective; by Nakayama, so is ϕ^{⩽ n+1}. □

If R = Q/I, then a minimal model of R over Q is a quasi-isomorphism Q[Y] ⟶ R, where Q[Y] is a minimal DG algebra.

Proposition 7.2.4. Each residue ring R = Q/I has a minimal model over Q. Any two minimal models are isomorphic DG algebras over Q.

Proof. Going through the construction in 2.1.10 of a resolvent of R over Q and strictly observing the Third Commandment, one gets a quasi-isomorphism Q[Y] ⟶ R; the DG algebra Q[Y] is minimal by Lemma 7.2.2. If Q[Y′] ⟶ R is a quasiisomorphism from a minimal DG algebra, then by the lifting property of Proposition 2.1.9 there is a quasi-isomorphism Q[Y] ⟶ Q[Y′] of DG algebras over Q; it is an isomorphism by Lemma 7.2.3. □

Remark. The proposition is from Wolffhardt [159], where minimal models are called ‘special algebra resolutions’. The ‘model’ terminology is introduced in [23] so as to reflect the similarity with the DG algebras (over ℚ) used by Sullivan [148] to encode the rational homotopy type of finite CW complexes. This parallel will bear fruits in Section 8.2.

Minimal models are not to be confused with acyclic closures: If R = Q/I, then the minimal model and acyclic closure of R over Q coincide in two cases only—when I is generated by a regular sequence, or when R ⊇ ℚ.

It is proved in [159] when R contains a field (by using bar constructions) and in [18] in general (by using Hopf algebras) that the deviations of R can be read off a minimal model. More generally, we have:

Proposition 7.2.5. Let (Q, 𝔫, k) be a regular local ring.

If Q[Y] is a semi-free extension with Y = Y_{⩾ 1} and H(Q[Y]) = R, then for each n ∈ ℤ there is an inequality card Y_{n} ≥ ε_{n+1}(R); equalities hold for all n if and only if ∂(Y_{1}) ⊆ 𝔫^{2} and Q[Y] is a minimal DG algebra.

Recall that two DG algebras A and A′ are quasi-isomorphic if there exists a sequence of quasi-isomorphisms of DG algebras A ≃ A^{1} ≃ ⋯ ≃ A^{m} ≃ A′, pointing in either direction. The next result is from Avramov [23].

Theorem 7.2.6. Let R⟨X⟩ be an acyclic closure of k over R, let R̂ ≅ Q/I be a minimal Cohen presentation, and let Q[Y] be a minimal model of R̂ over Q.

For each n ≥ 1 the DG algebras R⟨X_{⩽ n}⟩ and k[Y_{⩾ n}] = Q[Y]/(Y_{< n}) are quasiisomorphic, and card Y_{n} = card X_{n+1} = ε_{n+1}(R) for n ≥ 0 .

Thus, a minimal model of R over Q contains essentially the same information as an acyclic closure of k over R; the model has an advantage: it is defined over a regular ring, where relations are easier to compute than over R.

In view of Proposition 6.3.8, minimal models are homologically nilpotent:

Corollary 7.2.7. If e = edim R and Q[Y] is a minimal model of R̂, then for each n ≥ 1 the product of any (e + 1) elements of H_{⩾ 1}(k[Y_{⩾ n}]) is equal to 0. □

Wiebe [158] proves the next result through a lengthy computation.

Corollary 7.2.8. If R̂ = Q/I is a minimal regular presentation and E is the Koszul complex on a minimal generating set of I, then ε_{3}(R) = ν_{Q}(H_{1}(E)) .

Proof. Remark 7.1.2, the theorem, and Lemma 7.2.2 yield ε_{3}(R) = ε_{3}(R̂) = card Y_{2} = ν_{Q}(H_{1}(E)). □

Part of Theorem 7.2.6 is generalized in

Proposition 7.2.9. If Q[Y] is a minimal DG algebra over a regular local ring Q, then there exists an acyclic closure Q[Y]⟨X⟩ of k over Q[Y], such that X = {x_{y} : y ∈ Y, |x_{y}| = |y| + 1} and for each y ∈ Y there is an inclusion

∂(x_{y}) − y ∈ ∑_{j=0}^{n−1}(Y_{j})Q[Y_{< n}]⟨X_{⩽ n}⟩ where n = |y| .

We start the proofs of the theorems with a couple of general lemmas.

Lemma 7.2.10. For a (surjective) morphism of DG algebras ϕ: A ⟶ A′ and a set {z_{λ}}_{λ∈Λ} ⊆ Z(A) there is a unique (surjective) morphism of DG algebras

ϕ⟨X⟩: A⟨{x_{λ}}_{λ∈Λ} | ∂(x_{λ}) = z_{λ}⟩ ⟶ A′⟨{x_{λ}}_{λ∈Λ} | ∂(x_{λ}) = ϕ(z_{λ})⟩
with ϕ⟨X⟩ |_{A} = ϕ and ϕ⟨X⟩(x_{λ}) = x_{λ} for all λ ∈ Λ .

If ϕ is a quasi-isomorphism, then so is ϕ⟨X⟩.

Proof. The first assertion is clear. Since homology commutes with direct limits, for the second one we may assume that X is finite. By induction, it suffices to treat the case X = {x}. The result then follows from the homology exact sequences of Remarks 6.1.5 and 6.1.6, and the Five-Lemma. □

Lemma 7.2.11. Let A = k[Y′] be a minimal DG algebra over k, and let kY′ ⊂ A denote the span of the variables. For a linearly independent set Z = {z_{λ}}_{λ∈Λ} ⊂ kY′ ∩ Z(A) the canonical morphism of DG algebras

ξ: B = A⟨{x_{λ}}_{λ∈Λ} | ∂(x_{λ}) = z_{λ}⟩ ⟶ A/(Z)
with ξ(x_{λ}^{(i)}) = 0 for all λ ∈ Λ and i > 0

is a surjective quasi-isomorphism.

Proof. Consider the subalgebra C = k[Z]⟨{x_{λ}}_{λ∈Λ} | ∂(x_{λ}) = z_{λ}⟩ ⊆ B. By (an easy special case of) Proposition 6.1.7, the canonical projection ϵ: C ⟶ k is a quasiisomorphism. Since B is a semi-free DG module over C, Proposition 1.3.2 shows that π = B ⊗_{C} ϵ: B ⟶ A/(Z) is a quasi-isomorphism. □

Proof of Theorem 7.2.6. The canonical map R⟨X⟩ ⟶ R̂ ⊗_{R} R⟨X⟩ = R̂⟨X⟩ is a quasi-isomorphism, and induces quasi-isomorphisms R⟨X_{⩽ n}⟩ ⟶ R̂⟨X_{⩽ n}⟩ for n ≥ 1, so we assume that R = Q/I and take a minimal model ρ: Q[Y] ⟶ R.

Choose a minimal generating set Y_{0} = {t_{1}, …, t_{e}} of 𝔪 and pick s_{1}, …, s_{e} ∈ Q with ρ(s_{i}) = t_{i}. Lemma 7.2.10 yelds a quasi-isomorphism

R⟨X_{1} | ∂(x_{i}) = t_{i}⟩ ⟵^{ρ⟨X_{1}⟩} Q[Y]⟨X_{1} | ∂(x_{i}) = s_{i}⟩

for a set of Γ-variables X_{1} = {x_{1}, …, x_{e}}. On the other hand, as s_{1}, …, s_{e} is a Q–regular sequence, so Q⟨X_{1}⟩ ⟶ k is a quasi-isomorphism, and then Proposition 1.3.2 yields a quasi-isomorphism π^{1} : Q[Y]⟨X_{1}⟩ ⟶ k[Y].

Let n ≥ 1, and assume by induction that we have constructed surjective quasiisomorphisms of DG algebras

R⟨X_{⩽ n}⟩ ⟵^{ρ^{n}} Q[Y]⟨X_{⩽ n}⟩ ⟶^{π^{n}} k[Y_{⩾ n}]

The classes of {∂(x) | x ∈ X_{n+1}} form a basis of H_{n}(R⟨X_{⩽ n}⟩) by (6.3.1.2). Since ρ^{n} is a surjective quasi-isomorphism, these cycles are images of cycles in Q[Y]⟨X_{⩽ n}⟩, so by Lemma 7.2.10 we get a surjective quasi-isomorphism

R⟨X_{⩽ n+1}⟩ = R⟨X_{⩽ n}⟩⟨X_{n+1}⟩ ⟵^{ρ^{n}⟨X_{n+1}⟩} Q[Y]⟨X_{⩽ n}⟩⟨X_{n+1}⟩ = Q[Y]⟨X_{⩽ n+1}⟩ .

The same lemma yields a surjective quasi-isomorphism

Q[Y]⟨X_{⩽ n+1}⟩ = Q[Y]⟨X_{⩽ n}⟩⟨X_{n+1}⟩ ⟶^{π^{n}⟨X_{n+1}⟩} k[Y_{⩾ n}]⟨X_{n+1}⟩ .

As H_{n}(k[Y_{⩾ n}]) = kY_{n} by Remark 7.2.1, the differential ∂_{n+1} induces an isomorphism kX_{n+1} ⟶ kY_{n}. Lemma 7.2.11 yields a surjective quasi-isomorphism ξ^{n+1} : k[Y_{⩾ n}]⟨X_{n+1}⟩ ⟶ k[Y_{⩾ n+1}]. We set π^{n+1} = ξ^{n+1}∘π⟨X_{n+1}⟩, and note that card(X_{n+1}) = ε_{n+1}(R) by Theorem 7.1.3.

To complete the induction step set ρ^{n+1} = ρ^{n}⟨X_{n+1}⟩. □

Proof of Proposition 7.2.5. Choose g = g_{1}, …, g_{s} ∈ I and f = f_{1}, …, f_{r} ∈ I such that f ∪ g is a minimal set of generators of I, and g maps to a basis of I/(I ∩ 𝔫^{2}). The sequence g is then Q–regular, and R = Q′/I′ is a minimal regular presentation, with Q′ = Q/(g) and I′ = I/(g); set 𝔫′ = 𝔫/(g).

After a linear change of the variables of degree 1, we may assume that there is a sequence of variables y = y_{1}, …, y_{s} in Y_{1}, such that ∂(y_{i}) = g_{i} for 1 ≤ i ≤ s The Koszul complex Q[y | ∂(y_{i}) = g_{i}] is then a resolvent of Q′, so Q[Y] ⟶ Q[Y]/(g, y) = Q′[Y′] is a quasi-isomorphism by Proposition 1.3.2. As card(Y′_{n}) = card(Y_{n}) − s for n = 0, 1, and card(Y′_{n}) = card(Y_{n}) otherwise, we may assume that R = Q/I is a minimal presentation, and Q[Y] is a semi-free extension with H(Q[Y]) = R; we then have card(Y_{0}) = ε_{1}(R).

Let Q[Y′] be a minimal model of R over Q, so that card(Y′_{n}) = ε_{n+1}(R) by Theorem 7.2.6. Proposition 2.1.9 yields morphisms γ: Q[Y′] ⟶ Q[Y] and β: Q[Y] ⟶ Q[Y′] of DG algebras over Q, such that H(β) = id^{R}. By Lemma 7.2.3, βγ is an automorphism of Q[Y′], so for each n ≥ 1, the composition of γ^{⩾ n}: k[Y′_{⩾ n}] ⟶ k[Y_{⩾ n}] with β^{⩾ n}: k[Y_{⩾ n}] ⟶ k[Y′_{⩾ n}] is then bijective. In particular, β^{⩾ n} is onto, so card(Y_{n}) ≥ card(Y′_{n}) = ε_{n+1}(R) for n ≥ 1.

Assume that card(Y_{n}) = ε_{n+1}(R) for all n. The equalities for n ≤ 1 mean that I ⊆ 𝔫^{2} and ∂(Y_{1}) minimally generates I; equality for n ≥ 2 implies that H_{n}(β^{⩾ n}) is bijective, hence in k[Y_{⩾ n}] the differential ∂_{n+1} is trivial; this is equivalent to saying that in Q[X] the differential ∂_{n+1} is decomposable. □

Proof of Proposition 7.2.9. We first construct surjective quasi-isomorphisms

π^{n}: Q[Y]⟨X_{⩽ n}⟩ ⟶ k[Y_{⩾ n}] with X_{n} = {x_{y} : y ∈ Y_{n−1}} .

Let Y_{0} = {s_{1}, …, s_{e}} be a system of generators of 𝔫, set X_{1} = {x_{1}, …, x_{e}}, and Q[Y]⟨X_{1} | ∂(x_{i}) = s_{i}⟩. As in the proof of Theorem 7.2.6 we have a quasiisomorphism π^{1} : Q[Y]⟨X_{1}⟩ ⟶ k[Y].

Assume that π^{n} has been constructed for some n ≥ 1. Now each y ∈ Y_{n} is a cycle in k[Y_{⩾ n}], so y = π^{n}(z_{y}) for some cycle z_{y}; write z_{y} in the form y + ∑_{y′∈Y′_{n}} a_{y′}y′ +v, with a_{y′} ∈ 𝔫 for y′ ∈ Y′_{n} = Y_{n} ∖{y}, and v ∈ Q[Y_{< n}]⟨X_{⩽ n}⟩. Since a_{y′} = ∂(b_{y′}) for appropriate b_{y′} ∈ QX_{1}, after replacing z_{y} with the homologous cycle z_{y} − ∂( ∑_{y′∈Y′_{n}} b_{y′}y′), we can assume that z_{y} − y ∈ Q[Y_{< n}]⟨X_{⩽ n}⟩. For each y ∈ Y_{n} set ∂(x_{y}) = z_{y}. Lemma 7.2.10 then yields a surjective quasi-isomorphism

Q[Y]⟨X_{⩽ n+1}⟩ = Q[Y]⟨X_{⩽ n}⟩⟨X_{n+1}⟩ ⟶^{π^{n}⟨X_{n+1}⟩} k[Y_{⩾ n}]⟨X_{n+1}⟩ .

Lemma 7.2.11 yields a surjective quasi-isomorphismξ^{n}: k[Y_{⩾ n}]⟨X_{n+1}⟩ ⟶ k[Y_{⩾ n+1}].

In the limit, we get a quasi-isomorphism π: Q[Y]⟨X⟩ ⟶ k. As ∂(X_{1}) = Y_{0} minimally generates 𝔫, and ∂(X_{n}) minimally generates H_{n}(Q[Y]⟨X_{⩽ n}⟩) for n ≥ 2, the DG algebra Q[Y]⟨X⟩ is an acyclic closure of k over Q[Y]. By the choice of z_{y} above, and Theorem 6.3.4, we see that ∂(x_{y}) − y lies in

(Q[Y_{< n}]⟨X_{⩽ n}⟩ )_{n} ∩ ((Y)Q[Y]⟨X⟩ )_{n} = ∑_{j=0}^{n−1} ((Y_{j})Q[Y_{< n}]⟨X_{⩽ n}⟩)_{n} .

This is the desired condition on the differential. □

7.3. Complete intersections. Now we can ‘explain’ the term deviation.

Remark 7.3.1. By Corollary 7.1.5 ε_{1}(R) = edim R, so the following conditions are equivalent: (i) R is a field; (ii) ε_{1}(R) = 0 ; (iii) ε_{i}(R) = 0 for i ≥ 1 .

More generally, the regularity of a ring is detected by its deviations.

Theorem 7.3.2. The following conditions are equivalent.

(i) R is regular.

(ii) ε_{2}(R) = 0 .

(iii) ε_{n}(R) = 0 for n ≥ 2 .

Proof. When R is regular, the Koszul complex K^{R} is exact, yielding P_{k}^{R}(t) = (1 + t)^{dim R}; thus, (i) ⟹ (iii) by the uniqueness of the product decomposition (7.1.1). On the other hand, (ii) ⟹ (i) by Corollary 7.1.5. □

Recall that R is a complete intersection if R̂ has a minimal Cohen presentation Q/I, with I generated by a regular sequence; in that case, ε_{2}(R) = codim R by Corollary 7.2.8. Complete intersections of codimension 0 (respectively, ≤ 1) are precisely the regular (respectively, hypersurface) rings.

The next result gives characterizations of complete intersections in terms of vanishing of deviations, due to Assmus [15] for (iii) and to Gulliksen [78] for (iv) and [82] for (v). The use of minimal models in their proofs is new.

Theorem 7.3.3. The following conditions are equivalent.

(i) R is a complete intersection.

(ii) ε_{3}(R) = 0 .

(iii) ε_{n}(R) = 0 for n ≥ 3 .

(iv) ε_{n}(R) = 0 for n ≫ 0 .

(v) ε_{2i}(R) = 0 for i ≫ 0 .

Proof. We may take R ≅ Q/I with (Q, 𝔫, k) regular, g = f_{1}, …, f_{c} minimally generating I, and g ⊆ 𝔫^{2}, cf. Remark 7.1.2; let E be the Koszul complex on g.

(i) ⟹ (iii). If g is a Q–regular sequence, then c = codim R, so the deviations of R are computed by Theorem 7.3.2 and Proposition 7.1.6.

(ii) ⟹ (i). By Corollary 7.2.8 we have H_{1}(E) = 0, so g is Q–regular.

(v) ⟹ (iv) Take n big enough, so that ε_{2i}(R) = 0 for 2i ≥ n. By Theorem 7.2.6, the DG algebra k[Y_{⩾ n}] is a polynomial ring with variables of even degree. Their boundaries have odd degree, so are trivial, and thus H(k[Y_{⩾ n}]) = k[Y_{⩾ n}] is a polynomial ring. Each element of H_{⩾ 1}(k[Y_{⩾ n}]) is nilpotent by Corollary 7.2.7, so we conclude that Y_{⩾ n} = ∅.

(iv) ⟹ (i) Taking, as we may, f = f_{1}, …, f_{c} to be a maximal regular sequence in I, we set R′ = Q/(f). By Corollary 6.1.9, the R′–algebra k has a minimal free resolution of the form C = R′⟨x_{1}, …, x_{e+c}⟩. By the choice of f, the DG algebra C^{0} = C ⊗_{R′} R can be extended to an acyclic closure R⟨X⟩ of k by adjunction of Γ-variables of degree ≥ 2. We order them in such a way that |x_{i}| ≤ |x_{j}| for i < j, set C^{i} = C^{0}⟨x_{e+c+1}, …, x_{e+c+i}⟩, and s_{i} = sup{n | H_{n}(C^{i}) ≠ 0}. Assuming that R is not a complete intersection, we prove that X is infinite by showing that s_{i} = ∞ for each i ≥ 0.

Each element of I is a zero-divisor modulo (f), so pd_{R′}(R′/IR′) = ∞ by Proposition 1.2.7.2; as H(C^{0}) = Tor^{R′} (R, k) , we get s_{0} = ∞. For the induction step, set A = C^{i−1}, x = x_{e+c+i}, z = ∂(x), and u = cls(z), and assume that s_{i} = s < ∞. When |x| is even, the homology exact sequence in Remark 6.1.6 yields s_{i−1} ≤ s + |x|, contradicting the induction hypothesis. When |x| is odd the sequence in Remark 6.1.5 shows that H(A) = H_{⩽ s}(A) + u H(A). A simple iteration yields H(A) = H_{⩽ s+e|u|}(A) + u^{e+1} H(A). But u^{e+1} = 0 by Proposition 6.3.8, hence s_{i−1} ≤ s + e|u|; this contradiction proves that s_{i} = ∞. □

The last result is vastly generalized in Halperin’s [84] rigidity theorem:

Theorem 7.3.4. If ε_{n}(R) = 0 and n > 0 then R is a complete intersection. ■

A proof that uses techniques developed in Section 6.2, and extends the theorem to a relative situation, is given in [30].

7.4. Localization. The theme of the preceding section may be summarized as follows: The deviations of a local ring reflect the character of its singularity. Thus, one would expect that they do not go up under localization, and in particular that the complete intersection property localizes.

It is instructive to generalize the discussion by considering the number

cid(R) = ε_{2}(R) − ε_{1}(R) + dim R

that in view of the next lemma we^{28} call the complete intersection defect of R. The lemma also shows that in the definition of complete intersection the restriction to minimal presentations is spurious.

Lemma 7.4.1. If R̂ ≅ Q/I is a regular presentation, then

cid R = ν_{Q}(I) − height(I) ≥ 0 .

Furthermore, the following conditions are equivalent: (i) cid R = 0 ; (ii) I is generated by a regular sequence; (iii) R is a complete intersection,

Proof. Choose a regular sequence g = g_{1}, …, g_{s} as in the proof of Proposition 7.2.5. With Q′ = Q/(g) and I′ = I/(g) we have a minimal Cohen presentation R̂ ≅ Q′/I′. Corollary 7.1.5 provides the first equality below, the catenarity of Q′ yields the last one, and Krull’s Principal Ideal Theorem gives the inequality:

cid R = ν_{Q′}(I′) − ν_{Q′}(𝔫′) + dim R = ν_{Q}(I) − ν_{Q}(𝔫) + dim R
= ν_{Q}(I) − (dim Q − dim R) = ν_{Q}(I) − height(I) ≥ 0 .

The equivalence (i) ⟺ (ii) now follows from the Cohen-Macaulay Theorem. Applied to the minimal presentation R̂ ≅ Q′/I′, it yields (ii) ⟺ (iii). □

For the study of complete intersection defects and even deviations, the first part of the next theorem^{29} suffices; it is due to Avramov [20]. For odd deviations one needs the second part, due to André [9].

Theorem 7.4.2. If R ⟶ S is a faithfully flat homomorphism of local rings, then for each i ≥ 1 there is an integer δ_{i} ≥ 0, such that

ε_{2i}(R) ≤ ε_{2i}(S) = ε_{2i}(R) + ε_{2i}(S/𝔪S) − δ_{i};
ε_{2i−1}(S/𝔪S) ≤ ε_{2i−1}(S) = ε_{2i−1}(R) + ε_{2i−1}(S/𝔪S) − δ_{i} .

Furthermore, δ_{i} = 0 for i ≫ 0, and ∑_{i=0}^{∞} δ_{i} ≤ codepth(S/𝔪S). ■

As in [20], we deduce:

Theorem 7.4.3. If R ⟶ S is a flat local homomorphism, then

cid S = cid R + cid(S/𝔪S) .

In particular, S is a complete intersection if and only if both R and S/𝔪S are.

Proof. The first two equalities of Theorem 7.4.2 yield

ε_{2}(S) − ε_{1}(S) = ε_{2}(R) − ε_{1}(R) + ε_{2}(S/𝔪S) − ε_{1}(S/𝔪S) .

Classically, dim S = dim R + dim(S/𝔪S), so we have the desired result. □

Corollary 7.4.4. For each prime ideal 𝔭 of R, cid(R_{𝔭}) ≤ cid R.

Proof. As dim R = dim R̂, we have cid R = cid R̂ by Remark 7.1.2. Let R̂ ≅ Q/I be a regular presentation, and pick prime ideals 𝔭′ ⊆ R̂ and 𝔮 ⊆ Q, such that 𝔭′ ∩ R = 𝔭 and 𝔭′ = 𝔮R̂. As R̂_{𝔭′} ≅ Q_{𝔮}/I_{𝔮} is a regular presentation,

cid R̂ = ν_{Q}(I) − height(I) ≥ ν_{Q_{𝔮}}(I_{𝔮}) − height(I_{𝔮}) = cid(R̂_{𝔭′}) ,

28Kiehl and Kunz introduced it in [96] by the expression in Lemma 7.4.1, and called it the deviation of R; that was before an infinite supply of deviations appeared on the scene.

29For a more natural statement, cf. Remark 10.2.4.

with equalities coming from Lemma 7.4.1, and inequality from the obvious relations ν_{Q}(I) ≥ ν_{Q_{𝔮}}(I_{𝔮}) and height(I) ≤ height(I_{𝔮}). Finally, the theorem applied to the flat homomorphism R_{𝔭} ⟶ R̂_{𝔭′} yields cid(R̂_{𝔭′}) ≥ cid(R_{𝔭}). □

It is now clear that complete intersections localize, a fact initially proved in [19]. This is sharpened in the corollary of the next theorem from [20], [9], which represents a quantitative extension to arbitrary local rings of the classical localization of regularity, cf. Corollary 4.1.2.

Theorem 7.4.5. If 𝔭 is a prime ideal of R, then an inequality ε_{n}(R_{𝔭}) ≤ ε_{n}(R) holds for all even n and for almost all odd n. When R is a residue ring of a regular local ring the inequalities hold for all n.

Proof. If R is a residue ring of a regular local ring Q, let Q[Y] be a minimal model of R. If 𝔮 is the inverse image of 𝔭 in Q, then Q_{𝔮}[Y] is a semi-free extension of Q_{𝔮} with H(Q_{𝔮}[Y]) ≅ R_{𝔭}. Applying Proposition 7.2.5 first to Q[Y], then to Q_{𝔮}[Y], we get ε_{n+1}(R) = card(Y_{n}) ≥ ε_{n+1}(R_{𝔭}) .

In general, pick (by faithful flatness) a prime ideal 𝔭′ in R̂, such that 𝔭 = 𝔭′ ∩ R. By Remark 7.1.2 and the preceding case we then have ε_{n}(R) = ε_{n}(R̂) ≥ ε_{n}(R̂_{𝔭′}) for all n. On the other hand, Theorem 7.4.2 yields an inequality ε_{n}(R̂_{𝔭′}) ≥ ε_{n}(R_{𝔭}) for all even n, and almost all odd n. □

Corollary 7.4.6. If R is a complete intersection, then for each prime ideal 𝔭 of R the ring R_{𝔭} is a complete intersection with codim(R_{𝔭}) ≤ codim R. □

Proof. In view of Theorem 7.3.3, the inequalities of deviations for n = 2i ≫ 0 prove that R_{𝔭} is a complete intersection. Since codim R = ε_{2}(R) by Corollary 7.1.5, the inequality for n = 2 shows that codim(R_{𝔭}) ≤ codim R. □

For the first deviation, there is a more precise result of Lech [104]; a simpler proof is given by Vasconcelos [154].

Theorem 7.4.7. For each prime ideal 𝔭 of R, ε_{1}(R_{𝔭}) + dim(R/𝔭) ≤ ε_{1}(R). ■

We spell out the obvious remaining problems. The first one has a positive solution when char(k) = 2, due to André [8].

Problem 7.4.8. Let R ⟶ S be a faithfully flat homomorphism of local rings. Does an equality ε_{n}(S) = ε_{n}(R) + ε_{n}(S/𝔪S) hold for each n ≥ 3 ?

Note that by Corollary 7.1.5 we have

ε_{1}(R) − ε_{1}(S) + ε_{1}(S/𝔪S) = edim(R) − edim(S) + edim(S/𝔪S) ≥ 0 .

The inequality is strict unless a minimal generating set of 𝔪 extends to one of 𝔫, so additivity may fail for n = 1 and hence, by Theorem 7.4.2, also for n = 2.

Problem 7.4.9. Does ε_{n}(R_{𝔭}) ≤ ε_{n}(R) hold for all 𝔭 ∈ Spec R and odd n ≥ 3 ?

It is easily seen from the proof of Theorem 7.4.5 that a positive solution of the first problem implies one for the second. Larfeldt and Lech [103] prove that the two problems are, in fact, equivalent.

8. Test modules

Ring are ‘non-linear’ objects, so some of their properties are easier to verify after translation into conditions on some canonically defined modules.

The Auslander-Buchsbaum-Serre Theorem provides a model: the regularity of a local ring (R, 𝔪, k) is tested by checking the finiteness of the projective dimension of k. In terms of asymptotic invariants, this is stated as

cx_{R} k = 0 ⟺ R is regular ⟺ curv_{R} k = 0 .

The first two sections establish similar descriptions of complete intersections:

cx_{R} k < ∞ ⟺ R is a complete intersection ⟺ curv_{R} k ≤ 1 .

For algebras essentially of finite type, another classical test for regularity is given by the Jacobian criterion. Section 3 discusses extensions to complete intersections, in terms of the homology of Kähler differentials. The results there are partly motivated by (still open in general) conjectures of Vasconcelos.

8.1. Residue field. In this section (R, 𝔪, k) is a local ring.

We start with a few general observations on the Betti numbers of k. They show that an extremal property of Poincaré series characterizes complete intersections— and places across the spectrum from Golod rings.

Remarks 8.1.1. Set e = edim R, r = rank_{k} H_{1}(K^{R}), and ε_{n} = ε_{n}(R). (1) There is an inequality of formal power series

P_{k}^{R}(t) ≽ \frac{(1 + t)^{e}}{(1 − t^{2})^{r}} .

Indeed, Corollary 7.1.5 yields P_{k}^{R}(t) = (1+t)^{e}(1−t^{2})^{−r}Q(t), with Q(t) = ∏_{i=2}^{∞}(1+ t^{2i−1})^{ε_{2i−1}}/ ∏_{i=2}^{∞}(1 − t^{2i})^{ε_{2i}}; also, Q(t) ≽ 1 by Corollary 7.1.4.

(2) Theorem 7.3.3 shows that equality holds in (1) if and only if R is a complete intersection. In that case, cx_{R} k = r = codim R, and

β_{n}^{R}(k) = ∑_{i=0}^{e−r} \binom{e − r}{i}\binom{n + r − 1 − i}{r − 1} for n ≥ 0.

(3) If R is not a hypersurface, then β_{n}^{R}(k) > β_{n−1}^{R}(k) for n ≥ 1. Indeed, then e ≥ 1 and r ≥ 2, so we have coefficientwise (in)equalities

∑_{n=0}^{∞} (β_{n}^{R}(k) − β_{n−1}^{R}(k))t^{n} = (1 − t) P_{k}^{R}(t) = \frac{(1 + t)^{e−1}}{(1 − t^{2})^{r−1}} Q(t) ≽ \frac{1}{(1 − t)} .

Gulliksen [78], [82] extends the Auslander-Buchsbaum-Serre Theorem in

Theorem 8.1.2. The following conditions are equivalent.

(i) R is a complete intersection (respectively, of codimension ≤ c).

(ii) cx_{R} M < ∞ (respectively, cx_{R} M ≤ c) for each finite R–module M .

(iii) cx_{R} k < ∞ (respectively, cx_{R} k ≤ c).

Proof. By Proposition 4.2.4 we have cx_{R} M ≤ cx_{R} k, and the complexity of k is equal to codim R by Remark 8.1.1.2, so (i) implies (ii).

When R is not a complete intersection Theorem 7.3.3 gives infinitely many indices i_{d} with ε_{2i_{d}}(R) > 0. Remark 7.1.1 then yields an inequality

P_{k}^{R}(t) ≽ \frac{1}{(1 − t^{2i_{1}}) ⋯ (1 − t^{2i_{d}})} ≽ \frac{1}{(1 − t^{2i})^{d}} = ∑_{n=0}^{∞} \binom{n + d − 1}{d − 1}t^{(2i)n}

with i = i_{1} ⋯ i_{d}. Thus, cx_{R} k ≥ d for each d ≥ 1, so (iii) implies (i). □

Remark 8.1.3. We get a new proof of Corollary 7.4.6 by recycling the classical argument for regularity. Proposition 4.2.4.1 and Remark 8.1.1.2 yield

cx_{R_{𝔭}} (R_{𝔭}/𝔭R_{𝔭}) ≤ cx_{R}(R/𝔭) ≤ cx_{R} k = codim R ,

so R_{𝔭} is a complete intersection of codimension ≤ codim R by the theorem.

We finish this section by a computation of the curvature of k in terms of the deviations of R; the purely analytical argument is from Babenko [38].

Proposition 8.1.4. If R is not a complete intersection, then

curv_{R} k = lim sup_{n} \sqrt[n]{ε_{n}(R)} .

Proof. Note that lim sup_{n} \sqrt[n]{ε_{n}(R)} = 1/η, where η is the radius of convergence of the series E(t) = ∑_{n=1}^{∞} ε_{n}(R)t^{n}. By the definition of curv_{R} k, we have to show that η = ρ, where ρ is the radius of convergence of the Poincaré series P (t) = P_{k}^{R}(t); note that ρ > 0 by Remark 4.2.3.5. By Corollary 7.1.4, we have ε_{n}(R) = ε_{n} ≥ 0, so by the product formula of Remark 7.1.1 we get a coefficientwise inequality P (t) ≽ E(t), hence η ≥ ρ > 0.

To prove that ρ ≥ η, we show that if 0 < γ < η, then P (t) converges at t = γ. We have η < 1, because E(t) has integer coefficients and ε_{n} > 0 for infinitely many n by Theorem 7.3.3. For j ≥ 1 we then get

0 < − ln(1 − γ^{j}) = ∑_{h=1}^{∞} \frac{γ^{jh}}{h} < ∑_{h=1}^{∞} γ^{jh} = \frac{γ^{j}}{1 − γ^{j}} < \frac{γ^{j}}{1 − η} .

By a similar computation, 0 < ln(1 + γ^{j}) < (1 − η)^{−1}γ^{j}, so

0 ≤ L(γ) = ∑_{i=1}^{∞} (ε_{2i−1} ln(1 + γ^{2i−1}) − ε_{2i} ln(1 − γ^{2i})) ≤ \frac{E(γ)}{(1 − η)} < ∞ .

The numerical series with non-negative coefficients L(γ) converges, so the product in 7.1.1 converges at t = γ, as desired. □

8.2. Residue domains. In this section (R, 𝔪, k) is a local ring.

The results that follow are from Avramov [23].

Theorem 8.2.1. If 𝔭 is a prime ideal of R such that R_{𝔭} is not a complete intersection, then there is a real number β > 1 with the property that

β_{n}^{R}(R/𝔭) ≥ β^{n} for n ≥ 0 .

The converse may fail: the ring R_{𝔭} in Example 5.2.6 is a complete intersection, but curv_{R}(R/𝔭) > 1 by Theorem 5.3.3.2. Finite modules over a complete intersection have curvature ≤ 1 by Proposition 4.2.4 and Remark 8.1.1.2, so Corollary 8.2.2. The following conditions are equivalent:

(i) R is a complete intersection.

(ii) curv_{R} M ≤ 1 for each finite R–module M .

(iii) curv_{R} k ≤ 1 . □

The key to the proof of the theorem is to look at deviations.

Theorem 8.2.3. When R is not a complete intersection there exist a sequence of integers 0 < s_{1} < ⋯ < s_{j} < … and a real number γ > 1, such that

ε_{s_{j}}(R) ≥ γ^{s_{j}} for j ≥ 1

and s_{j+1} = i_{j}(s_{j} − 1) + 2 with integers 2 ≤ i_{j} ≤ edim R + 1.

Remark. The last result and its proof are ‘looking glass images’—in the sense of [49], [33]—of a theorem of Félix, Halperin, and Thomas [66] on the rational homotopy groups π_{n}(X) ⊗_{ℤ} ℚ of a finite CW complex X; it relies heavily on Félix and Halperin’s [64] theory of rational Ljusternik-Schnirelmann category.

That theorem was used by Félix and Thomas [67] to prove Corollary 8.2.2 for graded rings over fields of characteristic 0, but the L.-S. category arguments do not extend to local rings or to positive characteristic. This is typical of a larger picture: a theorem in rational homotopy or local algebra raises a conjecture in the other field, but a proof usually requires new tools.

The arguments below use the properties of minimal models already established in Section 7.2, and the additional information contained in the next lemma, proved at the end of the section.

Lemma 8.2.4. Let k[Y] be a minimal DG algebra, such that H_{n}(k[Y]) = 0 for n ≥ m. If ϕ: k[Y] ⟶ k[U] is a surjective morphism of DG algebras, such that U is a set of exterior variables and ∂(U) = 0, then card(U) < m.

As the proof of the theorem for rational homotopy groups, the one of the theorem on deviations proceeds in three steps. The lemma is needed for the first claim, which (now) can be obtained directly from Theorem 7.3.4, or from its precursor in [34]: If R is not a complete intersection, then ε_{n}(R) ≠ 0 for n ≫ 0. We present the original argument in order to keep the notes self-contained, and because of its intrinsic interest. The exposition of the arguments for Claims 2 and 3 follows [30].

Proof of Theorem 8.2.3. As may assume that R is complete, we take a minimal regular presentation R ≅ Q/I and a minimal model Q[Y] of R over Q.

Note that the DG algebra k[Y] = Q[Y]/𝔫Q[Y] is minimal by Remark 7.2.1, that card Y_{n} = ε_{n+1}(R) for n ≥ 1 by Theorem 7.2.6, and that

H_{n}(k[Y]) ≅ Tor_{n}^{Q} (k, R) = 0  for n ≥ m = edim R + 1 .  (∗)

Assuming that R is not a complete intersection, we show that the numbers

a(n) = card Y_{n}  and  s(n) = ∑_{j=n}^{2n} a(j)

satisfy the following list of increasingly stronger properties:

Claim 1. The sequence s(n) is unbounded.

Claim 2. The sequence a(n) is unbounded.

Claim 3. There exist positive integers r_{1}, r_{2}, … with r_{j+1} = i_{j}r_{j} + 1 and 2 ≤ i_{j} ≤ m, and a real number υ > 1, such that a(n) ≥ υ^{r_{j}} for each j ≥ 0 .

The last claim yields the theorem: with γ =\sqrt{υ} and s_{j} = r_{j} + 1 we have

ε_{s_{j}}(R) = a(r_{j}) > υ^{r_{j}} = γ^{2r_{j}} ≥ γ^{r_{j}+1} = γ^{s_{j}} for j ≥ 1 .

For the rest of the proof, we write Y_{[n]} for the span of ⋃_{j=n}^{2n} Y_{j}, and abuse notation by letting Y_{n} stand also for the k–linear span of the variables y ∈ Y_{n}; thus, Y_{[n]}^{i} is the k–linear span of all products involving i elements of Y_{[n]}.

Proof of Claim 1. Assume that there is a c ∈ ℕ such that s(n) ≤ c for all n ≥ 1.

We are going to construct for all r ≥ 1 and h ≥ 0 surjective morphisms of DG algebras ϕ_{h}^{r}: k[Y] ⟶ k[U_{h}^{r}], where each U_{h}^{r} is a set {u_{hr+1}^{r}, …, u_{hr+r}^{r}} of exterior variables subject to the restrictions

|u_{n+1}^{r}| > |u_{n}^{r}| + 1 for n ≥ 1 ;
|u_{hr+i}^{r}| > |u_{hr+1}^{r}| + ⋯ + |u_{hr+i−1}^{r}| + 1 for i = 2, …, r .

The second condition forces ∂(U_{h}^{r}) = 0, so in view of (∗) Lemma 8.2.4 implies that r < m. This contradiction establishes the unboundedness of s(n).

By Theorems 7.2.6 and 7.3.3, there is an infinite sequence y_{1}, y_{2}, ⋯ ∈ Y_{odd} with |y_{h+1}| > |y_{h}| + 1. Setting n_{h} = |y_{h}|, note that the compositions k[Y] ⟶ k[Y_{⩾ n_{h}}] ⟶ k[Y_{⩾ n_{h}}]/(Y_{⩾ n_{h}} \ {y_{h}}) have the desired properties for r = 1.

Assume by induction that morphisms ϕ_{h}^{r} have been constructed for some r ≥ 1. We fix n ≥ 0, simplify the notation by setting u_{ij} = u_{(n+i)r+j}^{r}, U_{i} = U_{n+i}^{r}, and ϕ_{i} = ϕ_{n+i}^{r}, for i = 0, …, c and j = 1, …, r, and embark on an auxilliary construction. Choose an index q > |u_{c1}| + ⋯ + |u_{cr}| + 1 such that Y_{q} ≠ ∅, and pick y ∈ Y_{q}. For i = 0, …, c the intervals

I_{i} = [(q + |u_{i1}| + 1) , (q + |u_{i1}| + ⋯ + |u_{ir}| + 1)]

are disjoint and contained in the interval [q, 2q].

Since s(q) ≤ c, we can choose an index i such that Y_{s} = ∅ for all s ∈ I_{i}. The restriction of ϕ_{i} to B = k[Y_{< q}] yields a surjective morphism of DG algebras B ⟶ k[U_{i}]. Tensoring it with k[Y] over B, we get a surjective morphism k[Y] ⟶ C = k[U_{i}] ⊗_{B} k[Y]. Note that C^{♮} = k[U_{i}] ⊗_{k} k[Y_{⩾ q}]^{♮} is equal to k[U_{i}] in degrees ≤ q−1, to kY_{q} in degree q, and has no algebra generators in degrees from I_{i}. As the differential ∂^{C} is decomposable, the ideal of C generated by the variables Y_{⩾ q} ∖{y} is closed under the differential of C.

We have now constructed a surjective morphism of DG algebras

k[Y] ⟶ C ⟶ C/(Y_{⩾ q} ∖ {y})C = k[u_{1} …, u_{r+1}]

where u_{j} = u_{ij} for j = 1, …, r and u_{r+1} is the image of y; clearly, the condition |u_{j}| > |u_{1}| + ⋯ + |u_{j−1}| + 1 holds for j = 2, …, r + 1. To end the auxilliary construction, choose an integer n′ such that rn′ > (n + c + 1)r. Setting n_{1} = 1 and n_{h} = n′_{h−1} for h ≥ 2, and applying the construction to n_{h} for h = 1, 2, …, we get a sequence of surjective morphisms ϕ_{h+1}^{r+1} with the desired properties.

Proof of Claim 2. We assume that there exists a number c such that rank_{k} Y_{n} ≤ c for all n, and work out a contradiction.

Fix for the moment an integer n ≥ 1. For every y ∈ Y_{⩾ n} there are uniquely defined α_{i}(y) ∈ Y_{[n]}^{i} ⊆ k[Y_{⩾ n}], such that

∂(y) ≡ ∑_{i ⩾ 2} α_{i}(y) mod ((Y_{> 2n})k[Y_{⩾ n}]) .

Clearly, the maps α_{i}: Y_{⩾ n} ⟶ Y_{[n]}^{i}, where y ↦ α_{i}(y), are k-linear. The minimality of k[Y] is inherited by k[Y_{⩾ n}], so there we have ∂(Y_{[n]}^{i}) = 0 for all i. Recalling from Corollary 7.2.7 that

( H_{⩾ 1}(k[Y_{⩾ n}]))^{m} = 0  for every n ≥ 1  (†)

we see that Y_{[n]}^{m} consists of boundaries, so we get an inclusion

∑_{i=2}^{m} Y_{[n]}^{m−i}α_{i}(Y_{⩾ n}) ⊇ Y_{[n]}^{m} .  (‡)

For degree reasons, α_{i}(Y_{j}) = 0 when j < in + 1 or j > i(2n) + 1, so

s(in + 1) = ∑_{j=in+1}^{2in+2} rank_{k} Y_{j} ≥ ∑_{j=in+1}^{i(2n)+1} rank_{k} α_{i}(Y_{j}) = rank_{k} α_{i}(Y_{⩾ n}) .  (§)

Set d = (2m)^{m} and choose by Claim 1 an integer n_{0} such that s(n_{0}) > (md)^{2}. Assume by induction on j that we have integers n_{0}, n_{1}, …, n_{j} such that

m(n_{h−1} + 1) ≥ n_{h} + 1 and s(n_{h}) ≥ (md)s(n_{h−1}) for 1 ≤ h ≤ j .  (¶)

Choose n_{j+1} = ln_{j} + 1 such that s(n_{j+1}) = max{s(in_{j} + 1) | 2 ≤ i ≤ m}. It is then clear that m(n_{j} + 1) ≥ n_{j+1} + 1. Using (§) and (‡), we get

(m − 1)s(n_{j})^{m−2}s(n_{j+1}) ≥ ∑_{i=2}^{m} s(n_{j})^{m−i}s(in_{j} + 1)

≥ ∑_{i=2}^{m} ( rank_{k} Y_{[n_{j}]})^{m−i} rank_{k} α_{i}(Y_{⩾ n_{j}})

≥ rank_{k} ( ∑_{i=2}^{m} Y_{[n_{j}]}^{m−i}α_{i}(Y_{⩾ n_{j}})) ≥ rank_{k} Y_{[n_{j}]}^{m}

≥ \binom{s(n_{j})}{m} ≥ \frac{s(n_{j})^{m}}{(2m)^{m}} = \frac{s(n_{j})}{d}s(n_{j})^{m−1}

≥ (m^{2}d)s(n_{j})^{m−1}

so s(n_{j+1}) ≥ (md)s(n_{j}), completing the induction step. Clearly, (¶) implies that m^{j}(n_{0} +1) ≥ n_{j} +1 and s(n_{j}) ≥ m^{j}s(n_{0})d^{j} hold for j ≥ 1. Thus, we get c(n_{j} +1) ≥ s(n_{j}), and hence c(n_{0} + 1) ≥ s(n_{0})d^{j} for all j. This is absurd.

Proof of Claim 3. Set b = (2m)^{m+1}, choose r_{1} so that a(r_{1}) = a > b, and assume by induction that r_{1}, …, r_{j} have been found with the property that

r_{h} = i_{h−1}r_{h−1} + 1 with 2 ≤ i_{h−1} ≤ m  and  a(r_{h}) ≥ \frac{a(r_{h−1})^{i_{h−1}}}{b}

for 1 ≤ h ≤ j. The condition β(y) ≡ ∂(y) mod ((Y_{> r_{j}})k[Y_{⩾ r_{j}}]) defines a k– linear homomorphism β: Y_{⩾ r_{j}} ⟶ ∑_{i ⩾ 2} Y_{r_{j}}^{i}. Noting that β(y) = 0 unless |y| ≡ 1 (mod r_{j}), and using the fact that k[Y] is minimal and satisfies condition (†), we obtain ∑_{i=2}^{m} Y_{r_{j}}^{m−i}β(Y_{ir_{j}+1}) ⊇ Y_{r_{j}}^{m} as in the the proof of the preceding claim. It follows that

∑_{i=2}^{m} a(r_{j})^{m−i}a(ir_{j} + 1) ≥ \binom{a(r_{j})}{m} ≥ \frac{a(r_{j})^{m}}{(2m)^{m}} = (2m)\frac{a(r_{j})^{m}}{b} .

The assumtion that a(ir_{j} + 1) < a(r_{j})^{i}/b for 2 ≤ i ≤ m, leads to the impossible inequality (m − 1)a(r_{j})^{m} > (2m)a(r_{j})^{m}. Thus, a(ir_{j} + 1) ≥ (r_{j})^{i}/b for some i = i_{j}, so the induction step is complete with r_{j+1} = i_{j}r_{j} + 1.

The quantities P_{j} = (i_{j} ⋯ i_{1}) and S_{j} = ∑_{h=2}^{j}(i_{j} ⋯ i_{j−h}) satisfy

P_{j} > P_{j}(\frac{1}{2} + ⋯ + \frac{1}{2^{j}} ) ≥ P_{j}(\frac{1}{i_{1}} + ⋯ + \frac{1}{i_{1} ⋯ i_{j}} ) = S_{j} .

Thus, P_{j}r_{1} > S_{j}r_{1} + 1 = r_{j+1}, and hence P_{j} > r_{j+1}/r_{1}. Since a > b, we have

a(r_{j+1}) ≥ \frac{a(r_{j})^{i_{j}}}{b} ≥ \frac{a(r_{j−1})^{(i_{j}i_{j−1})}}{b^{1+i_{j}}} ≥ ⋯ ≥ \frac{a^{P_{j}}}{b^{S_{j}}} > (\frac{a}{b})^{P_{j}} > (\frac{a}{b})^{\frac{r_{j+1}}{r_{1}}} .

To finish the proof of the claim, note that υ > 1 and set υ = \sqrt[r_{1}]{a/b}. □

Proof of Theorem 8.2.1. Since β_{n}^{R}(R/𝔭) ≥ β_{n}^{R_{𝔭}}(R_{𝔭}/𝔭R_{𝔭}) for each n, we may assume that 𝔭 = 𝔪; set β_{n} = β_{n}^{R}(R/𝔪) and e = edim R.

As R is not a complete intersection, Theorem 8.2.3 provides an infinite sequence s_{1}, s_{2}, … with (e+1)s_{j} > s_{j+1}, such that ε_{s_{j}}(R) ≥ γ^{s_{j}} for some real number γ > 1. For n ≥ 2 we have β_{n} > β_{1} = e > 1 by Remark 8.1.1.3, hence

β = min \{ \sqrt[e+1]{γ} , β_{1} , …, \sqrt[s_{1}]{β}_{s_{1}}\} > 1

and β_{n} ≥ β^{n} for s_{1} ≥ n ≥ 0. If s_{j+1} ≥ n > s_{j} with j ≥ 1, then

β_{n} > β_{s_{j}} ≥ ε_{s_{j}}(R) ≥ γ^{s_{j}} ≥ β^{(e+1)s_{j}} > β^{s_{j+1}} > β^{n}

so the desired inequality β_{n} ≥ β^{n} holds for all n ≥ 0. □

Proof of Lemma 8.2.4. Since ϕ^{♮}: k[Y]^{♮} ⟶ k[U] is a surjective homomorphism of graded free k–algebras, the ideal Ker ϕ^{♮} has a linearly independent generating set Y′ = {y′_{1}, …, y′_{j}, … } ⊂ kY, which we can assume ordered in such a way that |y′_{j+1}| ≥ |y′_{j}| for j ≥ 1. As Ker ϕ is a DG ideal, we have ∂(y′_{1}) = 0 and ∂(y′_{j+1}) ∈ (y′_{1}, …, y′_{j}) for j ≥ 1. Assume that for some j ≥ 1 the morphism ϕ factors through a quasi-isomorphism

π^{j}: A^{j} = k[Y]⟨x′_{1}, …, x′_{j}⟩ ⟶^{π^{j}} k[Y]/(y′_{1}, …, y′_{j}) = B^{j}

that maps x′_{i}^{(n)} to zero for 1 ≤ i ≤ j and n ≥ 1. As the \overline{y}′_{j+1} = π^{j}(y′_{j+1}) is a cycle in B^{j}, there is a cycle z_{j+1} ∈ A^{j} such that π^{j}(z_{j+1}) = π^{j}(y′_{j+1}). By Lemma 7.2.10, π^{j} extends to a quasi-isomorphism

π^{j}⟨x′_{j+1}⟩: A^{j+1} = A^{j}⟨x′_{j+1} | ∂(x′_{j+1}) = z_{j+1}⟩ ⟶ B^{j}⟨x′_{j+1} | ∂(x′_{j+1}) = \overline{y}′_{j+1}⟩ .

Lemma 7.2.11 shows that the map ξ^{j}: B^{j}⟨x′_{j+1}⟩ ⟶ B^{j}/(y′_{j+1}) that sends ∑_{i} b_{i}x′_{j+1}^{(i)} to b_{0} + (y′_{j+1}) is a quasi-isomorphism; thus, so is

π^{j+1} = ξ^{j}∘π^{j}⟨x′_{j+1}⟩: A^{j+1} = B^{j}⟨x′_{j+1}⟩ ⟶ B^{j}/(y′_{j+1}) = B^{j+1} .

In the limit, we obtain a factorization of ϕ in the form

k[Y] ↪ k[Y]⟨X′⟩ ⟶^{π} k[U]

with a surjective quasi-isomorphism π, such that Ker π is generated by Ker ϕ and x′_{j}^{(i)} with i, j ≥ 1. By Lemma 7.2.10, π extends to a quasi-isomorphism

π⟨X′′⟩: k[Y]⟨X′⟩⟨X′′⟩ ⟶ k[U]⟨X′′ | ∂(x′′_{j}) = u_{j}⟩ .

where X′′ = {x′′_{1}, …, x′′_{m}}. The DG algebra on the right is quasi-isomorphic to k, so we get a semi-free resolution W′ = k[Y]⟨X′ ∪X′′⟩ of k over k[Y]. Another semi free resolution W = k[Y]⟨X⟩ of k over k[Y], such that X = {x_{y} : |x_{y}| = |y|+1 , y ∈ Y }, and ∂(k[Y]⟨X⟩) ⊆ (Y)W, is given by Proposition 7.2.9 (applied with Q = k). By Propositions 1.3.1 and 1.3.2, the vector spaces V′ = k ⊗_{k[Y]} W′ = k⟨X′ ∪ X′′⟩ and V = k ⊗_{k[Y]} W = k⟨X⟩ are quasi-isomorphic. As ∂^{V} = 0, we get (in)equalities of formal power series

\frac{∏_{i=1}^{∞}(1 + t^{2i−1})^{card(X_{2i−1})}}{∏_{i=1}^{∞}(1 − t^{2i})^{card(X_{2i})}} = ∑_{n} rank_{k} V_{n}t^{n} = ∑_{n} rank_{k} H_{n}(V)t^{n}

= ∑_{n} rank_{k} H_{n}(V′)t^{n} ≼ ∑_{n} rank_{k} V′_{n}t^{n}

= \frac{∏_{i=1}^{∞}(1 + t^{2i−1})^{card(X′_{2i−1})}}{∏_{i=1}^{∞}(1 − t^{2i})^{card(X′_{2i})}} · \frac{∏_{i=1}^{∞}(1 + t^{2i−1})^{card(X′′_{2i−1})}}{∏_{i=1}^{∞}(1 − t^{2i})^{card(X′′_{2i})}} .

On the other hand, by construction we have for each j an equality

card X_{j+1} = card Y_{j} = card Y′_{j} + card U_{j} = card X′_{j+1} + card X′′_{j+1} .

It follows that H(V′) = V′, that is, that ∂(W′) ⊆ (Y)W′.

As W′^{♮} is a free module over k[Y]⟨X′⟩^{♮}, and H_{⩾ 1}(W) = 0, we see that

Z_{⩾ 1}(k[Y]⟨X′⟩) = Z_{⩾ 1}(W′) ∩ (k[Y]⟨X′⟩) = ∂(W′) ∩ (k[Y]⟨X′⟩)
⊆ (Y)W′ ∩ k[Y]⟨X′⟩ = (Y)k[Y]⟨X′⟩ .

Since π: k[Y]⟨X′⟩ ⟶ k[U] is a surjective quasi-isomorphism, we can find z_{1}, …, z_{m} ∈ Z(k[Y]⟨X′⟩) with π(z_{i}) = u_{i}. For them we have

z_{1} ⋯ z_{m} ∈ (Z_{⩾ 1}(k[Y]⟨X′⟩))^{m} ⊆ Z((Y)^{m}k[Y]⟨X′⟩ ) ⊆ Z(Jk[Y]⟨X′⟩)

where J ⊂ k[Y] is defined by

J_{n} = \begin{cases}0 & for n < m; \\ ∂(k[Y]_{m+1}) & for n = m; \\ k[Y]_{n} & for n > m . \end{cases}

For degree reasons, J is a DG ideal of k[Y]. By hypothesis H_{n}(k[Y]) = 0 for n ≥ m, so H(J) ≅ H(k[Y]) = 0. Thus, the projection τ: k[Y] ⟶ k[Y]/J is a quasiisomorphism; Proposition 1.3.2 then shows that the induced map k[Y]⟨X′⟩ ⟶ k[Y]⟨X′⟩ /Jk[Y]⟨X′⟩ is one, hence

Z(Jk[Y]⟨X′⟩) = ∂(Jk[Y]⟨X′⟩) ⊆ ∂(k[Y]⟨X′⟩)

hence cls(z) = 0. The computation 0 = H(π)(cls(z_{1}) ⋯ cls(z_{m})) = u_{1} … u_{m} ≠ 0 now yields the desired contradiction. □

8.3. Conormal modules. In this section we fix a presentation R = Q/I, where (Q, 𝔫, k) is a local or graded ring, and I ⊆ 𝔫^{2} is minimally generated by f. The R–module I/I^{2} is called the conormal module of the presentation^{30}.

If f is a regular sequence, then it is well known and easy to see that the image of f modulo I^{2} is a basis of the conormal module, and the projective dimension pd_{Q} R is finite. The starting point of the present discussion is a well known converse, due to Ferrand [68] and Vasconcelos [154]: Theorem 8.3.1. If pd_{Q} R < ∞ and the R–module I/I^{2} is free, then f is a regular sequence. ■

Later, Vasconcelos [155] conjectured a considerably stronger statement: If pd_{Q} R < ∞ and pd_{R}(I/I^{2}) < ∞, then f is a regular sequence. Various known cases of small projective dimension are surveyed in [156]; the one below is proved by Vasconcelos and Gulliksen.

Theorem 8.3.2. The conjecture holds if pd_{R}(I/I^{2}) ≤ 1.

Proof. In view of the preceding theorem, it suffices to assume that pd_{R}(I/I^{2}) = 1, and draw a contradiction.

For the Koszul complex E = Q⟨X_{1} | ∂(X_{1}) = f⟩, set Z = Z_{1}(E) and H = H_{1}(E). Tensoring the exact sequence 0 ⟶ Z ⟶ Q^{r} ⟶ I ⟶ 0 with R over Q, we get an exact sequence of R–modules Z/IZ ⟶ R^{r} ⟶ I/I^{2} ⟶ 0. As ∂(E_{2}) ⊆ IE_{1}, we have an induced exact sequence H ⟶ R^{r} ⟶ I/I^{2} ⟶ 0. The assumption pd_{R}(I/I^{2}) = 1 then implies that H contains a free direct summand R cls(z) ≅ R; note that z ∈ 𝔫E_{1}, because f minimally generates I.

Let E⟨X_{⩾ 2}⟩ = Q⟨X⟩ be an acyclic closure of R = H_{0}(E) over E, such that ∂(x) = z for some x ∈ X_{2}. The cokernel of the differential δ_{2} : RX_{2} ⟶ RX_{1} of the complex of indecomposables Ind_{Q}^{γ} Q⟨X⟩ is equal to H, so Proposition 6.2.7 yields a Q–linear Γ-derivation ϑ: Q⟨X⟩ ⟶ Q⟨X⟩ of degree −2, with ϑ(x) = 1.

The Γ-derivation θ = H(ϑ ⊗_{Q} k) of Q⟨X⟩ ⊗_{Q}k = k⟨X⟩ has θ(x) = 1. As ∂(x) = z ⊗ 1 = 0 ∈ E_{1}/𝔫E_{1}, each x^{(i)} is a cycle. Assuming that x^{(i)} = ∂(v), we get 1 = θ^{i}(x^{(i)}) = θ^{i}∂(v) = 0, which is absurd. Thus, 0 ≠ H_{2i}(k⟨X⟩) = Tor_{2i}^{Q} (R, k) for all i ≥ 0, contradicting the hypothesis that pd_{Q} R is finite. □

Next we present the results of Avramov and Herzog [35] on graded ring.

Theorem 8.3.3. Let Q = k[s_{1}, …, s_{e}] be a graded polynomial ring over a field k of characteristic 0, with variables of positive degree, let I be a homogeneous ideal of Q, and set R = Q/I. The following conditions are equivalent.

(i) R is a complete intersection.

(ii) pd_{R}(I/I^{2}) < ∞.

(iii) cx_{R}(I/I^{2}) < ∞.

(iv) curv_{R}(I/I^{2}) ≤ 1 . If R is not a complete intersection, then curv_{R} I/I^{2} = curv_{R} k.

30Or: of the embedding Spec(R) ⊆ Spec(Q).

The result is proved together with the next one:

Theorem 8.3.4. If R is as in the preceding theorem, and Ω_{R|k} is its module of Kähler differentials over k, then the following conditions are equivalent.

(i) R is a complete intersection.

(ii) cx_{R}(Ω_{R|k}) < ∞.

(iii) curv_{R}(Ω_{R|k}) ≤ 1 . If R is not a complete intersection, then curv_{R} Ω_{R|k} = curv_{R} k.

Remark. If pd_{R} Ω_{R|k} < ∞, then the theorem implies that R is a complete intersection—another conjecture of Vasconcelos—but there is more.

If 𝔭 is a minimal prime ideal, then Ω_{R_{𝔭}|k} ≅ (Ω_{R|k})_{𝔭} has finite projective dimension over R_{𝔭}. Thus, it is free, hence R_{𝔭} is regular by the Jacobian criterion, and so R is reduced by Serre’s criterion. Conversely, if R is a reduced complete intersection, then Ferrand [68] and Vasconcelos [154] prove that pd_{R} Ω_{R|k} ≤ 1.

The asymptotic results are easy consequences of more precise inequalities^{31} for the garded invariants described in Remark 1.2.10.

Theorem 8.3.5. In the notation of Theorem 8.3.3, for all n ≥ 0 and j ∈ ℤ there is an inequality between graded Betti numbers and deviations:

β_{nj}^{R}(Ω_{R|k}) ≥ ε_{n+1,j}(R)  and  β_{nj}^{R}(I/I^{2}) ≥ ε_{n+2,j}(R) .

Our proof proceeds through a structural result on the resolution of I/I^{2}, that depends on the grading and on the characteristic; the following is open:

Problem 8.3.6. When R is a local ring and R ≅ Q/I is a regular presentation, does an inequality β_{n}^{R}(I/I^{2}) ≥ ε_{n+2}(R) hold for each n ≥ 0 ?

In the arguments, we use graded versions of some basic constructions.

Remark 8.3.7. The first step in the construction of a minimal model of R over Q is a Koszul complex on the set f of minimal generators of I; we choose f to consist of homogeneous elements, so the first Koszul homology is a finite graded Q–module. Assume by induction that H_{n}(Q[Y_{⩽ n}]) has the same property for some n ≥ 1; to kill it we adjoin a minimal set of homogeneous generators, and assign to each variable y ∈ Y_{n+1} an internal degree, equal to that of ∂(y).

Thus, we get a graded minimal model Q[Y_{⩾ 1}] = k[Y] of R over Q. Similar considerations yield a graded acyclic closure R⟨X⟩ of k over R. The arguments in Sections 6.3 and 7.2 are compatible with the internal gradings, so the ‘obvious’ graded versions of the results proved there are available.

Remark 8.3.8. Proposition 6.2.3 can be repeated for ordinary (that is, not subject to a condition involving divided powers) k–linear derivations of the DG algebra k[Y] over k, to produce a DG module of differentials Diff_{k} k[Y] over k[Y]. It is semi-free with basis {dy : |dy| = |y|; deg(dy) = deg(y)}_{y∈Y}, where deg(a) is the internal degree of a; the map y ↦ dy extends to a universal derivation d: k[Y] ⟶ Diff_{k} k[Y]; the differential is determined by ∂(dy) = d(∂(y)); each k–linear derivation of k[Y] into a DG module U over k[Y] factors uniquely as the composition of d with a homomorphism of DG modules Diff_{k} k[Y] ⟶ U.

31Equalities hold for n = 0 by Corollary 7.1.5, but it appears that the other inequalities are strict unless R is a (reduced) complete intersection.

Consider the complex of free R–modules L = R ⊗_{k[Y]} Diff_{k} k[Y]. (Using Lemma 7.2.3 on the uniqueness of minimal models and a functorial construction of Diff_{k} k[Y], it can be shown that this complex is defined uniquely up to isomorphism by the k–algebra R; we do not use that here, and refer to [35] for details.) For g ∈ QY_{1}, an easy computation shows that the differential

∂_{1} : L_{1} ⟶ L_{0} acts by ∂_{1}(1 ⊗ g) = 1 ⊗ ∑_{i=1}^{e} \frac{∂f}{∂y_{i}} dy_{i} where ∂_{1}(g) = f ∈ Q .

On the other hand, the ‘second fundamental exact sequence’ for the module Ω_{R|k} of Kähler differentials of the k–algebra R has the form

I/I^{2} ⟶^{δ} R ⊗_{Q} Ω_{Q|k} ⟶ Ω_{R|k} ⟶ 0 with δ(f + I^{2}) = 1 ⊗ ∑_{i=1}^{e} \frac{∂f}{∂y_{i}} dy_{i} .

As Ω_{Q|k} is free with basis {dy_{1}, …, dy_{e}}, we conclude that H_{0}(L) = Ω_{R|k}.

Recall from Remark 4.1.7, that an augmentation ϵ: F ⟶ N of a complex of free R–modules F is essential, if for some lifting α: F ⟶ G to a minimal resolution G of N, the map k ⊗_{R} α is injective. In that case, α maps F isomorphically onto a subcomplex of G, that splits off as a graded R–module.

Theorem 8.3.9. The augmentation ϵ^{L}: L ⟶ H_{0}(L) = Ω_{R|k} is essential.

A special morphism is at the heart of the arguments to follow.

Construction 8.3.10. Euler morphisms. The graded algebra R has an Euler derivation R ⟶ 𝔪, that multiplies each homogeneous element a ∈ R by its (internal) degree. By Proposition 1.3.1, the R–linear map γ: Ω_{R|k} ⟶ 𝔪 that it defines lifts to a morphism ω: Diff_{k} k[Y] ⟶ V of DG modules over k[Y], where ϵ^{V} : V ⟶ 𝔪 is a semi-free resolution of 𝔪 over k[Y]. We call such a lifting an Euler morphism; it is unique up to k[Y]–linear homotopy.

Lemma 8.3.11. Let k[Y] be a graded minimal model of R over k, and let U = k[Y]⟨X⟩ be a graded acyclic closure of k over k[Y], as in Remark 8.3.7.

The DG module V = Σ^{−1}(U/k[Y]) is a semi-free resolution of 𝔪 over k[Y], and there is an Euler morphism ω: Diff_{k} k[Y] ⟶ V, such that

ω(dy) ≡ − deg(y)x_{y} mod 𝔫X_{n+1} + (k[Y_{⩽ n+1}]⟨X_{⩽ n}⟩])_{n+1} for y ∈ Y_{n} .

Proof. Set D^{n} = ∐_{|y| ⩽ n} k[Y]dy ⊆ Diff_{k} k[Y] = D.

The map a ↦ deg(a)a is a k–linear chain Γ-derivation k[Y] ⟶ U. In degree zero homology it induces the zero map R ⟶ k, so it is homotopic to 0. If ξ: D ⟶ U is the k[Y]–linear morphism that corresponds to it by Proposition 6.2.3, then ξ is homotopic to 0. We set ξ^{n} = ξ|_{D^{n}} and by induction on n construct k[Y]–linear homotopies σ^{n}: D^{n} ⟶ U between ξ^{n} and 0, such that

\begin{matrix}σ^{n}|_{D^{n−1}} = σ^{n−1} ; \\ σ^{n}(dy) ≡ deg(y)x_{y} mod 𝔫X_{n} + (k[Y_{⩽ n}]⟨X_{< n}⟩])_{n} . \end{matrix}  (∗)

If |y| = 0, then set σ^{0}(dy) = deg(y)x_{y}: clearly, the formula above holds. Let n ≥ 1, and assume by induction that σ^{n−1} has been found. It is easy to check that ξ(dy) − deg(y)∂(x_{y}) − σ^{n−1}∂(dy) is a cycle; as n ≥ 1, it is a boundary, that we write as ∂(u_{y} + v_{y}) with u_{y} ∈ QX_{n+1}, and v_{y} ∈ k[Y_{⩽ n+1}]⟨X_{⩽ n}⟩_{n+1}. Because d is a derivation and ∂(Y) ⊆ (Y)^{2}k[Y], we get

d(∂Y) ⊆ d((Y)^{2}k[Y]) ⊆ (Y)d(k[Y]) = (Y)D

Since σ^{n−1} is k[Y]–linear, this implies:

σ^{n−1}∂(dy) = σ^{n−1}d(∂(y)) ∈ W_{n} = 𝔫Y_{n} + k[Y_{< n}]⟨X_{⩽ n}⟩_{n} .

By Proposition 7.2.9, we have ∂(v_{y}) ∈ W_{n}, hence

∂(u_{y}) = ξ(dy) − deg(y)∂(x_{y}) − σ^{n−1}∂(dy) − ∂(v_{y}) ∈ W_{n} .

By the same theorem, we conclude that u_{y} ∈ 𝔫X_{n+1} . The map σ^{n}: D^{n} ⟶ U, σ^{n}(dy) = deg(y)x_{y} + u_{y} + v_{y}, defines a homomorphism of DG modules over k[Y] that satisfies (∗). As for |y| ≤ n we have

∂σ^{n}(dy) + σ^{n}∂(dy) = deg(y)∂(x_{y}) + ∂(u_{y} + v_{y}) + σ^{n−1}∂(dy) = ξ(dy) ,

the induction step of the construction is complete.

In the limit, the maps σ^{n} define a homotopy σ: D ⟶ U between ξ and 0. Let ω: D ⟶ V be the composition of σ with the canonical k[Y]–linear, degree −1 homomorphism U ⟶ U/k[Y] ⟶ Σ^{−1}(U/k[Y]) = V. As Im ξ ⊆ k[Y], the equality ∂σ + σ∂ = ξ implies ∂ω = ω∂, so ω is a chain map D ⟶ V.

The homology exact sequence of 0 ⟶ k[Y] ⟶ U ⟶ U/k[Y] ⟶ 0 yields H_{n}(U/k[Y]) = 0 for n ≠ 1 and H_{1}(V) = 𝔪, so V = Σ^{−1}(U/k[Y]) is a semi-free resolution of 𝔪. For n = 0, formula (∗) shows that H_{0}(ω): Ω_{R|k} ⟶ 𝔪 is the homomorphism induced by the Euler derivation; for n ≥ 1, the formula yields a congruence ω(dy) ≡ − deg(y)x_{y} mod 𝔫X_{n+1} + (k[Y_{⩽ n+1}]⟨X_{⩽ n}⟩])_{n+1}. □

Proof of Theorem 8.3.9. Let ω: Diff_{k} k[Y] ⟶ V be the Euler morphism, constructed in the preceding lemma, and consider the induced morphism

ϖ: L = R ⊗_{k[Y]} Diff_{k} k[Y] ⟶^{R⊗ω} R ⊗_{k[Y]} V = G

of complexes of graded R–modules. The lemma yields congruences

(k ⊗_{R} ϖ)(1 ⊗ dy) ≡ 1 ⊗ deg(y)x_{y} mod (k⟨X_{⩽ n}⟩_{n+1}) for y ∈ Y_{n} and n ≥ 0 ,

which show^{32} that k ⊗_{R} ϖ is injective. Furthermore, H_{0}(ϖ) is the homomorphism γ: Ω_{R|k} ⟶ 𝔪 defined by the Euler derivation.

By Proposition 1.3.2, the quasi-isomorphism ρ: k[Y] ⟶ R induces a quasiisomorphism ρ ⊗ V : V = k[Y] ⊗_{k[Y]} V ⟶ R ⊗_{k[Y]} V = G, so G is a minimal free resolution of 𝔪 over R. Let F be a minimal free resolution of Ω_{R|k} over R, let α: L ⟶ F be a lifting of the identity map of Ω_{R|k}, and let β: F ⟶ G be a lifting of γ. Since H_{0}(βα) = γ, the morphisms ϖ and βα are homotopic. As noted in Remark 4.1.7, this yields

k ⊗_{R} ϖ = k ⊗_{R} (βα) = (k ⊗_{R} β)(k ⊗_{R} α) ,

so k ⊗_{R} α is injective. This is the desired assertion. □

Proof of Theorem 8.3.5. By construction, L_{n} is a free R–module with basis Y_{n}, and card(Y_{n}) = ε_{n+1}(R) by Theorem 7.2.6. The inequalities for the Betti numbers of Ω_{R|k} follow from the result that we have just proved.

The morphism ϖ used in its proof induces a morphism ϖ′: L′ = Σ^{−1}L_{⩾ 1} ⟶ Σ^{−1}G_{⩾ 1} = G′, such that ϖ′^{♮} is a split injection of R–modules. An easy computation shows that H_{0}(L′) = I/I^{2}, so replacing in the preceding argument F by a minimal resolution of I/I^{2}, we conclude that ϵ^{L′}: L′ ⟶ I/I^{2} is essential. That gives the second series of inequalities. □

32This is the only place where the hypothesis of characteristic 0 is used.

Proof of Theorem 8.3.3 and Theorem 8.3.4. In view of Corollary 8.2.2, in each case it suffices to prove the last assertion. Using Proposition 4.2.4.1, Theorem 8.3.5, and Proposition 8.1.4, we get

curv_{R} k ≥ curv_{R}(I/I^{2}) = lim sup \sqrt[n]{β_{n}^{R}(I/I^{2})} ≥ lim sup \sqrt[n]{ε_{n}(R)} = curv_{R} k .

We have Theorem 8.3.3. An identical argument yields Theorem 8.3.4. □

9. Modules over complete intersections

Currently, homological algebra over complete intersections is an active area of research on infinite free resolutions. This chapter describes some basic techniques and results. Most proofs depend on a remarkable higher level structure on resolutions, introduced in Section 1 under more general hypotheses. It is then applied to modules over complete intersections, to study Betti numbers in Section 2, and other homological problems in Section 3.

9.1. Cohomology operators. In this section R = Q/(f), where f = f_{1}, …, f_{r} is a regular sequence in a (not necessarily regular local) commutative ring Q. We denote E = Q[y_{1}, …, y_{r} | ∂(y_{j}) = f_{j}] the Koszul complex on f, and let κ: E ⟶ R be its canonical augmentation.

Extending Shamash’s [143] construction of resolutions over hypersurface sections, cf. Theorem 3.1.3, Eisenbud [57] produces (in a finite number of steps, if pd_{Q} M is finite) a free resolution of an R–module M starting from any free resolution of M over Q. Here we present a version from Avramov and Buchweitz [31]; the result is somewhat weaker, but easier to prove and sufficient for our purposes.

Theorem 9.1.1. Let M be a finite R–module, let ϵ^{U}: U ⟶ M be a DG module resolution of M over E such that U_{n} is a free Q–module for each n.

Let G = Q⟨v_{1}, …, v_{r}⟩ be a Q–module with basis {v^{(H)} = v_{1}^{(h_{1})} ⋯ v_{r}^{(h_{r})} : |v^{(H)}| = 2(h_{1}|v_{1}| + ⋯ + h_{r}|v_{r}|) , H = (h_{1}, …, h_{r}) ∈ ℕ^{r}}, and set

C_{n}(E, U) = ⨁_{i ⩾ 0} \overline{G}_{i} ⊗_{R} \overline{U}_{n−i},;

∂(v^{(H)} ⊗ u) = − ∑_{j=1}^{r} v^{(H_{j})} ⊗ y_{j}u + v^{(H)} ⊗ ∂(u)

where \overline{G}_{i} = R ⊗_{Q} G_{i}, \overline{U}_{j} = R ⊗_{Q} U_{j}, and H_{j} = (h_{1}, …, h_{j} − 1, …, h_{r}). Then (C(E, U), ∂) is a free resolution of M over R.

Remark. The Koszul complex K on a regular sequence s with (s) ⊇ f is a DG module over E; by inspection, C(E, K) = C, the resolution of Corollary 6.1.9.

Proof. Let μ: E ⊗_{Q} E ⟶ E be the morphism of DG algebras, given by the multiplication of the exterior algebra. An elementary computation shows that Ker μ is generated by y′_{j} = y_{j} ⊗ 1 − 1 ⊗ y_{j}, for j = 1, …, r. Thus, μ is the composition of (E ⊗_{Q} E) ↪ D = (E ⊗_{Q} E)⟨v_{1}, …, v_{r} | ∂(v_{j}) = y′_{j}⟩ with ν: D ⟶ E, where ν(v^{(H)}) = 0 if |H| > 0. By Proposition 1.3.2, the map

E ⊗_{Q} κ: E ⊗_{Q} E ⟶ E ⊗_{Q} R = R⟨y_{1}, …, y_{r} | ∂(y_{j}) = 0⟩

is a quasi-isomorphism. As (E⊗_{Q}κ)(y′_{j}) = y_{j}, we see that H(E⊗_{Q}E) is the exterior algebra on H_{1}(E ⊗_{Q} E), itself a free R–module with basis cls(y′_{1}), …, cls(y′_{r}). Thus, Proposition 6.1.7 applied to the Γ-extension E ⊗_{Q} E ↪ D, shows that ν is a quasiisomorphism of DG algebras.

Since ν is a morphism of semi-free DG modules over E for the action of E on the right, by Proposition 1.3.3 so is ν ⊗_{E} U: D ⊗_{E} U ⟶ E ⊗_{E} U = U, hence H(D ⊗_{E} U) ≅ M. On the other hand, (D ⊗_{E} U)^{♮} ≅ E^{♮} ⊗_{Q} G ⊗_{Q} U^{♮} is a semi-free DG module for the action of E on the left. Thus, by Proposition 1.3.2 the morphism κ ⊗_{E} U: D ⊗_{E} U ⟶ R ⊗_{E} D ⊗_{E} U is a quasi-isomorphism. Comparison shows that R ⊗_{E} D ⊗_{E} U = C(E, U) as complexes of R–modules. □

Construction 9.1.2. Cohomology operators. Let S = R[χ_{1}, …, χ_{r}] be a graded algebra with variables χ_{1}, …, χ_{r} of degree^{33} −2. In the notation of the preceding theorem, set χ_{j} · v^{(H)} = v^{(H_{j})} for 1 ≤ j ≤ r. These are R–linear endomorphisms of degree −2 of C(E, U)^{♮}. They clearly commute with each other, and a glance at the formula for the differential ∂ of the complex C(E, U) shows that they are chain maps: χ_{j}∂ = ∂χ_{j}. Thus, C(E, U) is a DG module over the graded algebra^{34} of cohomology operators S of the presentation R ≅ Q/(f).

The construction above, taken from [31], is a variant of that of Eisenbud [57], cf. Construction 9.1.5. The introduction of operators of degree −2 on (co)homology is due to Gulliksen [80]; other constructions have been given by Mehta [119] and Avramov [25]. For a long time, it had been held that they coincide, but a close reading of the published arguments has revealed serious flaws. In fact, they yield the same result, but only up to sign: this is proved in [37]; ironically, that proof introduces two new constructions.

Proposition 9.1.3. For each R–module N there are S–linear homomorphisms

χ_{j}: Tor_{n}^{R} (M, N) ⟶ Tor_{n−2}^{R} (M, N) χ_{j}: Ext_{R}^{n} (M, N) ⟶ Ext_{R}^{n+2} (M, N) for 1 ≤ j ≤ c and all n ,

which turn Tor^{R} (M, N) and Ext_{R} (M, N) into modules over S.

These structures depend only on f, are natural in both module arguments, and commute with the connecting maps induced by short exact sequences.

Proof. For the first statement, observe that for each R–module N, the complexes C(E, U)⊗_{R}N and Hom_{R} (C(E, U), N) have an induced structure of DG S–module. Naturality in N is clear, as is linearity of the connecting homomorphisms induced by an exact sequence 0 ⟶ N′ ⟶ N ⟶ N′′ ⟶ 0.

If β: M′ ⟶ M is a homomorphism of R–modules, and U′ is a resolution of M′ given by Construction 2.2.7, then by the lifting property of Proposition 1.3.1 there is a morphism α: U′ ⟶ U of DG modules over E such that H(α) = β. The expressions for the differential in Theorem 9.1.1, and for the action of χ_{j} in Construction 9.1.2 show that v^{(H)} ⊗ u′ ↦ v^{(H)} ⊗ β(u′) defines a morphism of DG S–modules C(E, α): C(E, U′) ⟶ C(E, U). All choices of α are homotopic, so the degree 0 maps of S–modules H(C(E, α) ⊗_{R} N) and H Hom_{R} (C(E, α), N) are uniquely defined, and equal respectively to Tor^{R} (β, N) and Ext_{R} (β, N) . This proves naturality in M, and independence from the choice of U.

Let 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 be a short exact sequence of R–modules, and choose a semi-free resolution U′′ of M′′ over E, such that U′′^{♮} is a free module over E^{♮}. By the usual ‘Horseshoe Lemma’ argument, there exists a differential on U^{♮} = U′^{♮} ⊕ U′′^{♮}, such that U becomes a DG module resolution of M over E, and the canonical exact sequence 0 ⟶ U′ ⟶ U ⟶ U′′ ⟶ 0 is one of DG modules over E. Due to the expression for the differential in Theorem 9.1.1, it gives rise to an exact sequence of DG modules over S:

0 ⟶ C(E, U′) ⟶ C(E, U) ⟶ C(E, U′′) ⟶ 0

33This will not be surprising, once the χ_{j}’s reveal their cohomological nature.

34The algebra S itself has a trivial differential; this nicely illustrates the fact that DG module structures are to be found in all walks of life.

that splits over R. It induces short exact sequences of DG modules over S

0 ⟶ C(E, U′) ⊗_{R} N ⟶ C(E, U) ⊗_{R} N ⟶ C(E, U′′) ⊗_{R} N ⟶ 0

0 ⟶ Hom_{R} (C(E, U′′), N) ⟶Hom_{R} (C(E, U), N) ⟶Hom_{R} (C(E, U′), N) ⟶ 0

Their connecting maps commute with the action of the operators χ_{j}. □

The importance of the algebra of cohomology operators stems from

Theorem 9.1.4. If M and N are finite modules over a noetherian ring R, such that R = Q/(f) for some Q–regular sequence f, then the S–module Ext_{R} (M, N) is finite if and only if Ext_{Q}^{n} (M, N) = 0 for n ≫ 0.

Remark. Most of the remaining results in this chapter are based on this theorem. Section 2 uses the ‘if’ part; different proofs for it are given in each one of the papers quoted in Construction 9.1.2; here we use an elementary argument to establish a special case, that suffices for many applications. Section 3 is based on the converse statement in the special case N = k, proved in [25]; the general result is established in [32].

Partial proof of Theorem 9.1.4. Assume that Q is noetherian, finite projective Q– modules are free, and pd_{R} M is finite. Proposition 2.2.8 then yields a DG module resolution U of M over E, which is a finite complex of free Q–modules. By the preceding result, we may use U to compute the action of S. As Hom_{R} (C(E, U), R) is a semi-free DG module over S with underlying module S ⊗_{R} Hom_{Q} (U, R) , we see that it suffices to prove the

Claim. If F is a semi-free S–module of finite rank, then for each finite R–module N the S–module H(F ⊗_{R} N) is noetherian.

The advantage is that now we can induce on n = rank_{S} F. If n = 1, then F is a shift of S, so H(F ⊗_{R} N) ≅ Σ^{r}S ⊗_{R} N is a finite S–module. If n > 0, then choose a basis element u ∈ F of minimal degree. As ∂(u) = 0 for degree reasons, Su is a DG submodule of F, and G = F/Su is semi-free of rank n − 1. The homology exact sequence now yields an exact sequence of degree zero homomorphisms of S– modules Su ⊗_{R} N ⟶ H(F ⊗_{R} N) ⟶ H(G ⊗_{R} N), where the two outer ones are noetherian by induction. The claim follows. □

Eisenbud [57] shows how to compute the operators from any resolution.

Construction 9.1.5. Eisenbud operators. A lifting to Q of a free resolution (F, ∂) of M over R is a pair (F̃, ˜∂) consisting of a free Q–module F̃ and a degree −1 endomorphism ˜∂ of F̃, such that (F, ∂) = (F̃ ⊗_{Q} R, ˜∂ ⊗_{Q} R).

Liftings always exist—just take arbitrary inverse images in Q of the elements of the matrices of the differentials ∂_{n}. The relation ∂^{2} = 0 yields ˜∂^{2}(F̃) ⊆ (f)F̃, hence for j = 1, …, r there are degree −2 endomorphisms of Q–modules τ^{j}: F̃ ⟶ F̃, such that ˜∂^{2} = ∑_{j=1}^{r} f_{j}τ^{j}.

Each lifting produces a family of Eisenbud operators

τ = {τ^{j} = τ^{j} ⊗_{Q} R: F ⟶ F }_{1 ⩽ j ⩽ r} .

Proposition 9.1.6. Let τ be a family of Eisenbud operators defined by f.

For 1 ≤ j ≤ r the maps τ^{j} are chain maps of degree −2, that are defined uniquely up to homotopy, commute with each other up to homotopy, commute up to homotopy with any comparison of resolutions F′ ⟶ F constructed over a homomorphism of R–modules β: M′ ⟶ M, and satisfy

H(Hom_{R} (τ^{j}, N)) = −χ_{j} .

Proof. Let (F̃′, ˜∂′) be a lifting of a free resolution (F′, ∂′) of an R–module M′, chosee a family of maps τ̃′ = {τ′^{j}}: F′ ⟶ F′ as above, and set τ′ = {τ′^{j} = τ′^{j} ⊗_{Q} R}. If α: F′ ⟶ F is a chain map and α̃: F̃′ ⟶ F̃ is a map of Q–modules such that α̃ ⊗_{Q} R = α, then the equality ∂α = (−1)^{|α|}α∂′ implies that for 1 ≤ j ≤ r there exist Q–linear homomorphisms σ^{j}: F̃′ ⟶ F̃ with |σ^{j}| = |α| − 1 and ˜∂α̃ − (−1)^{|α|}α̃˜∂′ = ∑_{j=1}^{r} f_{j}σ^{j}. Thus, we have

∑_{j=1}^{r} f_{j}(τ^{j}α̃ − α̃τ′^{j}) = ˜∂^{2}α̃ − α̃˜∂′^{2}

= ( ∑_{j=1}^{r} ˜∂f_{j}σ^{j} + (−1)^{|α|} ˜∂α̃˜∂′) + (−1)^{|α|}( ∑_{j=1}^{r} f_{j}σ^{j} ˜∂′ − ˜∂α̃˜∂′)

= ∑_{j=1}^{r} f_{j}(˜∂σ^{j} − (−1)^{|σ^{j}|}σ^{j} ˜∂′) .

Since the elements of f are linearly independent modulo I^{2}, we get

τ^{j}α − ατ′^{j} = ∂(σ^{j} ⊗_{Q} R) − (−1)^{|σ^{j}|}(σ^{j} ⊗_{Q} R)∂′ for 1 ≤ j ≤ r ,

that is, σ^{j} ⊗_{Q} R: F′ ⟶ F is a homotopy from τ^{j}α to ατ′^{j}. We can now get most of the desired assertions by suitably specializing the maps chosen above.

First, letting α = ∂′ = ∂ and α̃ = ˜∂′ = ˜∂, we can set σ^{j} = 0 for 1 ≤ j ≤ r, and so conclude that each τ^{j} is a chain map. Next, taking α = id^{F} and varying τ′, we see that τ^{1}, …, τ^{r} are defined uniquely up to homotopy. Then, keeping α = τ^{j} and τ′ = τ, we see that τ^{j} commutes up to homotopy with each τ^{i}. Finally, choosing α to be a lifting of a homomorphism of R–modules β: M′ ⟶ M, we obtain that τ^{j}α and ατ′^{j} are homotopic for each j.

The resolution F = (C(E, U), ∂) of Theorem 9.1.1 has an obvious lifting:

F̃ = U^{♮} ⊗_{Q} G with ˜∂(v^{(H)} ⊗ u) = − ∑_{j=1}^{r} v^{(H_{j})} ⊗ y_{j}u + v^{(H)} ⊗ ∂(u) .

From it we get ˜∂^{2}(v^{(H)} ⊗ u) = − ∑_{j=1}^{r} v^{(H_{j})} ⊗ f_{j}u = − ∑_{j=1}^{r} f_{j}χ_{j}(v^{(H)} ⊗ u) and hence H(Hom_{R} (τ^{j}, N)) = −χ_{j} for 1 ⩽ j ⩽ r. □

Remark 9.1.7. For any integer d with 1 ≤ d ≤ r, the operators χ_{1}, …, χ_{d} act on Ext_{R}^{∗} (M, k) in two ways: the initial one, from R = Q/(f), and a new one, from the presentation R = P/(f_{1}, …, f_{d}) with P = Q/(f_{d+1}, …, f_{r}).

These actions coincide. Indeed, if (F̃, ˜∂) is a lifting to Q of a free resolution (F, ∂) of M over R, then it is clear that (F̃ ⊗_{Q} P, ˜∂⊗_{Q} P) is a lifting of (F, ∂) to P. In this case we have (˜∂ ⊗_{Q} P)^{2} = ∑_{j=1}^{d} f_{j}(τ^{j} ⊗_{Q} P). Thus, we may use τ^{j} ⊗_{Q} P to compute the operation of χ_{j} coming from the new presentation. It remains to observe that (τ^{j} ⊗_{Q} P) ⊗_{P} R = τ^{j} ⊗_{Q} R.

9.2. Betti numbers. Our method for studying homology over complete intersections is to use the action of the algebra of cohomology operators, in order to replace ‘degree by degree’ computations by ‘global’ considerations.

At that level, we are essentially dealing with finite graded modules over polynomial rings. This converts homological algebra back into commutative algebra, and opens the door to the use geometric methods to study cohomology. Such an approach was pioneered by Quillen [132] for cohomology of groups, and has evolved into a powerful tool of modular representation theory, cf. Benson [41] and Evens [62] for monographic expositions. Geometric methods are used in [25] to study resolutions over commutative rings.

For reference and comparison, the next theorem is presented along the lines of Theorem 5.3.3. It is compiled from four papers: the fact that P_{M}^{R}(t) is rational with denominator (1 − t^{2})^{codim R} is from Gulliksen [80]; the comparison of the orders of the poles, and (3), are from Avramov [25]; the first part of (5) comes from Eisenbud [57], the second from Avramov, Gasharov, and Peeva [32].

Theorem 9.2.1. Let R be a complete intersection with edim R = e and codim R = r. For a finite R–module M ≠ 0 with depth R − depth M = m and pd_{R} M = ∞, the following hold.

(1) There is a polynomial p(t) ∈ ℤ[t] with p(±1) ≠ 0, such that

P_{M}^{R}(t) = \frac{p(t)}{(1 + t)^{c}(1 − t)^{d}} with c < d .

(2) cx_{R} M = d ≤ codim R and curv_{R} M = 1 .

(3) β_{n}^{R}(M) ∼ \frac{b}{2^{c}(d − 1)!} n^{d−1} where b = p(1) > 0.

(4) lim_{n→∞} \frac{β_{n+1}^{R}(M)}{β_{n}^{R}(M)} = 1 .

(5.1) \frac{β_{n+1}^{R}(M)}{β_{n}^{R}(M)} = 1 and Syz_{n+2}^{R} (M) ≅ Syz_{n}^{R} (M) for n > m if cx_{R} M = 1 .

(5.2) \frac{β_{n+1}^{R}(M)}{β_{n}^{R}(M)} > 1 and Syz_{n+2}^{R} (M) ↠ Syz_{n}^{R} (M) for n ≫ 0 if cx_{R} M ≥ 2 .

Remark. A more precise version of the last inequality is proved in [32]: there are polynomials h_{±}(t) of degree d − 2 with leading terms a_{±} > 0, such that for n ≫ 0 the difference β_{n+1}^{R}(M) − β_{n}^{R}(M) is equal to h_{+}(n) if n is even, and to h_{−}(n) if n is odd; however, it is possible that a_{+} ≠ a_{−}, cf. Example 9.2.4.

Example 9.2.2. By Remark 8.1.1.2, we have β_{n}^{R}(k) ∼ 2^{e−r}n^{r−1}/(r − 1)!, so c = − dim R, d = codim R, b = 1, and p_{k}(t) = 1.

Recall that if R is a complete intersection, then mult R ≥ 2^{codim R}.

Example 9.2.3. If mult(R) = 2^{r}, then for each M there is an integer valued polynomial b(t) ∈ ℚ[t] such that β_{n}^{R}(M) = b(n) for n ≫ 0, cf. [28]. This generalizes a well known property of complete intersections of quadrics.

Not all Betti sequences are eventually given by some polynomial in n.

Example 9.2.4. Let q = \binom{e+1}{2} − rank_{k} 𝔪^{2}/𝔪^{3} be the number of ‘quadratic relations’ of R. It is proved in [28] that

P_{R/𝔪^{2}}^{R}(t) = \frac{(1 − t)^{q} + (1 + t)^{e−q−1} · (et − 1)}{(1 − t)^{r} · (1 + t)^{r−q−1} · t} .

Thus, when q ≤ r − 2 the Poincaré series has poles at t = 1 and at t = −1, so the even and odd Betti numbers are each given by a different polynomial. For instance, if R = k[s_{1}, s_{2}]/(s_{1}^{a_{1}}, s_{2}^{a_{2}}), with a_{i} ≥ 3, then β_{n}^{R}(R/𝔪^{2}) is equal to \frac{3}{2}n + 1 if n is even, and to \frac{3}{2}n + \frac{3}{2} if n is odd.

As (5.1) shows, if a Betti sequence is bounded, then it stabilizes after at most depth R steps. However, if cx_{R} M ≥ 2, then there exist modules whose Betti sequence strictly decreases over an initial interval of any given length. This shows that no bound on the degree of the polynomial p(t) can be expressed as a function only of invariants of the ring R:

Example 9.2.5. Let R be a complete intersection of codimension c ≥ 2. Fix N = Syz_{n}^{R} (k), with n > dim R, and let F be its minimal free resolution. The module N is maximal Cohen-Macaulay, cf. 1.2.8, hence the complex

0 ⟶ N^{∗} ⟶ F_{0}^{∗} ⟶^{∂_{0}^{∗}} F_{1}^{∗} ⟶^{∂_{1}^{∗}} F_{2}^{∗} ⟶ …,

where −^{∗} = Hom_{R} (−, R) , is exact and minimal. Splice it to the right of a minimal free resolution of N^{∗}: now you are holding a ‘doubly infinite’ exact complex of finite free R–modules, that you can truncate at will. The cokernel of ∂_{s}^{∗} is guaranteed to have s + 1 strictly decreasing Betti numbers at the beginning of its resolution, cf. Remark 8.1.1.3.

Before starting on the proof, we make a general observation.

Remark 9.2.6. Let Q be a regular local ring, let f be a Q–regular sequence, and set R = Q/(f). If M is a finite R–module, then Ext_{R} (M, k) is a finite module over R[χ_{1}, …, χ_{r}] by Theorem 9.1.4.

Since 𝔪 annihilates Ext_{R} (M, k) , we see that M = Ext_{R} (M, k) is a finit module over the graded polynomial ring P = k[χ_{1}, …, χ_{r}]. In particular, the Hilbert-Serre Theorem applies to the graded P–module M, and shows that P_{M}^{R}(t) = q(t)/(1−t^{2})^{r} for some polynomial q(t) ∈ ℤ[t].

Proof of Theorem 9.2.1. The hypotheses of the theorem and its conclusions do not change if one replaces (R, M) by (R′, M ⊗_{R} R′), where R′ is the completion of the local ring R[u]_{𝔪[u]}. Thus, we assume that R = Q/(f), where Q is regular with infinite residue field k, and f is a regular sequence.

Let F be a minimal free resolution of M over R, and set β_{n} = β_{n}^{R}(M).

(1) Due to Remark 9.2.6, P_{M}^{R}(t) can be written in the form

P_{M}^{R}(t) = ∑_{j=0}^{d−1} \frac{m_{j}}{(1 − t)^{d−j}} + ∑_{i=0}^{c−1} \frac{ℓ_{i}}{(1 + t)^{c−i}} + f(t) ,

with max{c, d} ≤ r and f(t) ∈ ℚ[t]. Thus, for n ≫ 0, there are equalities

β_{n} = \begin{cases}\frac{m_{0}}{(d − 1)!} · n^{d−1} + \frac{ℓ_{0}}{(c − 1)!} · n^{c−1} + g_{+}(n) & for even n; \\ \frac{m_{0}}{(d − 1)!} · n^{d−1} − \frac{ℓ_{0}}{(c − 1)!} · n^{c−1} + g_{−}(n) & for odd n; \end{cases}  (∗)

with m_{0} ≠ 0, and polynomials g_{±}(t) of degree < max{c, d} − 1. As the Betti numbers of M are positive, we have d ≥ c, and d > 0.

Assume next that d = c, so that ℓ_{0} ≠ 0. The positivity of Betti numbers implies that m_{0} ± ℓ_{0} > 0, hence m_{0} > 0.

Set γ(j, 2s) = ∑_{i=−j}^{j}(−1)^{i}β_{2s−i}. Formula (∗) shows that for all s, h ≫ 0 the function 2s ↦ γ(2h, 2s) is given by a polynomial in 2s of degree d with leading coefficient a_{0} = ((4h+1)ℓ_{0} +m_{0})/(d−1)!, and the function 2s ↦ γ(2h+1, 2s) by a polynomial of the same degree with leading coefficient a_{1} = ((4h + 3)ℓ_{0} − m_{0})/(d − 1)!. Thus: if ℓ_{0} < 0, then a_{0} < 0 for h ≫ 0, so γ(2h, 2s) < 0; if ℓ_{0} > 0, then a_{1} > 0 for h ≫ 0, so γ(2h + 1, 2s) > 0.

Localization of F at a minimal prime ideal 𝔭 of R yields an exact sequence

0 ⟶ L_{s,j} ⟶ (F_{2s+j})_{𝔭} ⟶ … ⟶ (F_{2s})_{𝔭} ⟶ … ⟶ (F_{2s−j})_{𝔭} ⟶ N_{s,j} ⟶ 0 .

Counting lengths over R_{𝔭}, we get an equality

γ(j, 2s) · length(R_{𝔭}) = (−1)^{j}( length(L_{j}) + length(N_{j}))

which shows that γ(2h, 2s) > 0 and γ(2h + 1, 2s) < 0, regardless of the sign of ℓ_{0}. We have a contradiction, so we conclude that d > c.

(3) Since d > c, formula (∗) yields lim_{n→∞} β_{n}/n^{d−1} = m_{0}/(d − 1)! . On the other hand, m_{0} = lim_{t→1}(1 − t)^{d} P_{M}^{R}(t) = p(1)/2^{c}.

(2) and (4) are trivial consequences of (1) and (3).

(5) By Theorem 9.1.4, Ext_{R} (M, k) is a finite graded module over the polynomial ring k[χ_{1}, …, χ_{r}]. Thus, its graded submodule

{μ ∈ Ext_{R} (M, k) | (χ_{1}, …, χ_{r})^{m}μ = 0 for some m}

is finite-dimensional, and hence is trivial, say, in degrees > s. Since k is infinite, we can find a linear combination χ of χ_{1}, …, χ_{r}, that is a non-zero-divisor on Ext_{R}^{>s} (M, k) . Thus, the operator χ is injective on Ext_{R}^{>s} (M, k) . Dualizing, we see that χ: Tor_{n+2}^{R} (M, k) ⟶ Tor_{n}^{R} (M, k) is surjective when n > s.

Changing bases, we may assume that χ = χ_{1}, and switch attention to the presentation R = P/(f), where P = Q/(f_{2}, …, f_{r}) and f is the image of f_{1}; note that f is P –regular. Let (F̃, ˜∂) be a lifting of the complex (F, ∂) to P, and let τ̃: F̃ ⟶ F̃ and τ = τ̃⊗_{P} R: F ⟶ F be the degree −2 endomorphisms from Construction 9.1.5. By Remark 9.1.7, we have χ = Hom_{R} (τ, k) .

Since χ_{n} is surjective for n > s, so are the maps τ̃_{n+2} : F̃_{n+2} ⟶ F̃_{n} and τ_{n+2} : F_{n+2} ⟶ F_{n} by Nakayama. The chain map τ induces surjections Syz_{n+2}^{R} (M) ⟶ Syz_{n}^{R} (M) for n > s. Localize the defining exact sequence

0 ⟶ Syz_{n+2}^{R} (M) ⟶ F_{n+1} ⟶^{∂_{n+1}} F_{n} ⟶ Syz_{n}^{R} (M) ⟶ 0

at a minimal prime 𝔭 of R. For n > s a lengths count over R_{𝔭} yields

β_{n+1} − β_{n} = ( length Syz_{n+2}^{R_{𝔭}} (M_{𝔭}) − length Syz_{n}^{R_{𝔭}} (M_{𝔭}) )/ length(R_{𝔭}) ≥ 0 ,

Next we assume that β_{n} = β_{n+1} = b ≠ 0 for some n > s, and show that β_{n+2} = b. Since τ̃_{n+2} is surjective for n > s, we have F̃_{n+2} = E ⊕ G with E = Ker τ̃_{n+2}, and the restriction θ of τ̃_{n+2} to G is an isomorphism with F̃_{n}. Let ζ: G ⟶ F̃_{n+1} be the restriction of ˜∂_{n+2}. As ˜∂_{n+1}ζ is the restriction to G of ˜∂_{n+1} ˜∂_{n+2} = fτ̃_{n+2}, where P/(f) = R, we have ˜∂_{n+1}ζ = fθ, and hence

⋀^{b} ˜∂_{n+1} ⋀^{b} ζ = ⋀^{b}(˜∂_{n+1}ζ) = ⋀^{b}(fθ) = f^{b} ⋀^{b} θ .

Note that G, F̃_{n+1}, and F̃_{n} have rank b and fix isomorphisms of P with ⋀^{b}(G), ⋀^{b}(F̃_{n+1}), and ⋀^{b}(F̃_{n}). The maps ⋀^{b} ˜∂_{n+1}, ⋀^{b} ζ, and ⋀^{b} θ are then given by multiplication with elements of P, say y, z, and u, respectively. The equality above becomes yz = f^{b}u. As θ is bijective so is ⋀^{b}(θ), hence u is a unit in P. As f is P -regular, so is y, hence ˜∂_{n+1} is injective. From ˜∂_{n+1} ˜∂_{n+2}(E) = fτ̃_{n+2}(E) = 0 we now see that E ⊆ Ker ˜∂_{n+2}, so Im ˜∂_{n+2} is a homomorphic image of F̃_{n+2}/E ≅ G. Remarking that

(Coker ˜∂_{n+3}) ⊗_{P} R ≅ Coker ∂_{n+3} = Syz_{n+2}^{R} (M)

we conclude that Syz_{n+2}^{R} (M) is a homomorphic image of the free R–module G ⊗_{P} R ≅ R^{b}. It follows that β_{n+2} ≤ b = β_{n}. On the other hand, we already know that β_{n+2} ≥ β_{n+1} ≥ β_{n}, hence all three are equal to b. Thus, the sequence {β_{n}}_{n>s} is either strictly increasing or constant: we have proved (5.2).

If β_{n+2} = β_{n}, then rank_{P} F̃_{n+2} = rank_{P} F̃_{n}, so E = 0 and the surjective homomorphism τ_{n+2} is bijective. To finish the proof of (5.1), we show that for m = depth R − depth M the complex F_{> m} is periodic of period 2. It is a minimal resolution of N = Syz_{m+1}^{R} (M), and N is maximal Cohen-Macaulay by Proposition 1.2.8. Thus, F_{> m}^{∗} = Hom_{R} (F_{> m}, R) is acyclic, with H_{0}(F_{> m}^{∗}) = N^{∗}. Since F_{> m}^{∗} is minimal, N^{∗} is a syzygy of C_{n} = Coker ∂_{n}^{∗} for each n ≥ m. For n ≫ 0 the minimal resolution of C_{n} is periodic of period 2, hence so is F_{> m}^{∗}. □

9.3. Complexity and Tor. Let (R, 𝔪, k) be a local ring.

If R = Q/I is a complete intersection and Q is regular, then the finite global dimension of Q implies that all R–modules have finite complexity. However, to study a specific R–module, it often pays off to use an intermediate (singular) complete intersection P, that retains the crucial property pd_{P} M < ∞. With this approach, the following factorization theorem is proved in [25].

Theorem 9.3.1. Let R ≅ Q/I be a regular presentation with I generated by a regular sequence. If k is infinite, then for each finite R–module M the surjection Q ⟶ R factors as Q ⟶ P ⟶ R, with the kernels of both maps generated by regular sequences, pd_{P} M < ∞, and cx_{R} M = pd_{P} R.

Proof. As in Remark 9.2.6, consider the finite graded module M = Ext_{R}^{∗} (M, k) over the ring P defined by the presentation R = Q/I. Elementary dimension theory shows that the Krull dimension of M over P is equal to cx_{R} M = d. As k is infinite, we may choose a homogeneous system of parameters χ_{1}, …, χ_{d} for M, and extend it to a basis χ_{1}, …, χ_{r} of P^{2}, the degree 2 component of P.

It is not hard to see that I can be generated by a Q–regular sequence f = f_{1} …, f_{r} that defines the operators χ_{1}, …, χ_{r}. Remark 9.1.7 identifies k[χ_{1}, …, χ_{d}] ⊆ P with the ring P′ of cohomology operators of a presentation R = P/(f_{1}, …, f_{d}), where P = Q/(f_{d+1}, …, f_{r}). As M is finite over P′, Theorem 9.1.4 shows that Ext_{P}^{n} (M, k) = 0 for n ≫ 0, that is, pd_{P} M < ∞. □

To deal with intrinsic properties of the R–module M, a concept of virtual projective dimension is introduced in [25] by the formula^{35}

vpd_{R} M = inf \{ pd_{Q′} M̂ | \begin{matrix}Q′ is a local ring such that R̂ ≅ Q′/(f′) \\ for some Q′–regular sequence f′ \end{matrix}\} .

35If k is infinite; otherwise, R and M are replaced by R̃ = R[u]_{𝔪[u]} and M̃ = M ⊗_{R} R̃.

Clearly, vpd_{R} M < ∞ whenever R is a complete intersection.

Recall that pd_{R} M is finite if and only if cx_{R} M = 0. It is easy to see that in that case, vpd_{R} M = pd_{R} M. Thus, the following result extends the Auslander-Buchsbaum Equality.

Theorem 9.3.2. If M is a finite R–module and vpd_{R} M is finite, then

vpd_{R} M = depth R − depth_{R} M + cx_{R} M .

Proof. We may assume that R is complete with infinite residue field.

Choosing Q′ with pd_{Q′} M = vpd_{R} M, we have

vpd_{R} M = pd_{Q′} M = depth Q′ − depth M = pd_{Q′} R + depth R − depth M ≥ cx_{R} M + depth R − depth M

where the inequality comes from Corollary 4.2.5.4. On the other hand, the preceding theorem provides a ring P from which R is obtained by factoring out a regular sequence, and that satisfies pd_{P} R = cx_{R} M, so we get

vpd_{R} M ≤ pd_{P} M = depth P − depth M = pd_{P} R + depth R − depth M = cx_{R} M + depth R − depth M

where the inequality holds by definition. □

Next we study the vanishing of Tor functors over a local ring. The subject starts with a famous rigidity theorem of Auslander [16] and Lichtenbaum [112]:

Theorem 9.3.3. If M and N are finite modules over a regular ring R and

Tor_{i}^{R} (M, N) = 0 for some i > 0 ,

then Tor_{n}^{R} (M, N) = 0 for all n ≥ i. ■

Heitmann [85] proves that rigidity may fail, even with R Cohen-Macaulay and pd_{R} M finite. On the other hand, there are partial extensions of the theorem to complete intersections. The first one is due to Murthy [123].

Theorem 9.3.4. If M and N are finite modules over a complete intersection R of codimension r, and for some i > 0 there are equalities

Tor_{i}^{R} (M, N) = ⋯ = Tor_{i+r}^{R} (M, N) = 0

then Tor_{n}^{R} (M, N) = 0 for all n ≥ i.

In codimension 1, this is complemented by Huneke and Wiegand [91]:

Theorem 9.3.5. If M and N are finite modules over a hypersurface R, and

Tor_{i}^{R} (M, N) = Tor_{i+1}^{R} (M, N) = 0 for some i > 0 ,

then either M or N has finite projective dimension.

When the vanishing occurs outside of an initial interval, Jorgensen [95] draws the conclusion from the vanishing of fewer Tor’s.

Theorem 9.3.6. Let M be a finite module over a complete intersection R, such that cx_{R} M = d and depth R − depth M = m. For a finite R–module N, the following are equivalent.

(i) Tor_{n}^{R} (M, N) = 0 for n > m.

(ii) Tor_{n}^{R} (M, N) = 0 for n ≫ 0 .

(iii) Tor_{i}^{R} (M, N) = ⋯ = Tor_{i+d}^{R} (M, N) = 0 for some i > m.

The number of vanishing Tor’s in (iii) cannot be reduced further; the next example elaborates on a construction from [95].

Example 9.3.7. For i ≥ 1, R = k[[s_{1}, …, s_{2r}]/(s_{1}s_{r+1}, …, s_{r}s_{2r}), and N = R/(\overline{s}_{r+1}, …, \overline{s}_{2r}) there is a module M_{i}, such that cx_{R} M_{i} = r, Tor_{n}^{R} (M_{i}, N) = 0 for i < n ≤ i + r, but Tor_{n}^{R} (M_{i}, N) ≠ 0 for infinitely many n.

Corollary 6.1.9 yields a minimal resolution

F = R⟨x_{1}, …, x_{2r} | ∂(x_{j}) = \overline{s}_{j} , ∂(x_{r+j}) = \overline{s}_{r+j}x_{j} for 1 ≤ j ≤ r⟩

of M = R/(\overline{s}_{1}, …, \overline{s}_{r}) ≅ k[[s_{r+1}, …, s_{2r}]]. As M is maximal Cohen-Macaulay, F^{∗} = Hom_{R} (F, R) is exact in degrees ≠ 0. It is easy to see that the sequence

F_{1} ⟶^{∂_{1}} F_{0} ⟶^{σ} F_{0}^{∗} ⟶^{−∂_{1}^{∗}} F_{1}^{∗} with σ(x) = s_{r+1} ⋯ s_{2r}x

is exact. The splice of F with Σ^{−1}F^{∗} along σ is a doubly infinite complex (G, ∂) of free R–modules. By construction, H_{n}(G ⊗_{R} N) = Tor_{R}^{n} (M, N) for n ≥ 1 and H_{n}(G ⊗_{R} N) = Ext_{R}^{1−n} (M, N) for n ≤ −2; as σ ⊗_{R} N = 0 trivial, these equalities extend to n = 0 and n = −1, respectively.

For 1 ≤ j ≤ r, consider N_{j} = N/(\overline{s}_{j+1}, …, \overline{s}_{r})N ≅ k[[s_{1}, …, s_{j}]], note that Tor^{R} (M, N_{j}) and Ext_{R} (M, N_{j}) are annihilated by (\overline{s}_{1}, …, \overline{s}_{2r}), and set

T_{j}(t) = ∑_{n=0}^{∞} rank_{k} Tor_{n}^{R} (M, N_{j}) t^{n} and E_{j}(t) = ∑_{n=0}^{∞} rank_{k} Ext_{R}^{n} (M, N_{j}) t^{n} .

The exact sequences 0 ⟶ N_{j} ⟶^{s_{j}} N_{j} ⟶ N_{j−1} ⟶ 0 induce (co)homology sequences in which multiplication by s_{j} is the zero map, so

T_{j−1}(t) = T_{j}(t) + tT_{j}(t)  and  E_{j−1}(t) = E_{j}(t) + \frac{1}{t}E_{j}(t) .

Since and N_{0} ≅ k, we have T_{0}(t) = E_{0}(t) = P_{M}^{R}(t) = 1/(1 − t)^{r}, and hence

T_{r}(t) = \frac{T_{0}(t)}{(1 + t)^{r}} = \frac{1}{(1 − t^{2})^{r}}  and  E_{r}(t) = \frac{t^{r}E_{0}(t)}{(1 + t)^{r}} = \frac{t^{r}}{(1 − t^{2})^{r}} .

Now set M_{i} = Im ∂_{−r−i−1}; as Tor_{n}^{R} (M_{i}, N) ≅ H_{n−r−i}(G ⊗_{R} N) for n ≥ 1, these equalities establish the desired property.

We start the proofs with a couple of easy lemmas.

Lemma 9.3.8. Let f_{1}, …, f_{d} be a regular sequence in a commutative ring Q, and set R = Q/(f_{1}, …, f_{d}). If

Tor_{s}^{R} (M, N) = ⋯ = Tor_{t}^{R} (M, N) = 0

for integers s and t with s + d ≤ t, then there are isomorphisms

Tor_{s+d−1}^{Q} (M, N) ≅ Tor_{s−1}^{R} (M, N) ;
Tor_{s+d}^{Q} (M, N) = ⋯ = Tor_{t}^{Q} (M, N) = 0 ;
Tor_{t+1}^{Q} (M, N) ≅ Tor_{t+1}^{R} (M, N) .

Proof. The Cartan-Eilenberg change of rings spectral sequence 3.2.1 has

2E_{p,q} = Tor_{p}^{R} (Tor_{q}^{Q} (M, R) , N) ⟹ Tor_{p+q}^{Q} (M, N) .

If E is the Koszul complex resolving R over Q, then

Tor_{q}^{Q} (M, R) = H_{q}(M ⊗_{Q} E) = M ⊗_{Q} E_{q} = M^{\binom{d}{q}} ,

hence ^{2}E_{p,q} = Tor_{p}^{R} (M, N)^{\binom{d}{q}}. Thus, ^{2}E_{p,q} = 0 for s ≤ p ≤ t. It follows that the only possibly non-zero module in total degree s + d − 1 is ^{2}E_{s−1 , d} = Tor_{s+d−1}^{R} (M, N) , that all modules in total degree n for s + d ≤ n ≤ t are trivial, and that the only possibly non-zero module in total degree t + 1 is 2E_{t+1 , 0} = Tor_{t+1}^{R} (M, N) . For degree reasons, no non-trivial differential can enter or quit these modules. This gives the desired isomorphisms. □

Proof of Theorem 9.3.4. By the lemma, Tor_{i+r}^{Q} (M, N) = 0 and Tor_{i+r+1}^{Q} (M, N) ≅ Tor_{i+r+1}^{R} (M, N) , so Theorem 9.3.3 yields Tor_{i+r+1}^{Q} (M, N) = 0, and we conclude that Tor_{i+r+1}^{R} (M, N) = 0. Iteration yields Tor_{n}^{R} (M, N) = 0 for n > i + r. □

C. Miller [120] provides a simple proof of Theorem 9.3.5, based on

Lemma 9.3.9. Let M, N be finite modules over a complete intersection R. If Tor_{n}^{R} (M, N) = 0 for n ≥ 1, then cx_{R} M + cx_{R} N = cx_{R}(M ⊗_{R} N). If Tor_{n}^{R} (M, N) = 0 for n ≫ 0, then cx_{R} M + cx_{R} N ≤ codim R.

Proof. As P_{M⊗_{Q}N}^{R}(t) = P_{M}^{R}(t) · P_{N}^{R}(t), cf. the proof of Proposition 4.2.4.6, comparison of orders of poles at t = 1 and Theorem 9.2.1.2 yield the first assertion. For the second one, replace M by a high syzygy M′; then cx_{R} M′ + cx_{R} N = cx_{R}(M′ ⊗_{R} N) ≤ codim R, the inequality coming from loc. cit. □

Proof of Theorem 9.3.5. By Theorem 9.3.4, Tor_{n}^{R} (M, N) = 0 for n ≥ i; thus, cx_{R} M + cx_{R} N ≤ 1 by lemma 9.3.9, so pd_{R} M or pd_{R} N is finite. □

We use the factorization theorem to give a short

Proof of Theorem 9.3.6. Only (iii) ⟹ (i) needs a proof. We may assume that R is complete with infinite residue field. By hypothesis, there are s, t ∈ ℕ, such that m < s < s + d ≤ t and Tor_{j}^{R} (M, N) = 0 for s ≤ j ≤ t. For the smallest such s, choose P as in Theorem 9.3.1; as pd_{P} M is finite,

pd_{P} M = depth P − depth M = depth R + d − depth M = m + d .

We see that if s > m + 1, then Tor_{s+d−1}^{P} (M, N) = 0; the first isomorphism in Lemma 9.3.8 yields Tor_{s−1}^{R} (M, N) = 0, contradicting the minimality of s. Thus, s = m + 1; it follows that t + 1 > pd_{P} M, and so Tor_{t+1}^{P} (M, N) = 0. The last isomorphism of the lemma yields Tor_{t+1}^{R} (M, N) = 0. Iterate… □

10. Homotopy Lie algebra of a local ring

It is a remarkable phenomenon that very sensitive homological information on a local ring is encrypted in a non-commutative object—a graded Lie algebra. We construct it, and show that its very existence affects the size of free resolutions, while its structure influences their form.

This chapter provides a short introduction to a huge area of research: the use of non-commutative algebra for the construction and study of free resolutions. We start by providing a self-contained construction of a graded Lie algebra, whose universal enveloping algebra is the Ext-algebra of the local ring.

10.1. Products in cohomology. We revert to a commutative ring 𝕜, and consider graded associative^{36} algebras over 𝕜. The primitive example of an associative algebra is a matrix ring. The graded version is the 𝕜–module of homogeneous homomorphisms Hom_{𝕜} (C, C) , with composition as product and the identity map as unit. If C is a complex, then the derivation on Hom_{𝕜} (C, C) , used since Section 1.1, turns it into an associative DG algebra. It appears in

Construction 10.1.1. Ext algebras. When ϵ: F ⟶ L is a free resolution of a 𝕜–module L, Proposition 1.3.2 yields an isomorphism

H Hom_{𝕜} (F, ϵ) : H Hom_{𝕜} (F, F) ≅ H Hom_{𝕜} (F, L) = Ext_{𝕜} (L, L) .

Thus, F defines a structure of graded 𝕜–algebra on Ext_{𝕜} (L, L) .

In degree zero, it is the usual product of Hom_{𝕜} (L, L) , an invariant of the 𝕜– module L. To see that all of it is invariant, take a resolution F′ of L, and choose morphisms α: F ⟶ F′ and α′: F′ ⟶ F, lifting the identity of L. As α′α also is such a morphism, there is a homotopy σ with α′α = id^{F} +∂σ + σ∂. Define ϕ: Hom_{𝕜} (F, F) ⟶ Hom_{𝕜} (F′, F′) by ϕ(β) = αβα′. If β and γ: F ⟶ F are chain maps, then

ϕ(βγ) = αβγα′ = αβ(α′α)γα′ − αβ(∂σ)γα′ − αβ(σ∂)γα′
= (αβα′)(αγα′) − (−1)^{|β|}∂(αβσγα′) − (−1)^{|γ|}(αβσγα′)∂
= ϕ(β)ϕ(γ) + ∂τ + (−1)^{|τ|}τ∂

with τ = −(−1)^{|β|}ϕ(βσγ). In homology, this shows that H(ϕ) is an homomorphism of algebras. As ϕ is a quasi-isomorphism by Propositions 1.3.2 and 1.3.3, H(ϕ) is an isomorphism. It is also unique: all choices for α and α′ are homotopic to the original ones, producing homotopic maps ϕ, and hence the same H(ϕ).

We have finished the construction of the Ext algebra of the 𝕜–module L, with the composition product^{37}.

The next structure^{38} might at first seem complicated.

36That is, not assumed positively graded or graded commutative.

37Another pairing is the Yoneda product, that splices exact sequences representing elements of Ext; they differ by a subtle sign, treated with care by Bourbaki [45].

38An early appearance is in the form 𝔤^{n+1} = π_{n}(X), the n’th homotopy group of a topological space X, with bracket given by the Whitehead product; the proof by Uehara and Massey [152] of the Jacobi identity for the Whitehead product was the first major application of the (then) newly discovered Massey triple product.

Remark 10.1.2. A graded Lie algebra ^{39} over 𝕜 is a graded 𝕜–module 𝔤 = {𝔤^{n}}_{n∈ℤ} equipped with a 𝕜–bilinear pairing, called the Lie bracket

[ , ]: 𝔤^{i} × 𝔤^{j} ⟶ 𝔤^{i+j}  for i, j ∈ ℤ ,  (ϑ, ξ) ↦ [ϑ, ξ] ,

such that for all ϑ , ξ , ζ ∈ 𝔤 signed versions of the classical conditions hold: (1) [ϑ, ξ] = −(−1)^{|ϑ||ξ|}[ξ, ϑ] (anti-commutativity)

(2) [ϑ, [ξ, ζ]] = [[ϑ, ξ], ζ] + (−1)^{|ϑ||ξ|}[ξ, [ϑ, ζ]] (Jacobi identity) To deal with deviant behavior over rings without \frac{1}{6}, we extend the definition by

requiring, in addition^{40}, that (1\frac{1}{2}) [ϑ, ϑ] = 0 for ϑ ∈ 𝔤^{even} .

(2\frac{1}{3}) [υ, [υ, υ]] = 0 for υ ∈ 𝔤^{odd} . and that 𝔤 be endowed with a reduced square^{41}

𝔤^{2h+1} ⟶ 𝔤^{4h+2}  for h ∈ ℤ ,  υ ↦ υ^{[2]} ,

such that the following conditions are satisfied:

(3) (υ + ω)^{[2]} = υ^{[2]} + ω^{[2]} + [υ, ω] for υ, ω ∈ 𝔤^{odd} with |υ| = |ω|;

(4) (aυ)^{[2]} = a^{2}υ^{[2]} for a ∈ 𝕜 and υ ∈ 𝔤^{odd} ;

(5) [υ^{[2]}, ϑ] = [υ, [υ, ϑ]] for υ ∈ 𝔤^{odd} and ϑ ∈ 𝔤. A Lie subalgebra is a subset of 𝔤 closed under brackets and squares; with the induced operations, it is a graded Lie algebra in its own right. A homomorphism β: 𝔥 ⟶ 𝔤 of graded Lie algebras is a degree zero 𝕜–linear map of the underlying graded 𝕜–modules, such that β[ϑ, ξ] = [β(ϑ), β(ξ)] and β(ϑ^{[2]}) = β(ϑ)^{[2]}.

One way to get a Lie structure is to partly forget an associative one. Let B be a graded associative algebra over 𝕜. The underlying module of B, with bracket [x, y] = xy − (−1)^{|x||y|}yx (the graded commutator) and reduced square v^{[2]} = v^{2} for v ∈ B^{odd}, is a graded Lie algebra, denoted Lie(B): the axioms are readily verified by direct computations. The non-triviality of the operations measures how far the algebra B is from being graded commutative.

There is also a vehicle to go from Lie to associative algebras. A universal enveloping algebra of 𝔤 is a graded associative 𝕜–algebra U together with a degree 0 homomorphism of graded Lie algebras ι: 𝔤 ⟶ Lie(U) with the following property: for each associative algebra B and each Lie algebra homomorphism β: 𝔤 ⟶ Lie(B), there is a unique homomorphism of associative algebras β′: U ⟶ B, such that β = β′ι; we call β′ the universal extension of β.

Remark 10.1.3. For the first few statements on enveloping algebras, one just needs to exercise plain abstract nonsense:

(1) Any two universal enveloping algebras of 𝔤 are isomorphic by a unique isomorphism, hence a notation U_{𝕜}(𝔤) is warranted.

(2) Each homomorphism 𝔥 ⟶ 𝔤 of graded Lie algebras induces a natural homomorphism U_{𝕜}(𝔥) ⟶ U_{𝕜}(𝔤) of graded (associative) algebras.

39In postmodern parlance, a super Lie algebra.

40Anticommutativity implies 2[ϑ, ϑ] = 0 for ϑ ∈ 𝔤^{even}, so (1 \frac{1}{2} ) is superfluous when 𝕜 ∋ \frac{1}{2} .

Jacobi yields 3[ϑ, [ϑ, ϑ]] = 0 for all ϑ, so (2 \frac{1}{3} ) is redundant when 𝕜 ∋ \frac{1}{3} .

41Only needed if \frac{1}{2} ∉ 𝕜: conditions (3) and (4) imply that 2υ^{[2]} = [υ, υ]; when 2 is invertible, υ^{[2]} = \frac{1}{2} [υ, υ] satisfies condition (5), by the Jacobi identity.

(3) The graded 𝕜–algebra U_{𝕜}(𝔤) is isomorphic to the residue of the tensor algebra T_{k}(𝔤) modulo the two-sided ideal generated by

ϑ ⊗ ξ − (−1)^{|ϑ||ξ|}ξ ⊗ ϑ − [ϑ, ξ] for all ϑ, ξ ∈ 𝔤;
υ ⊗ υ − υ^{[2]} for all υ ∈ 𝔤^{odd} ;

the map ι is the composition of 𝔤 ⊆ T (𝔤) with the projection T (𝔤) ⟶ U.

(4) Assume that 𝔤^{n} = 0 for n ≤ 0, and that ϑ = {ϑ_{i}}_{i ⩾ 1} is a set of generators of 𝔤, linearly ordered so that |ϑ_{i}| ≤ |ϑ_{j}| for i < j. We consider indexing sequences I = (i_{1}, i_{2}, …) of integers i_{j} ≥ 0, such that i_{j} ≤ 1 if |ϑ_{j}| is odd and i_{j} = 0 for j ≫ 0. For each I, pick any q such that i_{j} = 0 for j > q, and form the (well defined) normal monomial ϑ^{I} = ϑ_{q}^{i_{q}} ⋯ ϑ_{1}^{i_{1}} ∈ U_{𝕜}(𝔤).

The normal monomials span^{42} U_{𝕜}(𝔤). Indeed, (3) shows that U_{𝕜}(𝔤) is spanned by all product of elements of ϑ. If such a product contains ϑ_{i}^{2} with |ϑ_{i}| odd, then replace ϑ_{i}^{2} by ϑ_{i}^{[2]}; if it contains ϑ_{i}ϑ_{j} with i < j, then replace it by ϑ_{j}ϑ_{i} ± [ϑ_{i}, ϑ_{j}]; express each ϑ_{i}^{2} and [ϑ_{i}, ϑ_{j}] as a linear combination of generators. Applying the procedure to each of the new monomials, after a finite number of steps one ends up with a linear combination of normal monomials.

Returning to homological algebra, we show how basic constructions of Lie algebras create cohomological structures. A DG Lie algebra over 𝕜 is a graded Lie algebra 𝔤 with a degree −1 𝕜-linear map ∂: 𝔤 ⟶ 𝔤, such that ∂^{2} = 0,

∂[ϑ, ξ] = [∂(ϑ), ξ] + (−1)^{|ϑ|}[ϑ, ∂(ξ)] , and ∂(ϑ^{[2]}) = [∂(ϑ), ϑ] for ϑ ∈ 𝔤^{odd} .

A morphism of DG Lie algebras is a homomorphism of the underlying graded Lie algebras, that is also a morphism of complexes. Homology is a functor from DG Lie algebras to graded Lie algebras.

Lie algebras of derivations are paradigmatic throughout Lie theory. Here is a DG version, based on the Γ-free extensions of Chapter 6.

Lemma 10.1.4. Let 𝕜 ↪ 𝕜⟨X⟩ be a semi-free Γ -extension. The inclusion Der_{𝕜}^{γ} (𝕜⟨X⟩, 𝕜⟨X⟩) ⊆ Lie(Hom_{𝕜} (𝕜⟨X⟩, 𝕜⟨X⟩) ) is one of DG Lie algebras.

Proof. The proof is a series of exercises on the Sign Rule.

If b, c, are elements of 𝕜⟨X⟩, and υ is a derivation of odd degree, then

υ^{2}(bc) =υ (υ(b)c + (−1)^{|b|}bυ(c))
=υ^{2}(b)c + (−1)^{|ϑ|(|ϑ|+|b|)}υ(b)υ(c) + (−1)^{|b|}υ(b)υ(c) + (−1)^{|b|+|b|}bυ^{2}(c)
=υ^{2}(b)c + bυ^{2}(c) .

If x is a Γ-variable of even degree, and ϑ, ξ are Γ-derivations, then

υ^{2}(x^{(i)}) = υ(υ(x)x^{(i−1)}) = υ^{2}(x)x^{(i−1)} − (υ(x))^{2}x^{(i−2)} = υ^{2}(x)x^{(i−1)}

42In fact, if ϑ is a basis of 𝔤, then the normal monomials form a basis of U_{𝕜}(𝔤): this is the contents of the celebrated Poincaré-Birkhoff-Witt Theorem. The original proof(s) provide one of the first applications of ‘standard basis’ techniques; for an argument in the graded framework, cf. Milnor and Moore [121]; for the case needed here, cf. Theorem 10.2.1.

and

[ϑ, ξ](x^{(i)}) =ϑξ(x^{(i)}) − (−1)^{|ϑ||ξ|}ξϑ(x^{(i)})
=ϑ(ξ(x)x^{(i−1)}) − (−1)^{|ϑ||ξ|}ξ(ϑ(x)x^{(i−1)})
=ϑξ(x)x^{(i−1)} + (−1)^{|ϑ||ξ|}ξ(x)ϑ(x)x^{(i−2)}
− (−1)^{|ϑ||ξ|}ξϑ(x)x^{(i−1)} − (−1)^{|ϑ||ξ|+|ξ||ϑ|}ϑ(x)ξ(x)x^{(i−2)}
= ([ϑ, ξ](x)) x^{(i−1)}

A lengthier computation shows that [ϑ, ξ] is a derivation, completing the verification that Der_{𝕜}^{γ} (𝕜⟨X⟩, 𝕜⟨X⟩) is a Lie subalgebra of Lie(Hom_{𝕜} (𝕜⟨X⟩, 𝕜⟨X⟩) ).

It remains to prove that the differential of the derivation complex satisfies the requirements for a graded Lie algebra. This is best done by ‘interiorizing’ it: as ∂ is a DG Γ-derivation of 𝕜⟨X⟩, it may be viewed as an element δ ∈ Der_{𝕜}^{γ} (𝕜⟨X⟩, 𝕜⟨X⟩), and then ∂(ϑ) = [δ, ϑ]. The conditions on ∂[ϑ, ξ] and ∂(υ^{[2]}) are now seen to be transcriptions of the Jacobi identity and its complement. □

It is tempting to mimic the construction of Ext algebras: Choose a Tate resolution 𝕜⟨X⟩ of a commutative 𝕜–algebra P, and associate with P the graded Lie algebra H Der_{𝕜}^{γ} (𝕜⟨X⟩, 𝕜⟨X⟩). If ℚ ⊆ P, then it is an invariant of P: this is proved by Quillen [133], as an outgrowth of his investigation of rational homotopy theory [131]. The general case is very different:

Example 10.1.5. The Tate resolution A = 𝕜⟨X⟩ of 𝕜 over itself, with X = ∅, yields H Der_{𝕜}^{γ} (A, A) = 0. If 𝔽_{2} ⊆ 𝕜, then another Tate resolution of 𝕜 is

B = 𝕜⟨u, {x_{i}, x′_{i}}_{i ⩾ 0} | ∂(u) = 0 , ∂(x_{i}) = u^{(2^{i})} , ∂(x′_{i}) = u^{(2^{i})}x_{i}⟩

with |u| = 2 (hence |x_{i}| = 2^{i+1} + 1 and |x′_{i}| = 2^{i+2} + 2). Using Corollary 6.2.4 and Construction 6.2.5, one gets H Der_{𝕜}^{γ} (B, B) ≅ Hom_{𝕜} (Ind_{𝕜}^{γ} B, 𝕜) , and Ind_{𝕜}^{γ} B is the free 𝕜–module with basis {x_{i}}_{i ⩾ 1} ∪ {x′_{i}}_{i ⩾ 0}.

10.2. Homotopy Lie algebra. In this section (R, 𝔪, k) is a local ring.

Some of the problems occurring in the last example may be circumvented, by using acyclic closures. We follows the ideas of Sjödin [144], and simplify the exposition by using complexes of derivations from Section 6.2.

Theorem 10.2.1. Let R⟨X⟩ be an acyclic closure of k over R, where X = {x_{i}}_{i ⩾ 1} and |x_{i}| ≤ |x_{j}| for i < j, and set π(R) = H Der_{R}^{γ} (R⟨X⟩, R⟨X⟩).

(1) π(R) is a graded Lie algebra over k.

(2) rank_{k} π^{n}(R) = ε_{n}(R) for n ∈ ℤ.

(3) π(R) has a k-basis Θ = {θ_{i} = cls(ϑ_{i}) | ϑ_{i} ∈ Der_{R}^{γ} (R⟨X⟩, R⟨X⟩) , ϑ_{i}(x_{j}) = δ_{ij} for j ≤ i}_{i ⩾ 1} .

(4) The normal monomials on Θ form a k–basis of U_{k}(π(R)).

(5) Der_{R}^{γ} (R⟨X⟩, R⟨X⟩) ⊆ Hom_{R} (R⟨X⟩, R⟨X⟩) induces an injective homomorphism of graded Lie algebras ι: π(R) ⟶ Lie(Ext_{R} (k, k)). Its universal extension is an isomorphism of associative algebras

ι′: U_{k}(π(R)) ≅ Ext_{R} (k, k) .

Remark. By Remark 6.3.9, different choices of acyclic closures yield the same Lie algebra π(R); it is called the homotopy Lie algebra of R.

Proof. (1) and (2). Let ϵ denote the quasi-isomorphism R⟨X⟩ ⟶ k, and let k ↪ k⟨X⟩ be the semi-free Γ-extension with trivial differential. Using Lemma 6.2.4, the minimality of R⟨X⟩, and Proposition 6.2.3.4, we get

π(R) = H Der_{R}^{γ} (R⟨X⟩, R⟨X⟩) ≅ H Der_{R}^{γ} (R⟨X⟩, k)

≅ H Der_{k}^{γ} (k⟨X⟩, k) = Der_{k}^{γ} (k⟨X⟩, k) ≅ Hom_{k} (kX, k) .

So π(R) is a k–module; by Theorem 7.1.3, rank_{k} π^{n}(R) = card(X_{n}) = ε_{n}(R).

(3) Lemma 6.3.3.1 provides a set of R–linear Γ-derivations {ϑ_{i}}_{i ⩾ 1}, with ϑ_{i}(x_{j}) = δ_{ij} for j ≤ i. As {x_{i}}_{i ⩾ 1} is a basis of kX, the isomorphisms above imply that Θ is linearly independent in π(R). Since Θ_{n} has ε_{n}(R) elements for each n, it is a basis by (2).

(4) and (5). In the commutative diagram

Der_{R}^{γ} (R⟨X⟩, R⟨X⟩) ⟶ Hom_{R} (R⟨X⟩, R⟨X⟩)
Der_{R}^{γ}^{(R⟨X⟩,ϵ)}↓^{≃}  ^{≃}↓^{Hom}_{R}^{(R⟨X⟩,ϵ)}
Der_{R}^{γ} (R⟨X⟩, k) ⟶^{ι} Hom_{R} (R⟨X⟩, k)

the left hand arrow is a quasi-isomorphism, as noted for (1); the right hand one is a quasi-isomorphism by Proposition 1.3.2.

Let θ^{I} ∈ U_{k}(π(R)) be a normal monomial on Θ. For a normal Γ-monomial x^{(H)} on X, by Lemma 6.3.3.3 we see that ι′(θ^{I})(x^{(H)}) = cls (ϑ^{I}(x^{(H)})) is equal to 0 if H < I, and to 1 if H = I. As the normal Γ-monomials on X form a basis of k⟨X⟩, the triangular form of the matrix implies that the images of the normal monomials on Θ form a basis of Hom_{R} (R⟨X⟩, k) = Ext_{R} (k, k) . Thus, the homomorphism ι′ is surjective, and the images of the normal monomials are linearly independent. By (4) and Remark 10.1.3.4, these monomials generate U_{k}(π(R)), so we conclude that ι′ is an isomorphism, as desired. □

Sjödin [144] shows how to compute the Lie operations on π^{1}(R).

Example 10.2.2. Let R = Q/I, where (Q, 𝔫, k) is regular, 𝔫 is minimally generated by s_{1}, …, s_{e}, and I is minimally generated by f_{1}, …, f_{r}, with

f_{j} = ∑_{1 ⩽ h ⩽ i ⩽ e} a_{hi,j}s_{h}s_{i} with a_{hi,j} ∈ Q for 1 ≤ j ≤ r .

Using overbars to denote images in R, and setting z_{j} = ∑_{h ⩽ i} \overline{a}_{hi,j}\overline{s}_{h}x_{i}, we see that the acyclic closure R⟨X⟩ of k over R is then obtained from

R⟨x_{1}, …, x_{e+r} | ∂(x_{i}) = \overline{s}_{i} for 1 ≤ i ≤ e; ∂(x_{e+j}) = z_{j} for 1 ≤ j ≤ r⟩

by adjunction of Γ-variables of degree ≥ 3. Let ϑ_{1}, … ϑ_{e} be the Γ-derivations of R⟨x_{1}, …, x_{e}⟩, defined by ϑ_{i}(x_{h}) = δ_{ih}. To extend them to Γ-derivations of R⟨x_{1}, …, x_{e+r}⟩, such that ∂ϑ_{i} = −ϑ_{i}∂, note that

ϑ_{i}∂(x_{e+j}) = ϑ_{i}( ∑_{h ⩽ i} \overline{a}_{hi,j}\overline{s}_{h}x_{i}) = ∑_{h=1}^{i} \overline{a}_{hi,j}\overline{s}_{h} = ∂( ∑_{h=1}^{i} \overline{a}_{hi,j}x_{h}) ,

and set ϑ_{i}(x_{e+j}) = − ∑_{h} \overline{a}_{hi,j}x_{h} for 1 ≤ i ≤ e and 1 ≤ j ≤ r. This yields

[ϑ_{h}, ϑ_{i}](x_{e+j}) = −\overline{a}_{hi,j} for h < i  and  ϑ_{i}^{[2]}(x_{e+j}) = −\overline{a}_{ii,j} .

Thus, on the basis elements of Theorem 10.2.1, the Lie bracket π^{1}(R) × π^{1}(R) ⟶ π^{2}(R) and reduced square π^{1}(R) ⟶ π^{2}(R) are given by

[θ_{h}, θ_{i}] = − ∑_{j=1}^{r} a′_{hi,j}θ_{e+j} for h < i  and  (θ_{i})^{[2]} = − ∑_{j=1}^{r} a′_{ii,j}θ_{e+j}

where a′ denotes the image in k of a ∈ R.

Consider the k–subspace of π^{2}(R), spanned by the commutators and squares of all elements of π^{1}(R) (in fact, the squares suffice, cf. 10.1.2). By the preceding computation, its rank q is equal to that of the \binom{e+1}{2} × r matrix (a_{hi,j}) reduced modulo 𝔫, that is q = \binom{e+1}{2}−rank_{k}(I/I ∩𝔪^{3}). In particular, π^{1}(R) generates π^{2}(R) if and only if the r quadratic forms f_{j} = ∑_{h ⩽ i} a_{hi,j}s_{h}s_{i} are linearly independent in gr_{𝔫}(Q). At the other extreme, the Lie subalgebra of π(R) generated by π^{1}(R) is reduced to π^{1}(R) itself if and only if I ⊆ 𝔪^{3}.

Example 10.2.3. By Theorems 10.2.1.2 and 7.3.3, if π(R) is finite dimensional, then R is a complete intersection and π(R) is concentrated in degrees 1 and 2, so the preceding example determines its structure. The Lie subalgebra π^{⩾ 2}(R) is central in π(R), and its universal enveloping algebra is the polynomial ring P = k[θ_{1}, …, θ_{r}]. An isomorphism of P–modules Ext_{R} (k, k) ≅ P ⊗_{k} E, where E is the vector space underlying the exterior algebra on π^{1}(R) ≅ Hom_{k} (𝔪/𝔪^{2}, k), refines the equality P_{k}^{R}(t) = (1 + t)^{e}/(1 − t^{2})^{r}.

Conversely, each graded Lie algebra 𝔤 with rank_{k} 𝔤^{1} ≥ rank_{k} 𝔤^{2} and 𝔤^{n} = 0 for n ≠ 1, 2 is of the form π(R) for an appropriate complete intersection: one starts by fixing the desired quadratic parts g_{j} = ∑_{h ⩽ i} a_{hi,j}s_{h}s_{i} of the relations, and uses a ‘prime avoidance’ argument to find elements p_{j} in a high power of 𝔪, such that the sequence g_{1} + p_{1}, …, g_{r} + p_{r} is regular, cf. [144].

We conclude with some general remarks on the homotopy Lie algebra. A detailed study belongs to a different exposition.

Remark 10.2.4. A local homomorphism of local rings φ: R ⟶ S induces a homomorphism of graded Lie algebra π(φ): π(S) ⟶ π(R) ⊗_{k} ℓ, where ℓ is the residue field of S. This yields a contravariant functor with remarkable properties. For example, if φ is flat, then for each i there is an exact sequence

[figure: 0; π^{2i−1}(S/𝔪S); π^{2i−1}(S); π^{2i−1}(R) ⊗_{k} ℓ; ð^{2i−1}; π^{2i}(S/𝔪S); π^{2i}(S); π^{2i}(R) ⊗_{k} ℓ; 0]

where ð^{2i−1} = 0 for almost all i, and ∑_{i=0}^{∞} rank ð^{2i−1} ≤ codepth(S/𝔪S): this is proved (in dual form) in [20] and [9]. The Lie algebra π(R) is a looking glass version of the Lie algebra of rational homotopy groups in algebraic topology, cf. [23] and [33] for a systematic discussion.

Remark. The original construction of π(R) proceeded in two steps.

The first, initiated by Assmus [15], and completed by Levin [108] and Schoeller [140], constructs a homomorphism Δ: Tor^{R} (k, k) ⟶ Tor^{R} (k, k) ⊗_{k} Tor^{R} (k, k) of Γ-algebras, giving Tor^{R} (k, k) a structure of Hopf algebra. The second identifies the composition product as the dual of Δ under the isomorphism of Hopf algebras Ext_{R} (k, k) = Hom_{k} (Tor^{R} (k, k) , k).

At that point, a structure theorem due to Milnor and Moore [121] in characteristic 0 and to André in characteristic p > 0 (adjusted by Sjödin [145] for p = 2) shows that such a Hopf algebra is the universal enveloping algebra of graded Lie algebra. In fact, these results prove much more: namely, an equivalence (of certain subcategories) of the categories of Γ-Hopf algebras and graded Lie algebras, given in one direction by the universal enveloping algebra functor.

Remark. A graded Lie algebra H^{∗}(R, k, k) is attached to R by the simplicially defined tangent cohomology of André and Quillen. There is a homomorphism of graded Lie algebras H^{∗}(R, k, k) ⟶ π(R), cf. [4]. It is bijective for complete intersections, or when char(k) = 0, cf. [133]; when char(k) = p > 0, this holds in degrees ≤ 2p, but not always in degree 2p+1, cf. [7]. The computation of H^{∗}(R, k, k) is very difficult in positive characteristic: for the small ring R = 𝔽_{2}[s_{1}, s_{2}]/(s_{1}^{2}, s_{1}s_{2}, s_{2}^{2}) it requires the book of Goerss [73]; for comparison, π(R) is the free Lie algebra on the 2-dimensional vector space π^{1}(R).

10.3. Applications. Once again, (R, 𝔪, k) denotes a local ring.

We relate this chapter to the bulk of the notes by discussing two kinds of applications of π(R) to the study of resolutions. The structure of π(R) is reflected in the Poincaré series of finite R–modules. To illustrate the point, we characterize Golod rings in terms of their homotopy Lie algebras.

Example 10.3.1. Let V be a vector space over k. A graded Lie algebra 𝔤 is free on V, if V ⊆ 𝔤 and each degree zero k–linear map from V to a graded Lie algebra 𝔥 extends uniquely to a homomorphism of Lie algebras 𝔤 ⟶ 𝔥.

It is easy to see that free Lie algebras exist on any V: just take 𝔤 to be the subspace of the tensor algebra T_{k}(V), spanned by all commutators of elements of V, and all squares of elements of V_{odd} . Using the universal property of T_{k}(V) one sees that 𝔤 is a free Lie algebra on V, and comparing it with that of U_{k}(𝔤) one concludes that these two algebras coincide.

Avramov [21] and Löfwall [113] prove that R is Golod if and only if π^{⩾ 2}(R) is free, and then it is the free Lie algebra on V = Hom_{k} (Σ H_{⩾ 1}(K^{R}), k).

This nicely ‘explains’ the formula (5.0.1) for the Poincaré series of k over a Golod ring: f(t) = (1 + t)^{e} is the Hilbert series of the vector space E underlying the exterior algebra on π^{1}(R) ≅ Hom_{k} (𝔪/𝔪^{2}, k); the expression g(t) = 1/(1−∑_{i} rank H_{i}(K^{R})t^{i+1}) is the Hilbert series of T , the tensor algebra on V, and P_{k}^{R}(t) = f(t)g(t) reflects an isomorphism Ext_{R} (k, k) ≅ T ⊗_{k} E of T –modules.

The bracket and the square in a free Lie algebra are as non-trivial as possible, so the cohomological descriptions of Golod rings above and of complete intersections in Example 10.2.3 put a maximal distance between them (compare Remark 8.1.1.3). However, there exists a level at which these descriptions coalesce: the Lie algebra π^{⩾ 3}(R) is free, because it is trivial in the first case, and because freeness is inherited by Lie subalgebras, cf. [105], in the second.

Remark 10.3.2. It is proved in [28], using results from [24], that the following conditions on a local ring R are equivalent:

(i) π(R) contains a free Lie subalgebra of finite codimension; (ii) for some r ∈ ℤ, the Lie algebra π^{⩾ r}(R) is free; (iii) for some s ∈ ℤ, the DG algebra R⟨X_{< s}⟩ admits a trivial Massey operation,

cf. Remark 5.2.1.

When they hold, the ring R is called generalized Golod (of level ≤ s). Such rings abound in small codepth: this is the case when edim R − depth R ≤ 3 (Avramov, Kustin, and Miller [36]), or when edim R − depth R = 4 and R is Gorenstein (Jacobsson, Kustin, and Miller [94]) or an almost complete intersection (Kustin and Palmer [102]). Kustin [99], [100] proves that certain determinantal relations define generalized Golod rings.

Theorem 10.3.3. Let R be a generalized Golod ring of level ≤ s.

(1) There is a polynomial den(t) ∈ ℤ[t], and for each finite R–module M there is a polynomial q(t) ∈ ℤ[t], such that P_{M}^{R}(t) = q(t)/den(t); the numerator for M = k divides ∏_{2i+1<s}(1 + t^{2i+1})^{ε_{2i+1}(R)}.

(2) If cx_{R} M = ∞, then curv_{R} M = β > 1 and there is a real number α such that β_{n}^{R}(M) ∼ αβ^{n}. ■

The first part is proved by Avramov [28], the second by Sun [150]. Both use, among other things, a theorem of Gulliksen [81] that extends Remark 9.2.6.

Theorem 10.3.4. Let R⟨X⟩ be an acyclic closure of k over R. If M is a finite R–module, then for each n ≥ 1 there is a polynomial h_{n}(t) ∈ ℤ[t], such that

∑_{i=0}^{∞} rank_{k} H_{i}(M ⊗_{R} R⟨X_{⩽ n}⟩)t^{i} = \frac{h_{n}(t)}{∏_{2j≤n}(1 − t^{2j})^{ε_{2j}(R)}} .  ■

Theorem 10.3.3, or results that it generalizes, has been used in essentially all cases when the Poincaré series is known to be rational for all finite modules. It is difficult to resist asking the next question; a positive answer would be unexpected and very useful; a negative one might be equally interesting, since it would most likely involve unusual constructions.

Problem 10.3.5. If P_{M}^{R}(t) is rational for each M, is then R generalized Golod?

Next we describe applications of π(R) that do not make specific assumptions on its form. They use the following easy consequence of Theorem 10.2.1.

Remark 10.3.6. Let 𝔥 be a graded Lie subalgebra of π(R). By completing a basis of 𝔥 to one of π(R), and considering the corresponding basis of normal monomials of U_{k}(π(R)) = Ext_{R} (k, k) , cf. Theorem 10.2.1, one easily sees that there is an isomorphism U_{k}(𝔤) ≅ U_{k}(𝔥) ⊗_{k} V of left modules over U_{k}(𝔥), where V is the tensor product of the exterior algebra on (𝔤/𝔥)^{odd} with the symmetric algebra on (𝔤/𝔥)^{even}. By a simple count of basis elements,

H_{V}^{k} (t) = \frac{∏_{i=1}^{∞}(1 + t^{2i−1})^{ℓ^{2i−1}}}{∏_{i=1}^{∞}(1 − t^{2i})^{ℓ^{2i}}} with ℓ^{n} = rank_{k}(𝔤/𝔥)^{n} .

For each R–module M, the vector space Ext_{R} (M, k) is a left module over the universal enveloping algebra Ext_{R} (k, k) : it suffices to make the obvious changes in the construction of cohomology products in Section 10.1. This module structure is the essential tool in the proof of the next result. In fact, it has already been used throughout Chapter 9, in a different guise: over a complete intersection, the actions on Ext_{R} (M, k) of the graded algebras denoted P in Remark 9.2.6 and in Remark 10.2.3 are the same, cf. [37].

Motivated by Proposition 4.2.4.1, we say that a module L over a local ring (R, 𝔪, k) is extremal, if cx_{R} L = cx_{R} k and curv_{R} L = curv_{R} k. For instance, Theorems 8.3.3 and 8.3.4 show that (in the graded characteristic zero case) the conormal module and the module of differentials are extremal when R is not a complete intersection. Results from [29] add more instances, among them:

Theorem 10.3.7. If R is a local ring of embedding dimension e, M is a finite R–module and L is a submodule such that L ⊇ 𝔪M, then

P_{L}^{R}(t) · (1 + t)^{e} ≽ P_{k}^{R}(t) · rank_{k} (\frac{𝔪M}{𝔪L} ) .

For 𝔪^{i}M ⊆ 𝔪^{i−1}M, we get a quantitative version of Levin’s characterization of regularity [108]: If 𝔪M ≠ 0 and pd_{R}(𝔪M) < ∞, then R is regular:

Corollary 10.3.8. If 𝔪^{i}M ≠ 0 for some i ≥ 1, then 𝔪^{i}M is extremal. □

Corollary 10.3.9. Each non-zero R–module M ≠ k may be obtained as an extension of an extremal R–module by another such module. In particular, the extremal R–modules generate the Grothendieck group of R. □

Here is another class of extremal modules from [29].

Remark 10.3.10. If N ≠ 0 is a homomorphic image of a finite direct sum of syzygies of k, then cx_{R} N = cx_{R} k and curv_{R} N = curv_{R} k. As a consequence, we get a result of Martsinkovsky [116]: pd_{R} N = ∞, unless R is regular.

Proof of Theorem 10.3.7. The commutative diagram of R–modules

0 ⟶ L ⟶ M ⟶ N ⟶ 0
↓  ↓  ∥
0 ⟶ \frac{L}{𝔪L} ⟶ \frac{M}{𝔪L} ⟶ N ⟶ 0

induces a commutative square of homomorphisms of graded left modules

Ext_{R} (L, k) ⟶^{ð′} Ext_{R} (N, k)
↑  ∥
Ext_{R} (\frac{L}{𝔪L}, k) ⟶^{ð} Ext_{R} (N, k)

over E = Ext_{R} (k, k) , where ð′ and ð are connecting maps of degree 1. As 𝔪 annihilates L/𝔪L and N, we have isomorphisms

Ext_{R} (\frac{L}{𝔪L}, k) ≅ E ⊗_{k} Hom_{k} (\frac{L}{𝔪L}, k)

Ext_{R} (N, k) ≅ E ⊗_{k} Hom_{k} (N, k)

of graded E–modules. By Remark 10.3.6, E ≅ U ⊗_{k} ⋀(π^{1}(R)) as graded left modules over the subalgebra U = U_{k}(π^{⩾ 2}R). Noting that π^{1}(R) = E^{1}, we can rewrite ð as the top map of the commutative diagram

U ⊗_{k} ⋀(E^{1}) ⊗_{k} Hom_{k} (\frac{L}{𝔪L}, k) ⟶^{ð} U ⊗_{k} ⋀(E^{1}) ⊗_{k} Hom_{k} (N, k)
⋃|↑  ↑^{⋃|}
U ⊗_{k} Hom_{k} (\frac{L}{𝔪L}, k)  ⟶ U ⊗_{k} E^{1} ⊗_{k} Hom_{k} (N, k)
∥  ↑≅
U ⊗_{k} Hom_{k} (\frac{L}{𝔪L}, k)  ⟶^{U⊗_{k}ð^{0}}  U ⊗_{k} Ext_{R}^{1} (N, k) .

of homomorphisms of graded left U–modules.

The preceding information combines to yield a commutative square

Ext_{R} (L, k)  ⟶^{ð′}  Ext_{R} (N, k)
↑  ↑
U ⊗_{k} Hom_{k} (\frac{L}{𝔪L}, k) ⟶^{U⊗_{k}ð^{0}} U ⊗_{k} Ext_{R}^{1} (N, k)

of homomorphisms of graded U–modules, with injective right hand vertical arrow. Thus, Ext_{R} (N, k) contains a copy of the free module Σ(U) ⊗_{k} Im ð_{0}. By the commutativity of the square, Ext_{R} (L, k) contains a copy of U ⊗_{k} Im ð^{0}.

On the other hand, a length count in the cohomology exact sequence

0 ⟶ Hom_{R} (N, k) ⟶ Hom_{R} (\frac{M}{𝔪L}, k) ⟶ Hom_{R} (\frac{L}{𝔪L}, k) ⟶^{ð^{0}} Ext_{R}^{1} (N, k)

yields rank_{k} ð^{0} = rank_{k}(𝔪M/𝔪L). Thus, P_{L}^{R}(t) ≽ rank_{k}(𝔪M/𝔪L) · H_{U}^{k}(t). To finish the proof, multiply this inequality by (1 + t)^{e}, then simplify the right hand side by using the equality (1 + t)^{e} · H_{U}^{k}(t) = P_{k}^{R}(t) from 10.3.6. □

References

[1] S. S. Abhyankar, Local rings of high embedding dimension, Amer. J. Math. 89 (1967), 1073–1077.

[2] J. Alperin, L. Evens, Representations, resolutions, and Quillen’s dimension theorem, J. Pure Appl. Algebra 22 (1981), 1–9.

[3] M. André, Méthode simpliciale en algèbre homologique et algèbre commutative, Lecture Notes Math. 32, Springer, Berlin, 1967.

[4] M. André, L’algèbre de Lie d’un anneau local, Symp. Math. 4, (INDAM, Rome, 1968/69), Academic Press, London, 1970; pp. 337–375.

[5] M. André, Hopf algebras with divided powers, J. Algebra 18 (1971), 19–50.

[6] M. André, Homologie des algèbres commutatives, Grundlehren Math. Wiss. 204, Springer, Berlin, 1974.

[7] M. André, La (2p + 1)ème déviation d’un anneau local, Enseignement Math. (2) 23 (1977), 239–248.

[8] M. André, Algèbre homologique des anneaux locaux à corps résiduels de caractéristique deux, Sém. d’Algèbre P. Dubreil, Paris, 1979 (M.-P. Malliavin, ed.), Lecture Notes Math. 740, Springer, Berlin, 1979; pp. 237–242.

[9] M. André, Le caractère additif des déviations des anneaux locaux, Comment. Math. Helv. 57 (1982), 648–675.

[10] D. Anick, Constructions d’espaces de lacets et d’anneaux locaux à séries de Poincaré-Betti non rationnelles, C. R. Acad. Sci. Paris Sér. A 290 (1980), 729–732.

[11] D. Anick, Counterexample to a conjecture of Serre, Ann. of Math. (2) 115 (1982), 1–33; Comment, ibid. 116 (1982), 661.

[12] D. Anick, Recent progress in Hilbert and Poincaré series, Algebraic topology. Rational homotopy, Louvain-la-Neuve, 1986 (Y. Félix, ed.), Lecture Notes Math. 1318, Springer, Berlin, 1988; pp. 1–25.

[13] D. Anick, T. H. Gulliksen, Rational dependence among Hilbert and Poincaré series, J. Pure Appl. Algebra 38 (1985), 135–158.

[14] A. G. Aramova, J. Herzog, Koszul cycles and Eliahou-Kervaire type resolutions, J. Algebra 181 (1996), 347–370.

[15] E. F. Assmus, Jr., On the homology of local rings, Illinois J. Math. 3 (1959), 187–199.

[16] M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631– 647.

[17] M. Auslander, D. A. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625–657; Corrections, ibid, 70 (1959), 395–397.

[18] L. L. Avramov, On the Hopf algebra of a local ring, Math USSR-Izv. 8 (1974), 259–284; [translated from:] Izv. Akad. Nauk. SSSR, Ser. Mat. 38 (1974), 253–277 [Russian].

[19] L. L. Avramov, Flat morphisms of complete intersections, Soviet Math. Dokl. 16 (1975), 1413–1417; [translated from:] Dokl. Akad. Nauk. SSSR, 225 (1975), 11–14 [Russian].

[20] L. L. Avramov, Homology of local flat extensions and complete intersection defects, Math. Ann. 228 (1977), 27–37.

[21] L. L. Avramov, Small homomorphisms of local rings, J. Algebra 50 (1978), 400–453.

[22] L. L. Avramov, Obstructions to the existence of multiplicative structures on minimal free resolutions, Amer. J. Math. 103 (1981), 1–31.

[23] L. L. Avramov, Local algebra and rational homotopy, Homotopie algébrique et algèbre locale; Luminy, 1982 (J.-M. Lemaire, J.-C. Thomas, eds.) Astérisque 113-114, Soc. Math. France, Paris, 1984; pp. 15–43.

[24] L. L. Avramov, Golod homomorphisms, Algebra, algebraic topology, and their interactions; Stockholm, 1983 (J.-E. Roos, ed.), Lecture Notes Math. 1183, Springer, Berlin, 1986; pp. 56–78.

[25] L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), 71–101.

[26] L. L. Avramov, Homological asymptotics of modules over local rings, Commutative algebra; Berkeley, 1987 (M. Hochster, C. Huneke, J. Sally, eds.), MSRI Publ. 15, Springer, New York 1989; pp. 33–62.

[27] L. L. Avramov, Problems on infinite free resolutions, Free resolutions in commutative algebra and algebraic geometry; Sundance, 1990 (D. Eisenbud, C. Huneke, eds.), Res. Notes Math. 2, Jones and Bartlett, Boston 1992, pp. 3–23.

[28] L. L. Avramov, Local rings over which all modules have rational Poincaré series, J. Pure Appl. Algebra 91 (1994), 29–48.

[29] L. L. Avramov, Modules with extremal resolutions, Math. Res. Lett. 3 (1996), 319–328.

[30] L. L. Avramov, Locally complete intersection homomorphisms, and a conjecture of Quillen on the vanishing of cotangent homology, Ann. of Math. (to appear).

[31] L. L. Avramov, R.-O. Buchweitz, Homological algebra modulo a regular sequence with special attention to codimension two, Preprint, 1998.

[32] L. L. Avramov, V. N. Gasharov, I. V. Peeva, Complete intersection dimension, Publ. Math. I.H.E.S. 86 (1997), 67–114.

[33] L. L. Avramov, S. Halperin, Through the looking glass: A dictionary between rational homotopy theory and local algebra, Algebra, algebraic topology, and their interactions; Stockholm, 1983 (J.-E. Roos, ed.), Lecture Notes Math. 1183, Springer, Berlin, 1986; pp. 1–27.

[34] L. L. Avramov, S. Halperin, On the non-vanishing of cotangent cohomology, Comment. Math. Helv. 62 (1987), 169–184.

[35] L. L. Avramov, J. Herzog, Jacobian criteria for complete intersections. The graded case, Invent. Math. 117 (1994), 75–88.

[36] L. L. Avramov, A. R. Kustin, and M. Miller, Poincaré series of modules over local rings of small embedding codepth or small linking number, J. Algebra 118 (1988), 162–204.

[37] L. L. Avramov, L.-C. Sun, Cohomology operators defined by a deformation, J. Algebra 204 (1998), 684–710.

[38] I. K. Babenko, On the analytic properties of Poincaré series of loop spaces, Math. Notes 29 (1980), 359–366; [translated from:] Mat. Zametki 27 (1980), 751–765 [Russian].

[39] I. K. Babenko, Problems of growth and rationality in algebra and topology, Russian Math. Surv. 29 (1980), no. 2, 95–142; [translated from:] Uspekhi Mat. Nauk 41 (1986), no. 2, 95–142 [Russian].

[40] D. Bayer, M. E. Stillman, Macaulay, A computer algebra system for computing in Algebraic Geometry and Computer Algebra, 1990; available via anonymous ftp from

    zariski.harvard.edu.

[41] D. Benson, Representations and cohomology. I; II, Cambridge Stud. Adv. Math. 31; 32, Cambridge Univ. Press, Cambridge, 1991.

[42] A. Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra 21 (1993), 2317–2334.

[43] A. Blanco, J. Majadas, A. G. Rodicio, On the acyclicity of the Tate complex, J. Pure Appl. Algebra (to appear).

[44] R. Bøgvad, Gorenstein rings with transcendental Poincaré series, Math. Scand. 53 (1983), 5–15.

[45] N. Bourbaki, Algèbre, X. Algèbre homologique, Masson, Paris, 1980.

[46] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1993.

[47] D. A. Buchsbaum, D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), 447–485.

[48] R.-O. Buchweitz, G.-M. Greuel, F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II, Invent. Math. 88 (1987), 165–182.

[49] L. Carroll, Through the looking glass and what Alice found there, Macmillan, London, 1871.

[50] H. Cartan, Algèbres d’Eilenberg-MacLane, Exposés 2 à 11, Sém. H. Cartan, Éc. Normale Sup. (1954–1955), Secrétariat Math., Paris, 1956; [reprinted in:] Œvres, vol. III, Springer, Berlin, 1979; pp. 1309–1394.

[51] H. Cartan, S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, NJ, 1956.

[52] S. Choi, Betti numbers and the integral closure of ideals, Math. Scand. 66 (1990), 173–184.

[53] S. Choi, Exponential growth of Betti numbers, J. Algebra 152 (1992), 20–29.

[54] J. A. Eagon, M. Fraser, A note on the Koszul complex, Proc. Amer. Math. Soc. 19 (1968), 251–252.

[55] S. Eilenberg, Homological dimension and syzygies, Ann. of Math. (2) 64 (1956), 328–336.

[56] S. Eilenberg, S. MacLane, On the groups H(Π, n). I, Ann. of Math. (2) 58 (1953), 55–106.

[57] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.

[58] D. Eisenbud, Commutative algebra, with a view towards algebraic geometry, Graduate Texts Math. 150, Springer, Berlin, 1995.

[59] D. Eisenbud, S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89–133.

[60] S. Eliahou, M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 1–25.

[61] E. G. Evans, P. Griffith, Syzygies, London Math. Soc. Lecture Notes Ser. 106, Cambridge Univ. Press, Cambridge, 1985.

[62] L. Evens, The cohomology of groups, Oxford Math. Monographs, Clarendon Press, Oxford, 1991.

[63] C. T. Fan, Growth of Betti numbers over noetherian local rings, Math. Scand. 75 (1994), 161–168.

[64] Y. Félix, S. Halperin Rational L.-S. category and its applications, Trans. Amer. Math. Soc. 273 (1982), 1–37.

[65] Y. Félix, S. Halperin, C. Jacobsson, C. Löfwall, J.-C. Thomas, The radical of the homotopy Lie algebra, Amer. J. Math., 110 (1988), 301–322.

[66] Y. Félix, S. Halperin, J.-C. Thomas, The homotopy Lie algebra for finite complexes, Publ. Math. I.H.E.S. 56 (1982), 179–202.

[67] Y. Félix, J.-C. Thomas, The radius of convergence of the Poincaré series of loop spaces, Invent. Math. 68 (1982), 257–274.

[68] D. Ferrand, Suite régulière et intersection complète, C. R. Acad. Sci. Paris Sér. A 264 (1967), 427–428.

[69] R. Fröberg, T. H. Gulliksen, C. Löfwall, Flat families of local, artinian algebras, with an infinite number of Poincaré series, Algebra, algebraic topology, and their interactions; Stockholm, 1983 (J.-E. Roos, ed.), Lecture Notes Math. 1183, Springer, Berlin, 1986; pp. 56–78.

[70] V. N. Gasharov, I. V. Peeva, Boundedness versus periodicity over commutative local rings, Trans. Amer. Math. Soc, 320 (1990), 569–580.

[71] I. M. Gelfand, A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Publ. Math. I.H.E.S., 31 (1966), 509–523.

[72] F. Ghione, T. H. Gulliksen, Some reduction formulas for Poincaré series of modules, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), 82–91.

[73] P. G. Goerss, On the André-Quillen cohomology of commutative 𝔽_{2}–algebras, Astérisque 186, Soc. Math. France, Paris, 1990.

[74] E. S. Golod, On the homologies of certain local rings, Soviet Math. Dokl. 3 (1962), 745–748; [translated from:] Dokl. Akad. Nauk. SSSR, 144 (1962), 479–482 [Russian].

[75] E. H. Gover, M. Ramras, Increasing sequences of Betti numbers, Pacific J. Math. 87 (1980), 65–68.

[76] V. K. A. M. Gugenheim, J. P. May, On the theory and applications of differential torsion products, Mem. Amer. Math. Soc, 142, Amer. Math. Soc., Providence, RI, 1974.

[77] T. H. Gulliksen, A proof of the existence of minimal algebra resolutions, Acta Math. 120 (1968), 53–58.

[78] T. H. Gulliksen, A homological characterization of local complete intersections, Compositio Math. 23 (1971), 251–255.

[79] T. H. Gulliksen, Massey operations and the Poincaré series of certain local rings, J. Algebra 22 (1972), 223–232.

[80] T. H. Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167–183.

[81] T. H. Gulliksen, On the Hilbert series of the homology of differential graded algebras, Math. Scand. 46 (1980), 15–22.

[82] T. H. Gulliksen, On the deviations of a local ring, Math. Scand. 47 (1980), 5–20.

[83] T. H. Gulliksen, G. Levin, Homology of local rings, Queen’s Papers Pure Appl. Math. 20, Queen’s Univ., Kingston, ON, 1969

[84] S. Halperin, On the non-vanishing of the deviations of a local ring, Comment. Math. Helv. 62 (1987), 646–653.

[85] R. Heitmann, A counterexample to the rigidity conjecture for rings, Bull. Amer. Math. Soc. (New Ser.) 29 (1993), 94–97.

[86] J. Herzog, Komplexe, Auflösungen, und Dualität in der lokalen Algebra, Habilitationsschrift, Regensburg, 1973.

[87] J. Herzog, B. Ulrich, J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 (1991), 187–201.

[88] D. Hilbert, Über die Theorie der algebraischen Formen, Math. Ann. 36 (1890) 473–534; [reprinted in:] Gesammelte Abhandlungen, Band II: Algebra, Invariantentheorie, Geometrie, Springer, Berlin, 1970; pp. 199–257.

[89] M. Hochster, Topics in the homological study of modules over commutative rings, CBMS Regional Conf. Ser. in Math. 24, Amer. Math. Soc., Providence, RI, 1975.

[90] H. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. Algebra 21 (1993), 2335–2350.

[91] C. Huneke, R. Wiegand, Tensor products of modules, rigidity and local cohomology, Math. Scand. 81 (1997), 161–183.

[92] S. Iyengar, Free resolutions and change of rings, J. Algebra, 190 (1997), 195–213.

[93] C. Jacobsson, Finitely presented graded Lie algebras and homomorphisms of local rings, J. Pure Appl. Algebra 38 (1985), 243–253.

[94] C. Jacobsson, A. R. Kustin, and M. Miller, The Poincaré series of a codimension four Gorenstein ideal is rational, J. Pure Appl. Algebra 38 (1985), 255–275.

[95] D. A. Jorgensen, Complexity and Tor on a complete intersection, J. Algebra 195 (1997), 526–537.

[96] R. Kiehl, E. Kunz, Vollständige Durchschnitte und p-Basen, Arch. Math. (Basel), 16 (1965), 348–362.

[97] A. R. Kustin, Gorenstein algebras of codimension four and characteristic two, Comm. Algebra 15 (1987), 2417-2429.

[98] A. R. Kustin, The minimal resolution of a codimension four almost complete intersection is a DG algebra, J. Algebra 168 (1994), 371–399.

[99] A. R. Kustin, The deviation two Gorenstein rings of Huneke and Ulrich, Commutative algebra, Trieste, 1994 (A. Simis, N. V. Trung, G. Valla, eds.), World Scientific, Singapore, 1994; pp. 140-163.

[100] A. R. Kustin, Huneke-Ulrich almost complete intersections of Cohen-Macaulay type two, J. Algebra 174 (1995), 373–429.

[101] A. R. Kustin, M. Miller, Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four, Math. Z. 173 (1980), 171–184.

[102] A. R. Kustin, S. Palmer Slattery, The Poincaré series of every finitely generated module over a codimension four almost complete intersection is a rational function, J. Pure Appl. Algebra 95 (1994), 271–295.

[103] T. Larfeldt, C. Lech, Analytic ramification and flat couples of local rings, Acta Math. 146 (1981), 201–208.

[104] C. Lech, Inequalities related to certain couples of local rings, Acta Math. 112 (1964), 69–89.

[105] J.-M. Lemaire, Algèbres connexes et homologie des espaces de lacets, Lecture Notes Math. 422, Springer, Berlin, 1974.

[106] J. Lescot, Asymptotic properties of Betti numbers of modules over certain rings, J. Pure Appl. Algebra 38 (1985), 287–298.

[107] J. Lescot, Séries de Poincaré et modules inertes, J. Algebra 132 (1990), 22–49.

[108] G. Levin, Homology of local rings, Ph. D. Thesis, Univ. of Chicago, Chicago, IL, 1965.

[109] G. Levin, Local rings and Golod homomorphisms, J. Algebra 37 (1975), 266–289.

[110] G. Levin, Lectures on Golod homomorphisms, Matematiska Istitutionen, Stockholms Universitet, Preprint 15, 1975.

[111] G. Levin, Modules and Golod homomorphisms, J. Pure Appl. Algebra 38 (1985), 299–304.

[112] S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220–226.

[113] C. Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Extalgebra, Algebra, algebraic topology, and their interactions; Stockholm, 1983 (J.-E. Roos, ed.), Lecture Notes Math. 1183, Springer, Berlin, 1986; pp. 291–338.

[114] C. Löfwall, J.-E. Roos, Cohomologie des algèbres de Lie graduées et séries de Poincaré-Betti non-rationnelles, C. R. Acad. Sci. Paris Sér. A 290 (1980), 733–736.

[115] S. MacLane, Homology, Grundlehren Math. Wiss. 114 Springer, Berlin, 1967.

[116] A. Martsinkovsky, A remarkable property of the (co)syzygy modules of the residue field of a non-regular local ring, J. Pure Appl. Algebra 111 (1996), 9–13.

[117] H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1986.

[118] J. P. May, Matric Massey products, J. Algebra 12 (1969), 533–568.

[119] V. Mehta, Endomorphisms of complexes and modules over Golod rings, Ph. D. Thesis, Univ. of California, Berkeley, CA, 1976.

[120] C. Miller, Complexity of tensor products of modules and a theorem of Huneke-Wiegand, Proc. Amer. Math. Soc. 126 (1998), 53–60.

[121] J. W. Milnor, J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264.

[122] J. C. Moore Algèbre homologique et homologie des espaces classifiants, Exposé 7, Sém. H. Cartan, Éc. Normale Sup. (1959–1960), Sectétariat Math., Paris, 1957.

[123] M. P. Murthy, Modules over regular local rings, Illinois J. Math. 7 (1963), 558–565.

[124] M. Nagata, Local rings, Wiley, New York, 1962.

[125] D. G. Northcott, Finite free resolutions, Tracts in Pure Math., 71, Cambridge Univ. Press, Cambridge, 1976.

[126] S. Okiyama, A local ring is CM if and only if its residue field has a CM syzygy, Tokyo J. Math. 14 (1991), 489–500.

[127] S. Palmer Slattery, Algebra structures on resolutions of rings defined by grade four almost complete intersections, J. Algebra 168 (1994), 371–399.

[128] K. Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), 564–585.

[129] I. Peeva, 0-Borel fixed ideals, J. Algebra 184 (1996), 945–984.

[130] I. Peeva, Exponential growth of Betti numbers, J. Pure. Appl. Algebra 126 (1998), 317–323.

[131] D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295.

[132] D. Quillen, The spectrum of an equivariant cohomology ring I; II, Ann. of Math. (2) 94 (1971), 549–572; 573–602.

[133] D. Quillen, On the (co-)homology of commutative rings, Applications of categorical algebra; New York, 1968 (A. Heller, ed.), Proc. Symp. Pure Math. 17, Amer. Math. Soc., Providence, RI, 1970; pp. 65–87.

[134] M. Ramras, Sequences of Betti numbers, J. Algebra 66 (1980), 193–204.

[135] M. Ramras, Bounds on Betti numbers, Can. J. Math. 34 (1982), 589–592.

[136] P. Roberts, Homological invariants of modules over commutative rings, Sém. Math. Sup., 72, Presses Univ. Montréal, Montréal, 1980.

[137] J.-E. Roos, Relations between the Poincaré-Betti series of loop spaces and local rings, Sém. d’Algèbre P. Dubreil; Paris, 1977-78 (M.-P. Malliavin, ed.), Lecture Notes Math. 740, Springer, Berlin, 1979; pp. 285–322.

[138] J.-E. Roos, Homology of loop spaces and of local rings, Proc. 18^{th} Scand. Congr. Math. Århus, 1980 (E. Balslev, ed.), Progress Math. 11, Birkhäuser, Basel, 1982; pp. 441–468.

[139] G. Scheja, Über die Bettizahlen lokaler Ringe, Math. Ann. 155 (1964), 155–172.

[140] C. Schoeller, Homologie des anneaux locaux noethériens, C. R. Acad. Sci. Paris Sér. A 265 (1967), 768–771.

[141] J.-P. Serre, Sur la dimension homologique des anneaux et des modules noethériens, Proc. Int. Symp., Tokyo-Nikko (1956), pp. 175-189.

[142] J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes Math. 11 Springer, Berlin, 1965.

[143] J. Shamash, The Poincaré series of a local ring, J. Algebra 12 (1969), 453–470.

[144] G. Sjödin, A set of generators for Ext_{R} (k, k) , Math. Scand. 38 (1976), 1–12.

[145] G. Sjödin, Hopf algebras and derivations, J. Algebra 64 (1980), 218–229.

[146] H. Srinivasan, The non-existence of a minimal algebra resolutions despite the vanishing of Avramov obstructions, J. Algebra 146 (1992), 251–266.

[147] H. Srinivasan, A grade five Gorenstein algebra with no minimal algebra resolutions, J. Algebra 179 (1996), 362–379.

[148] D. Sullivan, Infinitesimal computations in topology, Publ. Math. I.H.E.S. 47 (1978), 269– 331.

[149] L.-C. Sun, Growth of Betti numbers of modules over rings of small embedding codimension or small linkage number, J. Pure Appl. Algebra, 96 (1994), 57–71.

[150] L.-C. Sun, Growth of Betti numbers of modules over generalized Golod rings, J. Algebra 199 (1998), 88–93.

[151] J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–25.

[152] H. Uehara, W. S. Massey, The Jacobi identity for Whitehead products, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton, NJ, 1957; pp. 361–377.

[153] V. A. Ufnarovskij, Combinatorial and asymptotic methods in algebra, Encyclopaedia of Math. Sci. 57, Springer, Berlin, 1995; pp. 1–196; [translated from:] Current problems in mathematics. Fundamental directions, 57, Akad. Nauk SSSR, VINITI, Moscow, 1990. pp. 5– 177 [Russian].

[154] W. V. Vasconcelos, Ideals generated by R–sequences, J. Algebra 6 (1970), 309–316.

[155] W. V. Vasconcelos, On the homology of I/I^{2}, Comm. Algebra 6 (1978), 1801–1809.

[156] W. V. Vasconcelos, The complete intersection locus of certain ideals, J. Pure Appl. Algebra, 38 (1986), 367–378.

[157] J. Watanabe, A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227–232.

[158] H. Wiebe, Über homologische Invarianten lokaler Ringe, Math. Ann. 179 (1969), 257–274.

[159] K. Wolffhardt, Die Betti-Reihe und die Abweichungen eines lokalen Rings, Math. Z. 114 (1970), 66–78.

Index

acyclic closure, 57

graded, 78 anti-commutativity, 94 associative DG algebra, 12 Auslander-Buchsbaum Equality, 10

generalized, 96 Auslander-Buchsbaum-Serre

Theorem, 35

bar construction, 26 Betti number, 9

graded, 11

canonical bimodule structure, 55 chain map, 6 codimension, 33 codepth, 33 Cohen Structure Theorem, 34 complex, 6

bounded below, 6 minimal, 7

complexity, 36 complete intersection defect, 68 complete intersection ring, 66 composition product, 93 conormal module, 77 curvature, 37

decomposable differential, 62, 78 decomposable Γ-monomial, 56 degree, 6

internal, 11 derivation, Γ-, 55

chain, 55 universal, 56

deviation, 61 DG algebra, 11 DG algebra resolution, 15 DG Lie algebra, 95 DG module, 12 DG module resolution 19 differentials, 78 differentials, Γ- 56 divided powers variable, 51

Eisenbud operators, 84 embedding dimension, 33 essential augmentation, 36 Euler morphism, 79 Ext algebra, 93 exterior variable, 17 extremal module, 100

free resolution, 7

Golod ring, 43

generalized, 100 Golod module, 48 graded algebra, 12

module over, 12 graded commutator, 94 graded Lie algebra, 94

free, 99 graded module, 11 graded resolution, 11 graded ring, 11

higher homotopies, 25 Hilbert series, 59 homomorphism, 6, 12 homotopy, 6, 12 homotopy class, 12 homotopy Lie algebra, 96 hypersurface ring, 42

indecomposables, Γ-, 56 indexing sequence, 57, 95 irrelevant maximal ideal, 11

Jacobi identity, 94

Koszul complex, 7 Künneth Theorem, 14

Leibniz rule, 11 Lie bracket, 94 lifting property, 12, 17 local homomorphism, 9

mapping cone, 7 Massey operation, 44 matrix factorization, 42 minimal DG algebra, 62 minimal model, 63

graded, 78 minimal multiplicity ring, 47 minimal resolution, 8

graded, 11 morphism, 6, 12

normal monomial, 95 normal Γ-monomial, 55

obstructions, 29

Poincaré series, 30

two variable, 46 polynomial variable, 17 projective resolution, 7 projective dimension, 7

quasi-isomorphism, 6

rank, 31 reduced square 94 regular element, 51 regular presentation, 34 regular ring, 33 regular sequence, 53 resolvent, 18

semi-free DG module, 12 semi-free extension, 17 semi-free Γ-extension, 51 shift, 7 spectral sequence

Cartan-Eilenberg, 28 Eilenberg-Moore, 29 Lescot, 28

standard basis, 55 standard resolution, 26 syzygy, 9

Tate resolution, 52 tensor product, 6, 12 Tor algebra, 21 Tor module, 22 translate, 11

underlying module, 6 universal enveloping algebra, 94

variable, Γ , 51 virtual projective dimension , 89