HOMOLOGICAL DIMENSIONS AND RELATED INVARIANTS OF MODULES OVER LOCAL RINGS

LUCHEZAR L. AVRAMOV

Explanation

This paper evolved out of two lectures given at the Workshop of ICRA IX^{1}. In the lectures I tried to present to a group of experts in representation theory

of Artin algebras some recent developments in commutative algebra. To bridge the gap between subjects, I looked for theorems that could be appreciated directly off the blackboard in mathematically meaningful detail. The search ended close to home: many results on free resolutions over commutative local rings can be recast as statements on projective resolutions over artinian rings (that just happen to be commutative). A further slate of assertions becomes intelligible when some property of local rings (e.g. ‘Gorenstein’) is replaced by an adequate condition on artinian algebras (e.g. ‘quasi-Frobenius’). Thus, I chose to discuss with a notnecessarily-commutative audience problems whose solutions may be influenced by a commutativity hypothesis, but whose statements do not depend on one.

Community of interest and hand waving provided some coherence to the oral exposition. Work on the article rapidly convinced me that the written version needs a more robust structure. Trying to find one I deleted material, expanded a few themes, and embarked upon a series of rewritings. The text that emerged focuses on measures of resolutions such as length, size, or width, and on the ways that their values affect the homological life of modules.

I want to thank the organizers for an opportunity to present some favorite topics, the participants for their willingness to cross mathematical frontiers, and the editors for trusting, against all odds, that this paper will be completed.

Introduction

We discuss ramifications and extensions of the classical notions of projective dimension and injective dimension of modules.

Various hypotheses are made on the base ring at different places in the text. It is essentially arbitrary in Sections 2 and 4, noetherian in Section 3, and graded in Section 10. Everywhere else, ring is assumed to be local, which here is taken to mean ‘commutative noetherian ring with unique maximal ideal’, and modules are assumed to be finite, that is, finitely generated. For the convenience of some readers, in Appendix B we recall the definitions of a few basic notions of local algebra, and provide background.

Date: August 8, 2001; slightly revised October 28, 2001.

The author was partly supported by a grant from the NSF.

1For the uninitiated: ICRA stands for International Conference on Representation of Algebras; the ninth one was held in Beijing from August 20 to September 1, 2000.

Modules of finite projective dimension or of finite injective dimension over a local ring have rather special properties. Some highlights are recalled in Section 1 with a dual purpose in mind: to provide a historical perspective and to indicate the kind of results to be expected (or desired) for modules of finite generalized homological dimension. An important area of research in commutative algebra—the structure theory of finite free resolutions—is omitted because its results heavily depend on determinantal ideals, and so have no clear counterparts outside of the commutative context.

Early generalizations of the classical dimensions measured the minimal length of resolutions (respectively, coresolutions) constructed using modules from a specific class. If such a class does not consist of projective (respectively, injective) modules, then for satisfactory results one needs extra conditions either on the maps or on the modules used in the (co)resolutions. In Section 2 we review well known solutions to the problems that arise. Our conventions for handling complexes and resolutions are explained in Appendix A.

The best studied non-classical homological dimension is the Gorenstein dimension, which is the subject of Section 3. The not quite traditional approach there is based on explicit constructions of complexes, and has applications to computations of relative cohomology and Tate cohomology.

It is classically known that each module has a ‘minimal’ injective coresolutions, and that some modules—in particular, finite ones over local rings—also have minimal projective resolutions. In either case the corresponding complex is unique up to non-unique isomorphism, and splits off every (co)resolution as a direct summand with contractible direct complement. Minimal (co)resolutions are discussed in Section 4. We use a new approach, based on a notion of minimality applicable to arbitrary complexes. It allows one to define and study minimal resolutions that do not necessarily consist of projectives.

Over the last two decades, new invariants of modules have been introduced by taking into account the ‘size’ of modules appearing in resolutions or coresolutions. It is natural to measure a free module by its rank; the Betti numbers of a module are the ranks of the free modules in its minimal free resolution. A measure for the size of an injective module is defined by using the structure theorem for injective module over commutative noetherian rings; it leads to the notion of Bass numbers. In Section 5 we discuss bounds on sequences of Betti numbers and of Bass numbers.

In all cases when it is known, the asymptotic behavior of these sequences belongs to one of two types—polynomial or exponential. Sample results pertaining to these types of growth are discussed in Sections 6 and 9, respectively. A features of many of their proofs is the use of subtle analogies with ‘distant’ branches of mathematics, such as group cohomology or rational homotopy theory. The invariant used to measure the growth of resolutions on a polynomial scale is known as the complexity of the module, the one used to gauge growth on the exponential scale has been called its curvature. Besides providing classifying parameters in situations where other dimensions are infinite, the values of these asymptotic invariants impact the vanishing of (co)homology.

In Section 8 certain kinds of resolutions are paired off with specified changes of the base ring, so as to produce a variety of new homological dimensions. The rationale comes from the fundamental hierarchy of properties of local rings. Section 7 focuses on complete intersection dimension. When it is finite, precise results are available on the asymptotic behavior of free resolutions.

A graded module over a graded ring has graded Betti numbers, which can be used to measure the ‘width’ of its minimal free resolution. This new parameter leads to the notion of Castelnuovo-Mumford regularity of a module. It is the subject of the final Section 10, where we show that in the commutative case regularity behaves in some respects like another homological dimension.

Throughout the text, and to the best of my knowledge, I have tried to give references to the original publications. The intent was to provide readers from a different area with a sense of the dynamics of the developments reported below. On the other hand, systematic exposition in research monographs, textbooks, or survey articles often offer the best introduction to a field. At the end of the relevant sections I have collected references to such sources.

I want to thank Sasha Gerko, Srikanth Iyengar, and Oana Veliche for carefully reading the manuscript and providing many helpful comments.

1. Classical dimensions

In this section (R, 𝔪, k) denotes a local ring and M a finite R-module.

A hypothesis that a classical (that is, projective or injective) dimension of M is finite imposes stringent restrictions on the structure of M. We illustrate this fact and give a flavor of things to come by reviewing basic results.

1.1. Finite projective dimension. When a projective dimension is finite, its value is determined by a famous equality.

1.1.1. Theorem (Auslander-Buchsbaum [5]). If proj dim_{R} M is finite, then

proj dim_{R} M = depth R − depth_{R} M

All admissible values of proj dim_{R} M are indeed achieved.

1.1.2. Example. Set d = depth R, choose an R-regular sequence a_{1}, …, a_{d}, and set y_{i} = a_{1}, …, a_{i} for i = 0, …, d. By Theorem (B.1.2), the Koszul complex K(y_{i}; R) is exact, hence is a free resolution of the R-module N_{i} = R/(y_{i}). This resolution is minimal, so proj dim_{R}(N_{i}) = i, cf. (4.2).

1.2. Regular local rings. One of the most influential results in commutative algebra describes the local rings of finite global dimension as being the regular rings. It is a formal consequence of the following more precise statement.

1.2.1. Theorem (Auslander-Buchsbaum [4], [5]; Serre [64]).  The following

conditions are equivalent.

(i) R is regular.

(ii) proj dim_{R} N ≤ depth R for every finite R-module N.

(iii) proj dim_{R} k < ∞.

The implication (i) ⟹ (ii) may be viewed as the ultimate generalization of Hilbert’s Syzygy Theorem, cf. also (10.2.2). The implication (iii) ⟹ (i) is considerably strengthened below (note that proj dim_{R} 0 = −∞, cf. (2.2.3)).

1.2.2. Theorem (Levin-Vasconcelos [49]). If there exists a finite module N such that proj dim_{R}(𝔪^{i}N) is finite for some i ≥ 1, then R is regular.

Our discussion of injective dimension parallels that of projective dimension.

1.3. Finite injective dimension. The first result may surprise.

1.3.1. Theorem (Bass [22]). If inj dim_{R} M is finite, then

inj dim_{R} M = depth R

Some examples of modules of finite injective dimension are easy to produce.

1.3.2. Example. Every Cohen-Macaulay ring R has finite modules of finite injective dimension. Indeed, choose an R-regular sequence y of length d = depth R. By Example (1.1.2), the module N_{d} = R/(y) has a finite free resolution K. If E is an injective envelope of k over R, then the complex Hom_{R}(K, E) is a finite injective coresolution of N = Hom_{R}(N_{d}, E). Furthermore, length_{R}(N) = length_{R}(N_{d}) < ∞, with equality obtained by induction using the exactness of the functor Hom_{R}(−, E), and inequality due to the fact that the ring R is Cohen-Macaulay.

In 1963, Bass [22] asked if finite modules of finite injective dimension exist only over Cohen-Macaulay rings. It took 25 years to get a complete answer.

1.3.3. Theorem (Peskine-Szpiro [56], Hochster [47], Roberts [60]). If the module M has finite injective dimension, then the ring R is Cohen-Macaulay.

We go on to homological characterizations of the Gorenstein property; for descriptions in terms of other homological dimensions cf. Theorem (3.5.1).

1.4. Gorenstein local rings. The classical description runs as follows:

1.4.1. Theorem (Bass [22]). R is Gorenstein if and only if inj dim_{R} R < ∞.

This characterization is contained in the more precise result below.

1.4.2. Theorem (Foxby [40]). The following conditions are equivalent. (i) R is Gorenstein. (ii) Every finite module N with proj dim_{R} N < ∞ has inj dim_{R} N < ∞. (iii) There exists a finite module N such that proj dim_{R} N and inj dim_{R} N are

both finite.

Expositions. The monographs of Hochster [47], Northcott [54], Roberts [59], Evans and Griffith [38], Strooker [66], Roberts [61] deal extensively (some— exclusively) with modules of finite projective dimension or finite injective dimension over commutative noetherian rings. The textbooks of Matsumura [52], Bruns and Herzog [26], Eisenbud [35] contain many relevant results.

2. Dimensions from resolutions

Let R be an associative ring, and let M = M(R) be the category of R-modules. We fix an abelian subcategory F ⊆ M, a subcategory C ⊆ F, and let P ⊆ F denote the subcategory whose objects are the projective R-modules in F, cf. (A.1) for some general conventions on categories of modules.

Two standard procedures for replacing projective resolutions by more general ones lead to homological dimensions measuring rather different effects.

2.1. Relative dimension. Under this approach, modules other than projectives are allowed in resolutions, but not all homomorphisms can be used as differentials. The goal is to ensure the existence of comparison morphisms for resolutions, leading to a theory of relative cohomology functors. The payoff is that whenever a module admits a resolution, its homological dimension can be computed from the vanishing of an appropriate cohomological functor. The downside is that the class of modules of finite homological dimension may shrink, due to insufficiency of proper maps.

2.1.1. Proper resolutions. A sequence E of homomorphisms in F is proper exact (with respect to C) if the sequence Hom_{R}(C, E) is exact for all C ∈ C. A C-resolution γ: C → M is proper if it yields a proper exact sequence

⋯ ⟶ C_{n} ⟶^{∂_{n}} C_{n−1} ⟶ ⋯ ⟶ C_{1} ⟶^{∂_{1}} C_{0} ⟶^{γ_{0}} M ⟶ 0

When C = P we are in the absolute case, where every C-resolution is proper.

2.1.2. Relative cohomology. Let \overline{C} denote the subcategory of F whose objects are the R-modules admitting some proper C-resolution.

To construct cohomology functors one uses a Comparison Lemma for proper resolutions, proved by the usual arguments for the projective case.

2.1.3. Lemma. If ϵ: C → M and ϵ′: C′ → M′ are proper C-resolutions, then for every R-linear homomorphism μ: M → M′ there is a unique up to homotopy comparison morphism μ̃: C → C′ that that satisfies ϵ′μ̃ = μϵ.

Choosing for each M ∈ \overline{C} a proper resolution C → M, define for each n ∈ ℤ and each N ∈ M(R) a relative cohomology group

Ext_{C}^{n}(M, N) = H^{n} Hom_{R}(C, N)

By the Comparison Lemma, (M, N) ↦ Ext_{C}^{n}(M, N) yields a functor

Ext_{C}^{n} : \overline{C}(R)^{op} × M(R) ⟶ M(ℤ)

Choosing for each M ∈ \overline{C} a projective resolution π: P → M and a morphism of complexes ϕ: P → C lifting id_{M} one obtains homomorphisms

ε_{C}^{n}(M, N) = H^{n} Hom_{R}(ϕ, N): Ext_{C}^{n}(M, N) ⟶ Ext_{R}^{n}(M, N)

These maps produce morphisms of functors ε_{C}^{n}: Ext_{C}^{n} → Ext_{R}^{n} which commute with connecting homomorphisms. Furthermore, ε_{C}^{0} : Ext_{C}^{0} → Hom_{R} is an isomorphism and ε_{C}^{1} : Ext_{C}^{1} → Ext_{R}^{1} is a monomorphism.

Relative cohomology looks like a cohomological functor: every proper exact sequence in either argument induces a natural cohomology exact sequence.

2.1.4. Yoneda classes. For all M, N ∈ F, Yoneda congruence relations involving only proper short exact sequences define equivalence classes of proper exact sequences of length n, starting at N and ending at M. The result is an abelian group of proper extensions that depends functorially on M and N. For every M ∈ \overline{C} this group is canonically isomorphic with Ext_{C}^{n}(M, N). The map sending the class of each proper sequence to its congruence class with respect to Yoneda equivalence using all short exact sequences is a morphism of extension functors. For M ∈ \overline{C}, the canonical isomorphism above transforms this morphism into ε^{n}: Ext_{C}^{n} → Ext_{R}^{n}.

Using properties of relative Ext functors outlined above, one gets the next result by repeating verbatim the classical arguments for the absolute case.

2.1.5. Proposition-Definition. Let C be a subcategory of F.

For every M ∈ \overline{C} and for each g ∈ ℕ the following are equivalent.

(i) There exists a proper C-resolution C → M with C_{n} = 0 for all n > g.

(ii) Every proper C-resolution C → M has Ω^{g}C ∈ C.

(iii) Ext_{C}^{g+1}(M, −) = 0.

(iv) Ext_{C}^{n}(M, −) = 0 for all n > g.

When they hold M has relative dimension at most g with respect to C, denoted

rel dim_{C} M ≤ g

For M ∈ F ∖ \overline{C} one sets rel dim_{C} M = ∞.

2.2. Resolving dimension. In this approach no restrictions are placed on maps in resolutions. Thus, enlarging the class of modules allowed in resolutions one gets a corresponding enlargements of the class of modules of finite homological dimension. Here the caveat is: for a dimension to be computable from every resolution, the class of allowable modules cannot be arbitrary.

2.2.1. Resolving subcategories. A subcategory C ⊆ F is resolving for F if C is closed under extensions and under kernels of epimorphisms, and contains enough projectives for F, in the sense that every M ∈ F is the homomorphic image of some module from C ∩ P. Every module M ∈ F has a C-resolution.

2.2.2. Proposition-Definition (Auslander-Bridger [3]). Let C be a resolving subcategory for F.

For every M ∈ M and for each g ∈ ℕ the following are equivalent.

(i) There exists a C-resolution C → M with C_{n} = 0 for n > g.

(ii) Every C-resolution C → M has Ω^{g}C ∈ C.

(iii) There exists a P-resolution P → M with Ω^{g}P ∈ C. When they hold, M is said to have C-dimension at most g, denoted

C-dim M ≤ g

The next observations follow directly from the definitions.

2.2.3. Remark. If C is resolving for F, then for each M ∈ F one has:

C-dim M = −∞ ⟺ rel dim_{C} M = −∞ ⟺ M = 0
C-dim M = 0 ⟺ rel dim_{C} M = 0 ⟺ M ∈ C ∖ 0
C-dim M ≤ min{rel dim_{C} M , proj dim_{R} M}

Standard use of mapping cones and the Horseshoe Lemma lead to

2.2.4. Proposition. For every exact sequence 0 → M → M′ → M′′ → 0 in F there is an inequality

C-dim M ≤ max{C-dim M′, C-dim M′′ − 1}

If C-dim M′ ≠ C-dim M′′, then equality holds.

2.2.5. Example. The subcategory P is resolving for F if and only if P has enough projectives for F; in that case P-dim M = proj dim_{R} M for M ∈ F.

Expositions. In discussing relative homological algebra we have followed the memoir of Eilenberg and Moore [33] and the book of MacLane [50].

3. Gorenstein dimension

In this section R is a noetherian (on both sides) ring, and M a finite R-module. We let F ⊆ M denote the subcategory of finite R-modules, and P ⊆ F the subcategory of projective R-modules.

3.1. Gorenstein dimension. An R-module G is reflexive if it is finite and

G ≅ G^{∗∗}

This happens if and only if the canonical map G → G^{∗∗} is bijective, cf. (A.1). We say that G is totally reflexive if it is reflexive and satisfies

Ext_{R}^{n}(G, R) = 0 = Ext_{R^{o}}^{n}(G^{∗}, R) for all n ≥ 1

These modules were introduced in [3] as modules of G-dimension zero. Let G ⊆ F be the subcategory of totally reflexive modules.

3.1.1. Lemma. The subcategory G is resolving for F.

By Proposition (2.2.2) the number G-dim M defined in (2.2.1) can be computed from any G-resolution. This invariant, introduced by Auslander and Bridger [3], is called the Gorenstein dimension or G-dimension of M; the name is explained by Theorem (3.5.1). To use standard notation, we set

G-dim_{R} M = G-dim M

In the next result the first inequality is obtained by an easy induction on G-dim_{R} M, the second from Proposition (2.2.2). The conditions for equality establish the precise sense in which G-dimension refines projective dimension.

3.1.2. Theorem (Auslander-Bridger [3]). There are inequalities

sup{n ∈ ℕ | Ext_{R}^{n}(M, R) ≠ 0} ≤ G-dim_{R} M ≤ proj dim_{R} M

and equalities hold to the left of any finite dimension.

The equivalence (i) ⟺ (ii) below is proved in [3] over local rings, and in [19] in general. The implication (ii) ⟹ (iii) is due to Zaks [69].

3.1.3. Theorem. For each integer d ≥ 0 the following are equivalent. (i) Every finite module, left or right, has G-dimension at most d. (ii) inj dim_{R} R ≤ d and inj dim_{R^{o}} R ≤ d. (iii) inj dim_{R} R = inj dim_{R^{o}} R ≤ d.

3.2. Relative dimension. Let rel dim_{G} M be the relative dimension of M with respect to G, as in (2.1). To relate it to G-dimension, we say that a G-resolution G → M is strict if G_{n} is projective for all n ≥ 1.

3.2.1. Theorem. For each integer g ≥ 0 the following are equivalent. (i) G-dim_{R} M ≤ g, that is, M has a G-resolution of length ≤ g. (ii) rel dim_{G} M ≤ g, that is, M has a proper G-resolution of length ≤ g. (iii) M has a strict G-resolution of length ≤ g.

In particular, G-dim_{R} M = rel dim_{G} M.

The implication (ii) ⟹ (i) is trivial, while (i) ⟹ (iii) is shown by an explicit construction. Finally, (iii) ⟹ (ii) holds because strict resolutions are proper, by the next lemma, easily proved by induction on proj dim_{R} N.

3.2.2. Lemma. If G is a totally reflexive R-module, then Ext_{R}^{n}(G, N) = 0 for every R-module N of finite projective dimension and all n ≥ 1.

3.3. Tate cohomology. A totally acyclic complex is a complex T such that T_{n} is finite projective for each n ∈ ℤ and the following condition holds

H_{n}(T) = 0 = H_{n}(T^{∗}) for all n ∈ ℤ

A complete resolution of M is a diagram T ⟶^{ϑ} P ⟶^{π} M with a P-resolution π, a totally acyclic complex T, a morphism ϑ with ϑ_{n} bijective for n ≫ 0.

3.3.1. Theorem. For each integer g ≥ 0 the following are equivalent. (i) G-dim_{R} M ≤ g. (ii) M has a complete resolution ϑ: T → P with ϑ_{n} bijective for all n ≥ g. (iii) M has a complete resolution ϑ: T → P with ϑ_{n} bijective for all n ≥ g and

surjective for all n ∈ ℤ

The proof yields explicit constructions of the desired complexes.

To define cohomology functors on the basis of complete resolutions, one needs a corresponding Comparison Lemma; the one below is from [19].

3.3.2. Lemma. If T ⟶^{ϑ} P ⟶^{π} M and T′ ⟶^{ϑ′} P′ ⟶^{π′} M′ are complete resolutions, then for each R-linear map μ: M → M′ there exists a unique up to homotopy morphism \overline{μ}, making the right hand square of the diagram

\xymatrix{T \ar[d]^{μ̂} \ar[r]^{ϑ} & P \ar[d]^{\overline{μ}} \ar[r]^{π} & M \ar[d]^{μ} \\ _{T_{′}} \ar[r]^{ϑ′} & _{P_{′}} \ar[r]^{π′} & M′}

commute, and for each choice of \overline{μ} there exists a unique up to homotopy morphism μ̂, making the left hand square commute up to homotopy.

Let G̃ be the subcategory of F whose objects are the modules of finite Gdimension. Choosing for each M ∈ G̃ a complete resolution T ⟶^{ϑ} P ⟶^{π} M, one defines for every N ∈ M and every n ∈ ℤ a Tate cohomology group

Ex̂t_{R}^{n}(M, N) = H^{n} Hom_{R}(T , N)

By the Comparison Lemma, (M, N) ↦ Ex̂t_{R}^{n}(M, N) yields a functor

Ex̂t_{R}^{n}: G̃(R)^{op} × M(R) ⟶ M(ℤ)

for every n ∈ ℤ, and the homomorphisms

ε_{R}^{n}(M, N) = H^{n} Hom_{R}(ϑ, N): Ext_{R}^{n}(M, N) → Ex̂t_{R}^{n}(M, N)

yield morphisms of functors ε_{R}^{n}: Ext_{R}^{n} → Ex̂t_{R}^{n}, invertible for n > G-dim_{R} M. Clearly, Ex̂t_{R}^{n}(M, −) is a cohomological functor, each short exact sequence of modules in G̃ induces a natural exact sequence of Tate cohomology groups, and ε_{R}^{n} defines morphisms of the corresponding long exact sequences.

The prototype of Tate cohomology is precisely what the name suggests:

3.3.3. Example. Let Π be a finite group, R = ℤ[Π] its group ring, and ℤ the R-module with trivial action: gm = m for all g ∈ Π and m ∈ ℤ.

Choose a resolution ϵ: P → ℤ by finite free R-modules. For each i, the action (gα)(p) = α(g^{−1}p) turns Hom_{ℤ}(P_{i}, ℤ) into a free left Π-module. Furthermore, H_{0}(Hom_{ℤ}(P , ℤ)) = ℤ, and H_{i}(Hom_{ℤ}(P , ℤ)) = 0 for i ≠ 0. Splice P with the complex Hom_{ℤ}(P , ℤ), shifted one degree to the right, along the R-linear maps ϵ and n ↦ n( ∑_{g∈Π} g). The resulting complex T of finite free R-modules is contractible over ℤ and hence so is the complex Hom_{ℤ}(T , ℤ). The isomorphisms Hom_{ℤ}(T , ℤ) ≅ Hom_{R}(T , Hom_{ℤ}(R, ℤ)) ≅ Hom_{R}(T , R) show that T is totally acyclic. Thus, T is a Tate resolution of ℤ over R and Ex̂t_{R}^{n}(ℤ, N) = H^{n} Hom_{R}(T , N) is the classical Tate cohomology group Ĥ^{n}(Π, N) of the group Π, cf. [29].

By Theorems (3.2.1) and (3.3.1), both relative cohomology and Tate cohomology are defined on the category G̃ of modules of finite G-dimension. The relations between these theories and absolute cohomology, provided by the morphisms ε_{G}^{n} of (2.1.2) and ε_{R}^{n} of (3.3), turn out to be unexpectedly tight.

3.3.4. Theorem (Avramov-Martsinkovsky [19]). If G-dim_{R} M = g < ∞, then for each R-module N there exist natural homomorphisms δ_{R}^{n}(M, N), such that the following sequence is exact:

[figure: 0; Ext_{G}^{1}(M, N); ε_{G}^{1}(M,N); Ext_{R}^{1}(M, N); ⋯; Ext_{G}^{n}(M, N); ε_{G}^{n}(M,N); Ext_{R}^{n}(M, N); ε_{R}^{n}(M,N); Ex̂t_{R}^{n}(M, N); δ_{R}^{n}(M,N); Ext_{G}^{n+1}(M, N); ⋯; Ex̂t_{R}^{g}(M, N); 0]

The proof is based on a careful analysis of the homology exact sequence induced by the surjective morphism of complexes ϑ: T → P from (3.3.1.iii).

3.4. Approximations. A G-approximation of M is an exact sequence

0 ⟶ Y ⟶ X ⟶ M ⟶ 0

with X ∈ G and proj dim_{R} Y < ∞. For existence of approximations we have:

3.4.1. Proposition. M has a G-approximation if and only if G-dim_{R} M < ∞.

Proof. If G-dim_{R} M < ∞, then the strict G-resolution from condition (3.2.1.ii) provides an exact sequence 0 → ∂_{1}(G_{1}) → G_{0} → M → 0 with G_{0} ∈ G and proj dim_{R}(∂_{1}(G_{1})) < ∞. Conversely, if 0 → Y → X → M → 0 is a Gapproximation, then splice this exact sequence with a P-resolution of Y of finite length to obtain a G-resolution of finite length.

Over local rings, G-dimension has additional properties.

3.5. Local rings. Let (R, 𝔪, k) be a local ring.

The next result gives the reason for the name Gorenstein dimension. The equivalence of (i) and (ii) follows from Theorems (1.4.1), (3.1.3), and (3.5.2).

3.5.1. Theorem (Auslander-Bridger [3]). The following are equivalent. (i) R is Gorenstein. (ii) G-dim_{R} N ≤ depth R for every finite R-module N. (iii) G-dim_{R} k < ∞.

Finite projective dimension implies finite G-dimension by Theorem (3.1.2). It follows from the preceding theorem that the converse fails, so the next result properly contains Theorem (1.1.1).

3.5.2. Theorem (Auslander-Bridger [3]). If G-dim_{R} M < ∞, then

G-dim_{R} M = depth R − depth_{R} M

For every finite R-module N, Corollary (3.1.2) and Theorem (B.1.1.2) yield

grade_{R}(ann_{R}(N), R) ≤ G-dim_{R} N

If equality holds, then N is said to be G-perfect, by analogy with the notion described in (B.1.3); below we use the notation N_{R}^{†} also introduced there.

3.5.3. Theorem (Golod [44]). If R → S is a surjective homomorphism of rings, and the R-module S is G-perfect with S_{R}^{†} ≅ S, then every finite S-module N satisfies

G-dim_{R} N = G-dim_{R} S + G-dim_{S} N

This remarkable property of G-dimension is not shared by projective dimension: if R = k[[t]], S = R/(t^{2}), and N = k, then the complexes

0 ⟶ R ⟶^{t} R ⟶ 0
⋯ ⟶ S ⟶^{t} S ⟶^{t} S ⟶^{t} S ⟶ 0

provide minimal free resolutions of N = k over R and over S, respectively.

Expositions. Chapters III and IV of the memoir of Auslander and Bridger [3] constitute the canonical text on Gorenstein dimension. There exist several accounts of that material within the framework of modules over local rings: in the early lectures of Auslander [2], the recent survey of Maşek [51], the monographs of Enochs and Jenda [36] and of Christensen [31]. The paper [19] provides a self-contained treatment of the classical results over arbitrary noetherian rings, using explicit constructions of complexes.

4. Minimal (co)resolutions

In this section R is an associative ring and M is an R-module.

We describe a concept of minimality from [19], which is applicable to every complex of R-modules. It explains the similarity of properties of minimal injective coresolutions and minimal projective resolutions, whenever the latter exist. It is also useful in situations where a notion of ‘minimality’ had been defined ad hoc (like maximal Cohen-Macaulay approximations), or has been missing (like proper resolutions). Omitted proofs can be found in [19].

4.1. Minimality. Let B be a complex of R-modules.

4.1.1. Proposition-Definition. The following conditions are equivalent. (i) Every homotopy equivalence β: B → B is an isomorphism. (ii) Each morphism β: B → B homotopic to id_{B} is an isomorphism. (iii) If β: B → C and γ: C → B are homotopy inverses, then β is injective, γ

is surjective, Ker γ is contractible, and C = Im β ⊕ Ker γ. When they hold, the complex B is said to ne minimal.

The proofs of the proposition and of its corollary use only basic homological algebra—the interpretation of homotopy classes of morphisms as homology classes in complexes of homomorphisms.

4.1.2. Corollary. Let B and C be minimal complexes of R-modules. (1) Every homotopy equivalence β: B → C is an isomorphism.

(2) If B is minimal, and A is a contractible subcomplex of B such that for every n ∈ ℤ the R-module A_{n} is a direct summand of B_{n}, then A = 0.

A first example is psychologically satisfactory, but carries little substance.

4.1.3. Remark. Every complex concentrated in a single degree—in particular, every R-module—is a minimal complex.

Fortunately, less trivial examples of minimal complexes are at hand.

4.2. Minimal projective resolutions. We say that a projective resolution P → M is minimal if the complex P is minimal. The existence of such resolutions depends on the availability of some special homomorphisms.

4.2.1. Projective covers. A projective cover of an R-module M is an epimorphism ϵ: P → M from a projective R-module P, such that every homomorphism ϕ: P → P with ϵϕ = ϵ is bijective. Over a general ring R, not every module has a projective cover.

4.2.2. Proposition. Let P be a complex of projective R-modules.

If P_{n} → Coker(∂_{n+1}) is a projective cover for each n, then P is minimal; the converse holds when Coker(∂_{n+1}) has a projective cover for each n.

4.2.3. Corollary. If P → M is a minimal projective resolution, then

proj dim_{R} M = sup{n ∈ ℤ | P_{n} ≠ 0}

If P′ → M is a projective resolution, then each comparison morphism P′ → P is a split epimorphism with contractible kernel; it is bijective if P′ is minimal.

Proofs may be obtained by dualizing those below. For finite modules over local rings, the general notion of minimal free resolution specializes to the standard concept introduced by Eilenberg [32]:

4.2.4. Example. Let (R, 𝔪, k) be a local ring and M a finite R-module.

By Nakayama’s Lemma every finite projective R-module is free, an epimorphism ϵ: F → M is a projective cover if and only if F is finite free and Ker(ϵ) ⊆ 𝔪F, and M has a projective cover. In view of these facts and the results above, a projective resolution F → M is minimal if and only if each F_{n} is finite free and ∂(F) ⊆ 𝔪F, and every finite R-module has a unique up to isomorphism minimal free resolution.

4.3. Minimal injective coresolutions. We say that an injective coresolution M → I is minimal if the complex I is minimal in the sense of (4.1).

4.3.1. Injective envelopes. An injective envelope of M is a monomorphism η: M → I to an injective module, such that each homomorphism ι: I → I with ιη = η is bijective. Every R-module M has an injective envelope.

For injective coresolutions minimality coincides with the classical concept.

4.3.2. Proposition. A complex I of injective modules is minimal if and only if Ker(∂_{n}) → I_{n} is an injective envelope for each n ∈ ℤ.

Proof. Assume first each I_{n} is an injective envelope of Ker(∂_{n}), and let β: I → I be a morphism homotopic to id_{I}. We want to prove β_{n} is bijective. Since I_{n} is an injective envelope of Ker(∂_{n}), it suffices to show that the restriction of β_{n} to Ker(∂_{n}) is injective, that is, the module K_{n} = Ker(∂_{n}) ∩ Ker(β_{n}) is trivial. Choose a homotopy θ from β to id_{I}. Every x ∈ K_{n} satisfies x = ∂_{n+1}θ_{n}(x), hence θ_{n}(K_{n}) ≅ K_{n} and θ_{n}(K_{n}) ∩ Ker(∂_{n+1}) = 0. Since I_{n+1} is an injective envelope of Ker(∂_{n+1}), we conclude θ_{n}(K_{n}) = 0, so K_{n} = 0, as desired.

Assume next some I_{n} is not an injective envelope of Ker(∂_{n}), so I_{n} = I′_{n} ⊕ J_{n} with I′_{n} an injective envelope of Ker(∂_{n}) and J_{n} ≠ 0. Thus, ∂_{n} maps J_{n} bijectively to a submodule J_{n−1} of I_{n−1}. Setting J_{i} = 0 for i ≠ n, n − 1 we get a contractible subcomplex J ⊆ I. As each J_{i} is injective, it splits off I_{i}, so J is irrelevant, hence I is not minimal by Proposition (4.1.3).

4.3.3. Corollary. If M → I is a minimal injective coresolution, then

inj dim_{R} M = − inf{n ∈ ℤ | I_{n} ≠ 0}

If M → I′ is an injective coresolution, then each comparison morphism I′ → I is a split epimorphism with contractible kernel; it is bijective if I′ is minimal.

Every R-module has a minimal injective coresolution.

Proof. Injective coresolutions of M are homotopy equivalent by the corresponding Comparison Lemma, so the first part follows from Corollary (4.1.2). For the second part, form an injective coresolution M → I by choosing the maps η: M → I_{0}, Coker η → I_{−1}, and Coker(∂_{n+2}) → I_{n} for all n ≤ −2 to be injective envelopes. Proposition (4.3.2) shows that I is minimal.

In the applications below the descriptions of minimal resolutions and the proofs of their existence are more delicate than in the classical cases. However, uniqueness properties are again obtained by specializing Corollary (4.1.2).

4.4. Minimal proper resolutions. We say that a G-resolution γ: G → M is minimal if G is a minimal complex.

4.4.1. Theorem. Assume (R, 𝔪, k) is a local ring and G-dim_{R} M is finite.

The module M has a minimal G-proper resolution. More precisely, a quasiisomorphism γ: G → M is a minimal proper G-resolution if and only if

(a) G_{n} = 0 for all n < 0.

(b) G_{0} is totally reflexive.

(c) G_{n} is free of finite rank for all n ≥ 1.

(d) ∂_{n}(G_{n}) contains no non-zero free direct summand of G_{n−1} for all n ≥ 1. If G → M is a minimal G-proper resolution, then

G-dim_{R} M = sup{n ∈ ℤ | G_{n} ≠ 0}

If G′ → M is a G-proper resolution, then each comparison morphism G′ → G is a split epimorphism with contractible kernel; it is bijective if G′ is minimal.

4.5. Minimal approximations. We say that a G-approximation

0 ⟶ Y ⟶ X ⟶ M ⟶ 0

is minimal if the complex ⋯ → 0 → Y → X → 0 → ⋯ is minimal.

4.5.1. Theorem. Assume (R, 𝔪, k) is a local ring and G-dim_{R} M is finite.

The module M has a minimal G-approximation. More precisely, an exact sequence 0 → Y ⟶^{υ} X → M → 0 is a minimal G-approximation if and only if X is totally reflexive, Y has finite projective dimension, and υ(Y) contains no non-zero direct summand of X.

If 0 → Y′ → X′ → M → 0 is a G-approximation, then it is isomorphic to

0 ⟶ Y ⊕ F ⟶^{υ⊕id_{F}} X ⊕ F ⟶ M ⟶ 0

where F is a free R-module; it is minimal if and only if F = 0.

4.5.2. Remark. Let R be a Gorenstein local ring. A finite R-module M is totally reflexive if and only if it is maximal Cohen-Macaulay, that is, depth_{R} M = dim R. In particular, the G-approximations of M are precisely the maximal Cohen-Macaulay approximations of Auslander and Buchweitz [7]. The theorem shows, in particular, that our concept of minimality for G-approximations agrees with that of Auslander [8].

5. Betti numbers and Bass numbers

In this section (R, 𝔪, k) denotes a local ring and M a finite R-module.

First we recall the definition of a sequences of homological invariants used to measure the size of a minimal projective resolution of M.

5.1. Betti numbers. By Example (4.2.4), every finite R-module M has a unique up to a isomorphism minimal resolution F → M. Thus,

β_{n}^{R}(M) = rank_{R}(F_{n})

is an invariant of M, called the nth Betti number of M over R.

The minimality of F is characterized by the condition ∂(F) ⊆ 𝔪F, so the complexes F ⊗_{R} k and Hom_{R}(F , k) have trivial differentials. This leads to:

5.1.1. Remark. β_{n}^{R}(M) = rank_{k} Tor_{n}^{R}(M, k) = rank_{k} Ext_{R}^{n}(M, k).

Corollary (4.2.4) also determines the non-vanishing Betti numbers:

5.1.2. Remark. β_{n}^{R}(M) ≠ 0 if and only if 0 ≤ n ≤ proj dim_{R} M.

The equicharacteristic hypothesis appears in the next result because all known proofs use Hochster’s ‘big Cohen-Macaulay modules’ [47]: they are known to exist only if R is an algebra over a field, or dim R ≤ 2.

5.1.3. Theorem (Evans-Griffith [37]). If R is equicharacteristic and M has finite projective dimension, then there are inequalities

β_{n}^{R}(M) ≥ \begin{cases}2n + 1 & for n = 0, …, proj dim_{R} M − 2 \\ depth R − depth_{R} M & for n = proj dim_{R} M − 1 \\ 1 & for n = proj dim_{R} M \end{cases}

Examples of Bruns [24] show that the inequalities given by the syzygy theorem cannot be improved in general. On the other hand, much stronger bounds were conjectured in [27] for modules of finite length.

5.1.4. Conjecture (Buchsbaum-Eisenbud). If proj dim_{R} M and length_{R} M are both finite, then the following inequalities hold

β_{n}^{R}(M) ≥ \binom{dim R}{n} for all n

Here are some comments on the status of this long open problem.

5.1.5. Remark. The conjectured bounds are best possible in general. Indeed, by the argument in Example (1.3.2) the existence of a module of finite length and finite projective dimension implies the existence of a module of finite length and finite injective dimension. Theorem (1.3.3) then shows that R is Cohen-Macaulay, so dim R = depth R by Remark (B.3.3). For d = depth R the R-module N_{d} of Example (1.1.2) then has β_{R}(N_{d}) = \binom{d}{n} for all n.

The conjecture holds trivially for n = 0. It holds for n = 1 by Krull’s Theorem (B.3.1). As R is Cohen-Macaulay, the module M is perfect of finite length, hence so is the module M_{R}^{†}. Thus, the conjecture holds for M_{R}^{†} and n = 0, 1. By Remark (5.3.3) below, it then holds for M and n = d − 1, d.

The known cases imply the conjecture holds whenever dim R ≤ 4.

Next we describe sequences of invariants that measure the size of parts of a minimal injective resolution of M. There is one such sequence for each 𝔭 ∈ Spec R, where Spec R denotes the set of prime ideals of R. Invariants derived from injective resolutions are often more difficult to handle, due to the fact that injective modules are finite (if and) only if the ring R is artinian.

5.2. Bass numbers. Let M → I be a minimal injective coresolution, cf. Corollary (4.3.3). By a well known theorem of Matlis, every injective module I_{n} is a direct sum of indecomposables, every indecomposable is isomorphic to the injective envelope E_{R}(R/𝔭) for some 𝔭 in Spec R, and the isomorphism

I_{n} ≅ ⨁_{𝔭∈Spec R} E_{R}(R/𝔭)^{μ_{R}^{n}(𝔭,M)}

uniquely defines μ_{R}^{n}(𝔭, M), the nth Bass number of M over R at the prime 𝔭. When 𝔭 = 𝔪, reference to the prime ideal is suppressed from terminology and notation.

The results on Betti numbers discussed above have analogs for Bass numbers. Proofs use the following expression, which is not hard to establish.

5.2.1. Lemma. μ_{R}^{n}(M) = rank_{k} Ext_{R}^{n}(k, M) for all n ∈ ℤ.

Vanishing patterns for Bass numbers μ_{R}^{n}(M) are well understood. By Rees’ Theorem (B.1.1.2), the first non-vanishing one appears for n = depth_{R} M. Bass [22] proved the last non-vanishing number occurs for n = inj dim_{R} M. The positivity of all intermediate μ_{R}^{n}(M), although formally similar to the statement for Betti numbers in Remark (5.1.2), is a more delicate fact.

5.2.2. Theorem (Fossum-Foxby-Griffith-Reiten [39]).

μ_{R}^{n}(M) ≠ 0 if and only if depth_{R} M ≤ n ≤ inj dim_{R} M

In the next result, an equicharacteristic hypothesis on R appears for the same reason as in Theorem (5.1.3).

5.2.3. Theorem (Bruns [25]). If R is equicharacteristic, then

μ_{R}^{n}(M) ≥ \begin{cases}1 & for n = depth_{R} M \\ depth R − depth_{R} M & for n = depth_{R} M + 1 \\ 2(depth R − n) + 1 & for n = depth_{R} M + 2, …, depth R \end{cases}

5.3. Reductions. Certain adjustments can be made to facilitate computation of Betti numbers or Bass numbers. We provide some samples.

5.3.1. Remark. If φ: R → R′ is a faithfully flat homomorphism of local rings, then β_{n}^{R}(M) = β_{n}^{R′}(M ⊗_{R} R′) for all n ∈ ℤ.

Indeed, faithful flatness implies φ(𝔪) is contained in the maximal ideal of R′, so if F → M is a minimal free resolution of M over R, then F ⊗_{R} R′ → M ⊗_{R} R′ is a minimal free resolution of M ⊗_{R} R′ over R′.

5.3.2. Remark. β_{n}^{R}(M) = βR̂ ; n(M̂) and μ_{R}^{n}(M) = μn ; R̂(M̂) for all n ∈ ℤ.

Indeed, R → R̂ is faithfully flat by Theorem (B.4.1), so Remark (5.3.1) yields the equalities for Betti numbers. As 𝔪R̂ is the maximal ideal of R̂, we have Ext_{R}^{n}(k, M) ≅ Extn ; R̂(k, M ⊗_{R} R̂), whence the equalities for Bass numbers.

5.3.3. Remark. If M is perfect with proj dim_{R} M = g, then

β_{n}^{R}(M) = β_{g−n}^{R}(M_{R}^{†}) for all n ∈ ℤ

Indeed, if F → M is a minimal free resolution, then F^{∗} → M_{R}^{†} is a free resolution by the definition of perfection (B.1.3); it is clearly minimal.

If a submodules is ‘small’ in the 𝔪-adic topology, then its Betti are tightly linked to those of the module and of the residue module.

5.3.4. Proposition (Avramov [9]). There exists an integer s such that for each R-submodule L contained in 𝔪^{s}M there is an equality

β_{n}^{R}(M/L) = β_{n}^{R}(M) + β_{n−1}^{R}(L) for all n ≥ 0

Finally, we note that, most of the time, one can stay with Betti numbers.

5.3.5. Proposition (Foxby [41]). For every finite R-module M there exists an R̂- module M′ such that μ_{n}^{R}(M) = βR̂ ; n−d(M′) for all n > d = dim M.

Expositions. Proofs of the results in (5.1) and (5.2) are given in [26].

6. Complexity

In this section (R, 𝔪, k) is a local ring and M a finite R-module.

We start a discussion of asymptotic properties of the sequences of Betti numbers and of Bass numbers for R-modules of infinite projective or injective dimension, to be continued in Sections 7 and 9. In view of Theorem (1.2.1), the only simple R-module, its residue field k, has has infinite classical dimensions whenever the ring R is not regular.

We describe a measure of the growth of Betti numbers, on the polynomial scale, introduced by Alperin and Evens [1] in a slightly different manner for finite dimensional representations of finite groups. The definition below is from [11]. For variants applying to general rings see the appendix of [12].

6.1. Complexity. The module M has complexity c, denoted cx_{R} M = c, if c is the least non-negative integer d such that

β_{n}^{R}(M) ≤ βn^{d−1} for a real number β and all n ≫ 0

In particular, cx_{R} M = 0 if and only if proj dim_{R} M < ∞, so complexity offers a refinement of the dichotomy between finite and infinite projective dimension.

The first non-trivial results on projective dimension many a student sees concern the effect of factoring out a regular element. They have counterparts for complexity, but the next property of complexity has no antecedent.

6.1.1. Lemma. If R = Q/(x) where Q is a local ring and x is a Q-regular element, then

cx_{Q} M ≤ cx_{R} M ≤ cx_{Q} M + 1

Proof. Consider the classical change-of-rings spectral sequence

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

of Cartan-Eilenberg [29]. Isomorphisms

Tor_{q}^{Q}(M, R) ≅ \begin{cases}M & if q = 0, 1 \\ 0 & otherwise \end{cases}

are easily obtained using the resolution 0 → Q ⟶^{x} Q → 0 of R over Q. As ^{2}E_{p,q} = 0 for q ≠ 0, 1, the spectral sequence collapses to a long exact sequence

[figure: ⋯; Tor_{n−1}^{R}(M, k); Tor_{n}^{Q}(M, k); Tor_{n}^{R}(M, k); χ_{n}; Tor_{n−2}^{R}(M, k); Tor_{n−1}^{Q}(M, k); ⋯]

Estimates of vector space dimensions yield the desired inequalities.

Clearly, the first (respectively, second) inequality in the lemma becomes an equality if the map χ_{n} in the exact sequence above is trivial for all n (respectively, surjective for all n). Sufficient conditions are given in the next result; the original proofs proceed by explicit constructions of a minimal resolution of M over R, starting from a minimal resolution of M over Q.

6.1.2. Theorem. Assume R = Q/(x), where (Q, 𝔫, k) is a local ring and x is a Q-regular element.

(1) (Nagata [53]) If x ∉ 𝔫^{2}, then cx_{R} M = cx_{Q} M.

(2) (Shamash [65]) If x ∈ 𝔫 ann_{Q} M, then cx_{R} M = cx_{Q} M + 1.

Complete intersections take their turn for a homological characterization.

6.2. Complete intersection local rings. The next characterization of the complete intersection property parallels that of regularity in Theorem (1.2.1).

6.2.1. Theorem (Gulliksen [46]). The following are equivalent.

(i) R is complete intersection.

(ii) cx_{R} N ≤ edim R − depth R for every finite R-module N.

(iii) cx_{R} k < ∞.

Proof. (i) ⟹ (ii). Remark (5.3.2) yields cx_{R} N = cxR̂(N ⊗_{R} R̂), while for trivial reasons we have edim R̂ = edim R and depth R̂ = depth R. Thus, we may assume R is complete, and hence R ≅ Q/(x) for a regular local ring with edim Q = edim R and a Q-regular sequence x, cf. Remark (B.4.4). Theorem (1.2.1) yields cx_{Q} N = 0, so Lemma (6.1.1) gives the inequality below:

cx_{R} N ≤ card(x) = depth Q − depth R

= edim Q − depth R = edim R − depth R

The equalities come from the definitions of depth and of regularity.

(iii) ⟹ (i) is proved by very different arguments, cf. Section 7.

As already noted, a module has complexity 0 if and only if it has finite projective dimension, that is, its Betti numbers eventually vanish. The next simplest class, that of complexity at most 1, consists of those modules whose Betti numbers are bounded. Over complete intersections the minimal resolution of such a module is described by 6.2.2. Theorem (Eisenbud [34]). If R is complete intersection, cx_{R} M ≤ 1 and F → M is a minimal free resolution, then F has the form

⋯ ⟶^{β} G ⟶^{α} F ⟶^{β} G ⟶^{α} F ⟶^{∂_{g+1}} F_{g} ⟶^{∂_{g}} F_{g−1} ⟶ ⋯ ⟶ F_{1} ⟶^{∂_{1}} F_{0} ⟶ 0

where g = depth R − depth M and F, G are free R-modules of the same rank.

Next we describe complete intersections in the spirit of Theorem (1.1.1).

6.3. Lower complete intersection dimension. Let (R, 𝔪, k) be a local ring. Recall from Section 3 that G denotes the category of totally reflexive R-modules, and let B denote its subcategory consisting of those modules that have finite complexity. Clearly, B contains all finite free R-modules. Using free resolutions constructed via mapping cones and the Horseshoe Lemma, and the fact that G is a resolving subcategory for the category F of finite R-modules, cf. Lemma (3.1.1), one obtains:

6.3.1. Lemma. The subcategory B is resolving for F.

Following [43] (but changing terminology and notation), we use (2.2.2) to define a lower complete intersection dimension of M over R by the formula

CI_{∗}-dim_{R}M = B-dim(M)

6.3.2. Theorem (Gerko [43]). The following are equivalent.

(i) R is complete intersection.

(ii) CI_{∗}-dim_{R}N ≤ depth R for every finite R-module N.

(iii) CI_{∗}-dim_{R}k < ∞.

Proof. Assume first R is complete intersection. By Remark (B.2.1) it is Gorenstein, so for every finite R-module N Theorem (3.5.1) yields G-dim_{R} N ≤ d where d = depth R. If F → N is a minimal free resolution, then the R-module Ω^{d}F is totally reflexive by Theorem (2.2.2). On the other hand, cx_{R}(Ω^{d}F) = cx_{R} N < ∞, with the equality holding by definition, and the inequality by Theorem (6.2.1). Thus, we have shown Ω^{d}F is in B, hence CI_{∗}-dim_{R}N ≤ d by Proposition (2.2.2).

Assume next CI_{∗}-dim_{R}k = g < ∞. If P → k is a free resolutions, then Ω^{g}P is in B by Lemma (6.3.1) and Proposition (2.2.2), hence cx_{R} k = cx_{R}(Ω^{g}P) < ∞. From Theorem (6.2.1) we conclude that R is complete intersection.

Modules of finite lower complete intersection dimension over general rings may lack some of the nice properties of modules over complete intersections. One such discrepancy can be seen by comparing Theorem (6.2.2) with the following counterexample to a conjecture of Eisenbud [34].

6.3.3. Example (Gasharov-Peeva [42]). Let k be a field, let t_{1}, …, t_{4} be indeterminates over k, fix a ∈ k ∖ {0}, and set

R_{(a)} = \frac{k[t_{1}, t_{2}, t_{3}, t_{4}]}{(at_{1}t_{3} + t_{2}t_{3} , t_{1}t_{4} + t_{2}t_{4} , t_{3}t_{4} , t_{1}^{2} , t_{2}^{2} , t_{3}^{2} , t_{4}^{2})}

It is easy to check that the minimal complex of R_{(a)}-modules

T_{(a)} = ⋯ ⟶ R_{(a)}^{2} \underrightarrow{(\begin{matrix}t_{1} & a^{n}t_{3} + t_{4} \\ 0 & t_{2} \end{matrix})} R_{(a)}^{2} ⟶ ⋯

is exact, and so is the complex (T_{(a)})^{∗}. Thus, the R_{(a)}-module

M_{(a)} = Coker (\begin{matrix}t_{1} & t_{3} + t_{4} \\ 0 & t_{2} \end{matrix})

has β_{n}^{R_{(a)}}(M_{(a)}) = 2 for all n ≥ 0, and is in G by Theorem (3.3.1).

It is proved in [42] that T_{(a)} is periodic of period q if and only if q is the multiplicative order of a. Thus, k and a may be chosen so that the artinian (and hence Cohen-Macaulay) ring R_{(a)} has a module M_{(a)} with CI_{∗}-dim_{R_{(a)}}M_{(a)} = 0 and cx_{R_{(a)}} M_{(a)} = 1, whose minimal resolution is periodic of period q for any integer q ≥ 1, or is not periodic.

Similar examples are constructed in [42] also over Gorenstein rings.

Expositions. The survey [14] contains proofs for the results in this section.

7. Complete intersection dimension

In this section (R, 𝔪, k) is a local ring and M a finite R-module.

We describe a dimension whose finiteness for all modules characterizes complete intersection rings, and whose finiteness for an individual module imposes strict conditions on the structure of its minimal resolutions.

7.1. Complete intersection dimension. A diagram of homomorphisms R → R′ ← Q of local rings is a quasi-deformation if R → R′ is faithfully flat and R′ ← Q is surjective with kernel generated by a regular sequence. We set

CI-dim_{R} M = inf \{proj dim_{Q} (M ⊗_{R} R′) − proj dim_{Q} R′ | \begin{matrix}R → R′ ← Q is a \\ quasi-deformation \end{matrix}\}

This invariant, introduced in [17], is called the complete intersection dimension, or CI-dimension, of M over R. Here is one of its basic properties.

7.1.1. Remark. If CI-dim_{R} M is finite, then so is cx_{R} M by Lemma (5.3.1) and Remark (6.1.1). In fact, there is a common upper bound on the complexities of all R-modules of finite CI-dimension, cf. Theorem (7.3.3).

The next easy consequence of Theorem (6.2.1) is noted in [11], [17].

7.1.2. Theorem. The following are equivalent.

(i) R is complete intersection.

(ii) CI-dim_{R} N ≤ depth R for every finite R-module N.

(iii) CI-dim_{R} k < ∞.

Proof. (i) ⟹ (ii) Remark (B.4.4) yields a quasi-deformation R → R̂ ← Q with Q a regular local ring, so proj dim_{Q}(N ⊗_{R} R̂) < ∞ by Theorem (1.2.1).

(iii) ⟹ (i) Remark (7.1.1) yields cx_{R} k < ∞, so R is complete intersection by Theorem (6.2.1).

The study of modules of finite CI-dimension depends on the use of classical multiplicative structures carried by homology and cohomology over such rings.

7.2. Cohomology operators. Let Q be a commutative ring, let x = x_{1}, …, x_{c} be a Q-regular sequence, set J = (x), and R = Q/J. For the canonical action of R, the module J/J^{2} is free and the residue classes x_{1} + J^{2}, …, x_{c} + J^{2} form a basis.

Let x_{1}^{∗}, …, x_{c}^{∗} denote the dual basis of the free R-module Hom_{R}(J/J^{2}, R). We identify the symmetric algebra of this module with the polynomial ring

S = R[x_{1}^{∗}, …, x_{c}^{∗}]

graded by assigning cohomological degree 2 to the indeterminates. We call x_{1}^{∗}, …, x_{c}^{∗} the cohomology operators defined by the regular sequence x.

The next result subsumes the work of several authors, cf. [21] for details, and (A.3) for the algebra and module structures in cohomology used in it.

7.2.1. Theorem (Avramov-Sun [21]). For every R-module N there exist homomorphisms of graded R-algebras

ExtR^{•}(N, N) ⟵^{ζ_{N}} R[x_{1}^{∗}, …, x_{c}^{∗}] ⟶^{ζ_{M}} ExtR^{•}(M, M)

with images in the centers of the respective Ext algebras, and such that

ζ_{N}(x_{i}^{∗}) · ε = ε · ζ_{M}(x_{i}^{∗}) for all ε ∈ ExtR^{•}(M, N) and for i = 1, …, c
ζ_{N}(x_{i}^{∗}) · τ = −ζ_{M}(x_{i}^{∗}) · τ for all τ ∈ TorR_{•}(M, N) and for i = 1, …, c

For applications, it is crucial to understand the non-triviality of the action of the polynomial ring, resulting from the preceding result.

7.2.2. Theorem (Gulliksen [45]; Avramov-Gasharov-Peeva [17]). The R[x_{1}^{∗}, …, x_{c}^{∗}]-module ExtR^{•}(M, N) is finite if and only if Ext_{Q}^{n}(M, N) = 0 for all n ≫ 0.

Combining the preceding results, one gets

7.2.3. Corollary. If CI-dim_{R} M < ∞, then ExtR^{•}(M, k) is a finite graded module over a polynomial ring generated by central elements in Ext_{R}^{2}(k, k).

One of the reasons for introducing CI-dimension was to find, over arbitrary rings, modules whose resolutions have patterns of growth similar to that of resolutions over complete intersection rings.

7.3. Betti numbers. Assume CI-dim_{R} M is finite. The proofs of the next two theorems use the results in (7.2). The first one extends Theorem (6.2.2).

7.3.1. Theorem (Avramov [11]). If cx_{R} M ≤ 1, then each minimal free resolution F → M is periodic of period 2 after depth R − depth M steps and

β_{n+1}^{R}(M) = β_{n}^{R}(M) for all n > depth R − depth M

7.3.2. Theorem (Avramov-Gasharov-Peeva [17]). If cx_{R} M = d ≥ 2, then there exist integers b > 0 and 0 ≤ e ≤ d − 1 and polynomials

b_{±}(t) = \frac{b}{2^{e}(d − 1)!}t^{d−1} + lower order terms ∈ ℚ[t]

such that for all n ≫ 0 there are equalities

β_{n}^{R}(M) = \begin{cases}b_{+}(n) when n is even \\ b_{−}(n) when n is odd \end{cases}

Furthermore, both difference polynomials b_{±}(t + 1) − b_{∓}(t) have degree d − 2 and positive (but possibly different) leading coefficient, hence there are inequalities

β_{n+1}^{R}(M) > β_{n}^{R}(M) for all n ≫ 0

The next result extends the implication (i) ⟹ (ii) of Theorem (6.2.1).

7.3.3. Theorem (Avramov-Gasharov-Peeva [17]). There is an inequality

cx_{R} M ≤ edim R − depth R

Over complete intersections the inequality above follows from the fairly elementary Lemma (6.1.1), but the proof of the general case relies on deep structural properties of the algebra ExtR^{•}(k, k), cf. Remark (9.4).

7.4. Support varieties. Let (R, 𝔪, k) be a local complete intersection with

edim R − depth R = c

Let k̃ denote an algebraic closure of k, and let M, N be finite R-modules.

By Theorem (7.2.1), ExtR^{•}(M, N) ⊗_{R} k is a finite graded module over the graded polynomial ring k[x_{1}^{∗}, …, x_{c}^{∗}]. The support variety of the pair (M, N) is the algebraic set V_{R}^{∗}(M, N) ⊆ k̃^{c} defined by the vanishing of all polynomials f(x_{1}^{∗}, …, x_{c}^{∗}) that annihilate ExtR^{•}(M, N) ⊗_{R} k. The annihilator of a graded module is a graded ideal, so V_{R}^{∗}(M, N) is a cone with vertex at the origin. We set V_{R}^{∗}(M) = V_{R}^{∗}(M, k). The construction in two variables was introduced in [15]. It extends a direct construction of V_{R}^{∗}(M) in [11], itself partly inspired by support varieties associated to representations of finite groups; Benson’s book [23] provides a systematic account of the use of varieties in representation theory. We list three (among many) properties of support varieties.

7.4.1. Theorem (Avramov-Buchweitz [15]). Support varieties satisfy: (1) V_{R}^{∗}(M, M) = V_{R}^{∗}(M) = V_{R}^{∗}(k, M).

(2) V_{R}^{∗}(M, N) = {0} if and only if Ext_{R}^{n}(M, N) = 0 for n ≫ 0.

(3) V_{R}^{∗}(M, N) = V_{R}^{∗}(M) ∩ V_{R}^{∗}(N) = V_{R}^{∗}(N, M).

The dimension of the variety V_{R}^{∗}(M, N) measures ‘the size’ of Ext_{R}^{∗}(M, N). To make this precise, we define the complexity of the pair (M, N) by

cx_{R}(M, N) = inf \{ b ∈ ℕ | \begin{matrix}ν_{R}( Ext_{R}^{n}(M, N)) ≤ a n^{b−1} for some \\ real number a and for all n ≫ 0 \end{matrix} \}

where ν_{R}(E) denotes the minimal number of generators of an R-module E. Elementary facts from dimension theory of algebraic varieties give

cx_{R}(M, N) = dim V_{R}^{∗}(M, N)

On the other hand, cx_{R}(M, k) = cx_{R} M holds by definition, so basic results on intersection of homogeneous varieties turn the structural statements of the preceding theorem into numerical data on the growth of resolutions.

7.4.2. Corollary. The following (in)equalities hold:

(1) cx_{R}(M, M) = cx_{R} M.

(2) cx_{R}(M, N) = 0 if and only if Ext_{R}^{n}(M, N) = 0 for n ≫ 0.

(3) cx_{R} M + cx_{R} N − codim R ≤ cx_{R}(M, N)

= cx_{R}(N, M) ≤ min{cx_{R} M, cx_{R} N}.

The statements above are purely algebraic, but their proofs—and even the guess that they might hold—came from the geometric intuition provided by support varieties. The same remark applies to the following vanishing theorem for (co)homological functors over complete intersections.

7.4.3. Theorem (Avramov-Buchweitz [15]). The following are equivalent. (i) Ext_{R}^{n}(M, N) = 0 for h ≤ n ≤ h + c and some h > dim R. (ii) Ext_{R}^{n}(N, M) = 0 for i ≤ n ≤ i + c and some i > dim R. (iii) Tor_{n}^{R}(M, N) = 0 for j ≤ n ≤ j + c and some j > 0. (iv) Ext_{R}^{n}(M, N) = Ext_{R}^{n}(N, M) = Tor_{n}^{R}(M, N) = 0 for all n > dim R.

7.4.4. Remark. The theorem above yields an unexpected and remarkable feature of complete intersection local rings:

(ee) Every pair (M, N) of finite R-modules has the following property:

if Ext_{R}^{n}(M, N) = 0 for all n ≫ 0 then Ext_{R}^{n}(N, M) = 0 for all n ≫ 0. On the other hand, if condition (ee) holds for a local ring (R, 𝔪, k), then ap-

plied to the pair (R, k) it yields μ_{R}^{n}(R) = 0 for all n ≫ 0, so R is Gorenstein by Theorems (5.2.2) and (1.4.1). Thus, the class of rings satisfying condition (ee) lies between the complete intersection rings and the Gorenstein rings, cf. (B.2.1). Very recently, Huneke-Jorgensen [48] and Şega [63] have provided examples of noncomplete-intersection Gorenstein rings that satisfy (ee). It is not known whether all Gorenstein rings have this property.

8. Dimensions from deformations

In this section (R, 𝔪, k) denotes a local ring and M a finite R-module.

We describe new homological dimensions whose definitions involve two choices— that of a classical homological dimension, and that of a class of ring homomorphisms allowing descent without increasing the classical dimension. The procedure extends the one used for CI-dimension in (7.1). It partly formalizes the existence of Cohen presentations from Theorem (B.4.3), and the characterizations of properties of residue rings of regular local rings in terms of the defining ideal from Theorem (B.2.2).

8.1. Resolving categories and homomorphisms. Let C(R) be a resolving subcategory of the category F(R) a finite R-modules, for which the following holds: Whenever R → R′ is a faithfully flat homomorphism of local rings, M belongs to C(R) if and only if M ⊗_{R} R′ belongs to C(R′). The categories P(R) of projective R-modules, G(R) of totally reflexive R-modules, and B(R) of totally reflexive Rmodules of finite complexity all have this property.

Fixing a class h of surjective homomorphisms of local rings, we say that a diagram of local homomorphisms R → R′ ← Q is a h-quasi-deformation if R → R′ is flat extension and R′ ← Q belongs to h. We now set

h∥C-dim M = inf \{C(Q)-dim (M ⊗_{R} R′) − C(Q)-dim R′ | \begin{matrix}R → R′ ← Q is an \\ h-quasi-deformation \end{matrix}\}

Note that if h consists of isomorphisms, then h∥C-dim M = C-dim M, so the new batch of dimensions comprises those defined solely in terms of resolutions.

The dimension described in (7.1) also fits this mold.

8.2. Complete intersection dimension. If c denotes the class of surjective homomorphisms with kernel generated by a regular sequence, then

CI-dim_{R} M = c∥P-dim M

To introduce the next concept we use the notation of (B.1.3).

8.3. Upper Gorenstein dimension. Let g denote the class of surjective homomorphisms Q → R′ such that R′ is a perfect Q-module with R′_{Q}^{†} ≅ R′.

Veliche [68] introduces the upper Gorenstein dimension of M over R by

G^{∗}-dim_{R} M = g∥P-dim M

Below we use the notion of G-perfect module from (3.5).

8.4. Cohen-Macaulay dimension. Let p denote the class of all surjective homomorphisms Q → R′ making R′ into a G-perfect Q-module.

Gerko [43] defines the Cohen-Macaulay dimension of M over R to be

CM-dim_{R} M = p∥G-dim M

The new homological dimensions can be computed from any h-quasi-deformation that establishes its finiteness: this follows from standard change-of-rings results for depth, projective dimension, and G-dimension:

8.5. Lemma. If (h∥C) equals (c∥P) (respectively, (g∥P) or (p∥G)) and there is a h-quasi-deformation R → R′ ← Q with C(Q)-dim(M ⊗_{R} R′) < ∞, then

h∥C-dim M = C(Q)-dim(M ⊗_{R} R′) − C(Q)-dim R′

We now have a homological dimension of M over R, denoted H-dim_{R} M, defined for H = CI in (7.1), H = CI_{∗} in (6.3), H = G in (3.1), H = G^{∗} in (8.3), and H = CM (8.4). For uniformity of notation, in this section we set Reg-dim M = proj dim_{R} M. We say that the ring R has property Reg (respectively, CI, CI_{∗}, G, G^{∗}, CM) if it is regular (respectively, complete intersection, complete intersection, Gorenstein, Gorenstein, Cohen-Macaulay).

A feature common to all these dimensions is recorded in the next theorem: cf. Theorems (1.2.1), (7.1.2), (6.3.2), and (3.5.1) for the first four values of H, [68] for H = G^{∗}, and [43] for H = CM.

8.6. Theorem. For H equal to Reg (respectively, CI, CI_{∗}, G, G^{∗}, CM) the following conditions are equivalent.

(i) R has property H.

(ii) H-dim_{R} N ≤ depth R for every finite R-module N.

(iii) H-dim_{R} k < ∞.

In an omnibus statement, we collect properties common to most dimensions. They are textbook fare for H = Reg and come from [3] for H = G; once again, proofs for the remaining values of H are given in [17], [43], [68].

8.7. Theorem. Let (R, 𝔪, k) be a local ring.

For H equal to Reg (respectively, CI, CI_{∗}, G^{∗}, G, or CM) the homological dimension H-dim_{R}(M) has the following properties.

(0) H-dim_{R} M = −∞ if and only if M = 0.

(1) H-dim_{R} M = depth R − depth_{R} M if H-dim_{R} M < ∞.

(2) H-dim_{R} M ≤ H-dim_{R} k.

If F → M is a free resolution, then (3) H-dim_{R} M = max{H-dim_{R}(Ω^{1}F) − 1, 0} if M ≠ 0.

(4) H-dim_{R} M ≥ H-dim_{R_{𝔭}} M_{𝔭} for every prime ideal 𝔭 of R.

(5) H-dim_{R} M = H-dimR̂(M ⊗_{R} R̂).

If a local ring S is a faithfully flat R-algebra, then M′ = M ⊗_{R} S satisfies

(6) H-dim_{R} M ≤ H-dim_{S} M′
with equality if H-dim_{S}(M′) < ∞ or H ∈ {Reg, CI_{∗}, G}.

For an element y ∈ 𝔪 the module \overline{M} = M/yM and ring \overline{R} = R/(y) satisfy (7) H-dim_{R} M = H-dim_{R}(\overline{M}) − 1 when y is M-regular.

(8) H-dim_{R} M ≥ H-dim\overline{R}(\overline{M}) when y is M-regular and R-regular
with equality if H-dim_{R} M < ∞ or H ∈ {Reg, CI_{∗}, G}.
(9) H-dim_{R} M ≥ H-dim\overline{R} M + 1 when y is R-regular, yM = 0, H ≠ Reg
with equality if H-dim_{R} M < ∞ or H ∈ {CI_{∗}, G}.

H-dim_{R} M ≤ H-dim\overline{R} M + 1 when y is R-regular, yM = 0, H = Reg

with equality if H-dim\overline{R} M < ∞.

The two parts of (9) underscore the difference between projective and Gorenstein dimensions, recorded in Theorem (3.5.3). It is desirable to know if equalities always hold in (6), (8), and (9) for H equal to CI, G^{∗}, or CM, and whether these dimensions satisfy the inequality in Proposition (2.2.4). Next we bring the hierarchy of local rings (B.2.1) to the level of individual modules.

8.8. Theorem. Homological dimensions satisfy the inequalities

CM-dim_{R} M ≤ G-dim_{R} M ≤ CI_{∗}-dim_{R}M ≤ CI-dim_{R} M ≤ proj dim_{R} M

G-dim_{R} M ≤ G^{∗}-dim_{R} M ≤ CI-dim_{R} M

If any one of these dimensions is finite, then it is equal to those to its left.

The inequalities follow from the definitions, except for the upper bound for CI_{∗}-dim_{R}M and the lower bound for G^{∗}-dim_{R} M; they are proved in [17] and [68], respectively. The inequalities imply that if one of these dimensions is finite, then so are those to its left, which then are equal by Theorem (8.7.1). The next result illustrates the usefulness of various dimensions for classifying modules over general local rings (a different proof of (1) is given in [17]).

8.9. Theorem (Veliche [68]). Let (Q, 𝔫, k) be a local ring, which in part (n) below is assumed to satisfy depth Q ≥ n. There exists then a pair (J, N), where J is ideal contained in 𝔫^{2} such that the Q-module R = Q/J is perfect and N is a finite R-module, with the following property:

(1) grade_{Q} R ≥ 1 and 0 = CI-dim_{R} N < proj dim_{R} N = ∞.

(2) grade_{Q} R = 2 and 0 = CM-dim_{R} N < G-dim_{R} N = ∞.

(3) grade_{Q} R = 3, and 0 = G^{∗}-dim_{R} N < CI_{∗}-dim_{R}N = ∞. If Q is equicharacteristic, then furthermore

(4) grade_{Q} R = 4 and 0 = CI_{∗}-dim_{R}N < CI-dim_{R} N = ∞.

(5) grade_{Q} R = 5 and 0 = G^{∗}-dim_{R} M = CI_{∗}-dim_{R}N < CI-dim_{R} N = ∞.

No example of module N with CI_{∗}-dim_{R}N < G^{∗}-dim_{R} N or, more generally, with G-dim_{R} N < G^{∗}-dim_{R} N is known at present.

9. Curvature

In this section (R, 𝔪, k) denotes a local ring and M a finite R-module.

We first note that no sequence of Betti numbers can grow faster than exponentially, then exhibit classes of modules whose Betti sequences admit also an exponential lower bound. Many results on exponential growth have counterparts in rational homotopy theory, cf. Remark (9.4).

9.1. Upper bounds. We start with an easy computation.

9.1.1. Example. If R is artinian and length_{R} R = l, then

β_{n}^{R}(M) ≤ β_{0}^{R}(M)(l − 1)^{n} for each n ≥ 1

Indeed, if F → M is a minimal free resolution, then F_{n}/𝔪F_{n} ≅ (Ω^{n}F)/𝔪(Ω^{n}F) for n ≥ 0 and Ω^{n}F ⊆ 𝔪F_{n−1} for all n ≥ 1, hence

β_{n}^{R}(M) = length(F_{n}/𝔪F_{n}) ≤ length_{R}(Ω^{n}F)
≤ length(𝔪F_{n−1}) = (l − 1)β_{n−1}^{R}(M)

Iterating the inequality above, we obtain the desired exponential bound.

For non-artinian rings, the first inequality in the next proposition is somewhat harder to establish. For the second inequality, note that Proposition (5.3.4) with L = 𝔪^{s}M yields β_{n}^{R}(M) ≤ β_{n}^{R}(M/𝔪^{s}M) for all n, so it suffices to prove the assertion for modules of finite length. In that case a simple induction on length_{R} M shows that the desired inequality holds with λ = length_{R} M.

9.1.2. Proposition. The Betti numbers of M are bounded above as follows. (1) β_{n}^{R}(M) ≤ α^{n} for some real number α and each n ≥ 1.

(2) β_{n}^{R}(M) ≤ λ · β_{n}^{R}(k) for some real number λ and each n ≥ 0.

The inequality in Proposition (9.1.2.1) suggests the use of an exponential scale to measure the growth of Betti numbers over arbitrary local rings.

9.2. Curvature. The curvature of M over R is defined in [13] to be

curv_{R} M = lim sup \sqrt[n]{β_{n}^{R}(M)}

9.2.1. Remark. The first three relations below follow directly from the definitions, while the last one results from Proposition (9.1.2.2):

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

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

(3) cx_{R} M < ∞ ⟹ curv_{R} M ≤ 1.

(4) cx_{R} M ≤ cx_{R} k and curv_{R} M ≤ curv_{R} k < ∞.

A main open problem on the asymptotic behavior of Betti numbers is whether the converse of (3) holds. Whenever an answer is available, it is a consequence of some more precise statement on the growth of Betti numbers.

9.2.2. Theorem (Avramov [10], [14]). If 𝔭 is a prime ideal of R such that R_{𝔭} is not a complete intersection, then

β_{n}^{R}(R/𝔭) ≥ β^{n} for some β > 1 and for all n ≫ 0

The converse may fail: for R = k[t_{1}, t_{2}, t_{3}]/(t_{1}^{2}t_{3}, t_{2}^{2}t_{3}) and 𝔭 = (t_{1}, t_{2}) the ring R_{𝔭} is a complete intersection, but curv_{R}(R/𝔭) > 1.

An immediate application of the theorem is a characterization of complete intersections in terms of curvature, paralleling the one in terms of complexity in Proposition (6.2.1): that proposition provides the implication (i) ⟹ (ii) below, while the theorem yields (iii) ⟹ (i).

9.2.3. Corollary. The following conditions are equivalent:

(i) R is complete intersection.

(ii) curv_{R} N ≤ 1 for every finite R-module N.

(iii) curv_{R} k ≤ 1.

Over some classes of rings, the Betti numbers of all modules are known to satisfy the polynomial/exponential dichotomy.

9.2.4. Theorem (Avramov [12], Sun [67]). Set c = edim R − depth R. If c ≤ 3, or if R is Gorenstein and c ≤ 4, then either cx_{R} M ≤ c, or

β_{n}^{R}(M) ≥ β · β_{n−1}^{R}(M) for some β > 1 and for all n ≫ 0

9.2.5. Theorem (Peeva [55]). If length_{R} R ≤ 8, or if R is Gorenstein and length_{R} R ≤ 11, then either β_{n}^{R}(M) is eventually constant, or

β_{n}^{R}(M) ≥ β · β_{n−1}^{R}(M) for some β > 1 and for all n ≫ 0

9.3. Extremal modules. Motivated by Proposition (9.1.2.2), 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. By the next theorem, such modules are plentyful.

9.3.1. Theorem (Avramov [13]). If i ≥ 1 and 𝔪^{i}M ≠ 0, then the module 𝔪^{i}M is extremal.

The theorem is a quantitative sharpening of Theorem (1.2.2). It also implies that the Grothendieck group of R is generated by extremal R-modules:

9.3.2. Corollary. Set l = length_{R} M. If l = 1, then M is extremal. If l > 1, then there is an exact sequence 0 → L → M → N → 0 with extremal L, N.

Proof. Assume first 𝔪M = 0, so that M ≅ k^{l}; if l = 1 then M is extremal by definition; if l > 1 then there is an exact sequence 0 → k → M → k^{l−1} → 0 with the desired properties. Assuming 𝔪M ≠ 0, the theorem shows that 𝔪M is extremal, and the preceding discussion shows that so is M/𝔪M.

9.4. Remark. The proofs of many results in this section hinge on obtaining nontrivial information on the algebra ExtR^{•}(k, k) and on its action on ExtR^{•}(M, k), cf. (A.3). The cohomology algebra is itself the universal enveloping algebra of a functorial in R graded Lie algebra π^{•}(R), called the homotopy Lie algebra of R. It shares many properties with the graded Lie algebra of rational homotopy groups of a finite CW complex. This analogy is described in the papers [10], [18], along with applications of rational homotopy to local algebra, and vice versa.

Expositions. The survey [14] contains proofs of the results above.

10. Castelnuovo-Mumford regularity

We say that an algebra R over a field k is graded if R = ⨁_{n∈ℕ} R_{n} as a k-vector space, R_{h} · R_{i} ⊆ R_{h+i} for all h, i, and R_{0} = k.

10.1. Graded modules. An R-module M is graded if M = ⨁_{j∈ℤ} M_{j} with R_{i} · M_{j} ⊆ M_{i+j} for all i, j ∈ ℤ. For each s ∈ ℤ, form the graded submodule M_{⩾s} = ⨁_{j≥s} M_{j} of M, and define a graded module M(s) by setting M(s)_{j} = M_{s+j} for all j ∈ ℤ and endowing it with the obvious action of R. Set

inf M = inf{j ∈ ℤ | M_{j} ≠ 0} and sup M = sup{j ∈ ℤ | M_{j} ≠ 0}

We say that M is locally finite if rank_{k} M_{j} is finite for all j. Abusing notation, we set k = R/R_{⩾1}; up to shift, this is the only simple graded R-module.

Let M^{gr} denote the category of graded R-modules with morphisms the R-linear maps β: M → N satisfying β(M_{i}) ⊆ N_{i} for all i ∈ ℤ.

It is easy to produce for every M ∈ M^{gr} a graded free resolution F → M, with each F_{i} a direct sum of copies of R(−j). In some cases, more is known.

10.2. Minimal resolutions. Let M ∈ M^{gr} be a module with inf M > −∞.

10.2.1. Graded projective covers. Choose a k-linear section σ of the canonical projection M → M/R_{⩾1}M and form the map

R ⊗_{k} (M/R_{⩾1}M) ⟶ M  by setting  a ⊗ x ⟼ aσ(x)

It is easy to see that it is a projective cover of M in M^{gr}.

10.2.2. Minimal graded resolutions. Iterating the construction above, one obtains a free resolution F → M in which F_{i} → Ker(∂_{i−1}) is a graded projective cover for each i. By the graded version of Corollary (4.2.3) such a resolution is unique up to an isomorphism of complexes of graded R-modules.

It follows from the existence of minimal free resolutions that the global dimension of R is finite if and only if the projective dimension of k is finite. In the finitely generated commutative case, the algebras of finite global dimension are completely described by the next result, where (i) ⟹ (ii) is Hilbert’s Syzygy Theorem, while (iii) ⟹ (i) is a consequence of Theorem (1.2.1).

10.2.3. Theorem. If R is commutative and noetherian, then the following conditions are equivalent.

(i) R is a polynomial ring over k.

(ii) proj dim_{R} N < ∞ for every finite graded R-module N.

(iii) proj dim_{R} k < ∞.

The uniqueness of minimal free resolutions yields new numerical invariants.

10.2.4. Graded Betti numbers. The number β_{ij}^{R}(M) of summands of F_{i} isomorphic to R(−j) is an invariant of M, its ijth graded Betti number, so each graded R-module M module has a Betti diagram (not just a Betti sequence).

From the construction of graded projective covers one obtains equalities

β_{ij}^{R}(M) = 0  for all  j < i + inf M

and, if R and M are locally finite, inequalities β_{ij}^{R}(M) < ∞ for all i, j ∈ ℤ.

If R is noetherian and M is finite, then every module in a minimal free resolution of M has finite rank, so β_{ij}^{R}(M) = 0 for all j ≫ 0.

The next notion originates in algebraic geometry.

10.2.5. Castelnuovo-Mumford regularity. The number

reg_{R} M = sup {n ∈ ℤ | β_{i, i+n}^{R}(M) ≠ 0 for some i ∈ ℕ}

is known as the Castelnuovo-Mumford regularity of M over R.

In view of Hilbert’s Syzygy Theorem and the last remark in (10.2.4), finite graded modules over a graded polynomial ring have finite regularity. This invariant plays important role in computational commutative algebra, cf. [34].

The next result shows some respects in which regularity behaves like a homological dimension, cf. Theorem (8.7.2) and Proposition (2.2.4).

10.2.6. Proposition. There is an inequality reg_{R} M ≤ reg_{R} k · sup M If 0 → M → M′ → M′′ → 0 is an exact sequence in M^{gr}(R), then

reg_{R} M ≤ max{reg_{R} M′, reg_{R} M′′ − 1}

with equality if reg_{R} M′ ≠ reg_{R} M′′.

Proof. The second assertion is obtained by standard use of mapping cones and the Horseshoe Lemma. For the first assertion, one may assume both reg_{R} k and s = sup M are finite, then induce on s by applying the already proven statement to the exact sequence 0 → M_{s}(−s) → M → M/M_{s} → 0.

10.3. Koszul algebras. The algebra R is said to be Koszul if reg_{R} k = 0.

This is a simplification, widely adopted in texts on commutative algebra and algebraic geometry, of a notion introduced by Priddy [57]. Under this definition, a Koszul algebra is generated in degree 1 and has a set of quadratic defining relations. However, the Koszul property is much subtler and cannot be inferred from the knowledge of any finite number of Betti numbers of k.

10.3.1. Example (Roos [62]). Let t_{1}, …, t_{6} be commuting indeterminates of degree 1 over k. For each integer n form the graded algebra

R_{(n)} = \frac{k[t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}]}{\begin{matrix}( t_{1}t_{3} + nt_{3}t_{6} − t_{4}t_{6} , t_{1}t_{4} + t_{3}t_{6} + (n − 2)t_{4}t_{6} , \\ t_{1}^{2} , t_{2}^{2} , t_{3}^{2} , t_{4}^{2} , t_{5}^{2} , t_{6}^{2} , t_{1}t_{2} , t_{2}t_{3} , t_{3}t_{4} , t_{4}t_{5} , t_{5}t_{6} ) \end{matrix}}

If k has characteristic 0 and n ≥ 3, then there are equalities

β_{ij}^{R_{(n)}}(k) = 0 for all j ≠ i < n and β_{n,n+1}^{R_{(n)}}(k) ≠ 0

A graded resolution F → M is linear if there is an integer s, such that for each n ≥ 0 the free R-module F_{n} is generated in degree n + s. For instance, the Koszul property of R can be described by the condition that the residue field has a linear resolution. The following was conjectured by Kempf.

10.3.2. Theorem (Avramov-Eisenbud [16]). If R is commutative, noetherian, and Koszul, then for every finite R-module N there exists an integer m such that for each s ≥ m the module N_{⩾s} has a linear resolution.

It follows that M itself has finite regularity: the graded module L = M_{⩾s} has reg_{R} L ≤ s by the theorem, so one obtains reg_{R} M ≤ s by applying Proposition (10.2.6) to the exact sequence 0 → L → M → M/L → 0.

The last observation is generalized below.

10.4. Finite regularity. Let R be a commutative graded algebra.

The proof of the next result uses a graded version of Proposition (5.3.4).

10.4.1. Theorem (Avramov-Peeva [20]). If R is noetherian and reg_{R} k < ∞, then reg_{R} N < ∞ for every finite graded module N.

The next theorem, which does not require finite generation of R over k, pinpoints the relation of the Koszul property of R and the finiteness of reg_{R} k.

10.4.2. Theorem (Avramov-Peeva [20]). If reg_{R} k < ∞, then S = k[R_{1}] is a Koszul algebra and R is a polynomial ring in finitely many indeterminates over S.

Conversely, if R is a polynomial ring in indeterminates t_{1}, …, t_{ℓ} over a Koszul algebra, then reg_{R} k = ∑_{p=1}^{ℓ}(deg(t_{p}) − 1).

Combining the theorems above, we obtain a complete characterization of graded algebras over which all modules have finite regularity.

10.4.3. Corollary. If R is finitely generated as a k-algebra, then the following conditions are equivalent.

(i) R is a polynomial ring over a Koszul algebra.

(ii) reg_{R} N < ∞ for every finite R-module N.

(iii) reg_{R} k < ∞.

The form of the corollary brings to mind several results in this survey.

Appendix A. Complexes

Let R denote an associative ring, assumed to act on its modules from the left; right R-modules are treated as modules over the opposite ring, R^{o}.

A.1. Categories. We write M or M(R) for the category of R-modules. We only consider full additive subcategories of M, closed under isomorphisms.

For any module M, left or right, the dual M^{∗} = Hom_{R}(M, R) is given the canonical action on the other side. We let ( )^{∗} denote either functor

\xymatrix{M(R)^{op} \ar[r]^{Hom_{R}( ,R)} & M(R^{o})^{op} \ar[l]^{Hom_{R^{o}} ( ,R)}}

A.2. Complexes. A complex of R-modules is a sequence of R-linear maps

B = ⋯ ⟶ B_{n+1} ⟶^{∂_{n+1}^{B}} B_{n} ⟶^{∂_{n}^{B}} B_{n−1} ⟶ ⋯

with ∂_{n}^{B} ∘ ∂_{n+1}^{B} = 0 for all n. We call Ω^{n}B = Coker ∂_{n+1}^{B} the nth syzygy module of B. Whenever convenient, we identify a module M with the complex having M_{0} = M and M_{n} = 0 for n ≠ 0.

Every complex B of R-modules determines a complex of R^{o}-modules

B^{∗} = ⋯ ⟶ (B_{−n−1})^{∗} ⟶^{(∂_{−n}^{B})^{∗}} (B_{−n})^{∗} ⟶^{(∂_{−n+1}^{B})^{∗}} (B_{−n+1}^{∗})^{∗} ⟶ ⋯

where the module (B_{−n})^{∗} appears in degree n.

A morphism β: B → C of complexes is a sequence of R-linear maps β_{n}: B_{n} → C_{n} with ∂_{n}^{C}β_{n} = β_{n−1}∂_{n}^{B} for all n ∈ ℤ. It is a quasiisomorphism if the induced map H_{n}(β): H_{n}(B) → H_{n}(C) is bijective for every n.

If β, β′: B → C are morphisms, then a homotopy from β to β′ is a sequence θ of R-linear maps θ_{n}: B_{n} → C_{n+1} such that β′_{n} = β_{n} + ∂_{n+1}^{C}θ_{n} + θ_{n−1}∂_{n}^{B} for all n ∈ ℤ. If such a homotopy exists, then β′ is said to be homotopic to β, denoted β′ ∼ β; when this is the case, H_{n}(β) = H_{n}(β′) for all n ∈ ℤ.

A complex A is said to be contractible (or split exact) if id_{A} ∼ 0_{A}.

A morphism β is a homotopy equivalence if it has a homotopy inverse, that is, a morphism γ such that β γ ∼ id_{C} and γ β ∼ id_{B}. The following implications, non-invertible in general, obviously hold for every morphism:

isomorphism ⟹ homotopy equivalence ⟹ quasiisomorphism

Let M be an R-moduled and let C be a subcategory of M(R). A C-resolution (respectively, C-coresolution) of M is a quasiisomorphism γ: C → M (respectively, M → C) where C a complex C with C_{n} ∈ C for all n ∈ ℤ and C_{n} = 0 for all n < 0 (respectively, for all n > 0).

A.3. Products in (co)homology. Let L, M, N be R-modules.

Comparison morphisms for projective resolutions provide natural maps

Ext_{R}^{i}(M, N) ⊗_{ℤ} Ext_{R}^{j}(L, M) ⟶ Ext_{R}^{i+j}(L, N)

These pairings are associative, and so give ExtR^{•}(N, N) ExtR^{•}(M, M) structures of graded R-algebras, and ExtR^{•}(M, N) a structure of bimodule with ExtR^{•}(N, N) acting from the left and ExtR^{•}(M, M) from the right.

Comparison morphisms also define natural homomorphisms of R-modules

Ext_{R}^{i}(M, M) ⊗_{ℤ} Tor_{j}^{R}(M, N) ⟶ Tor_{j−i}^{R}(M, N)
Tor_{j}^{R}(M, N) ⊗_{ℤ} Ext_{R}^{i}(N, N) ⟶ Tor_{j−i}^{R}(M, N)

that turn Tor_{j}^{R}(M, N) into a bimodule.

Appendix B. Local rings

In this Appendix (R, 𝔪, k) denotes a local ring, that is, R is a commutative noetherian ring, 𝔪 is its unique maximal ideal, and k is its residue field R/𝔪. Furthermore, M denotes an R-module which is assumed to be finite, in the sense that it is finitely generated over R. We define some basic numerical invariants attached to M and R, and to use them to give concise definitions of the main classes of commutative local rings.

B.1. Depth. This concept extends the notion of non-invertible-non-zero-divisor: A sequence y = y_{1}, …, y_{r} of elements of R is M-regular if yM ≠ M, and for i = 1, …, r multiplication with y_{i} gives an injective map

M/(y_{1}, …, y_{i−1})M ⟶^{y_{i}} M/(y_{1}, …, y_{i−1})M

Let I be an ideal of R. Due to the noetherian character of R, every M-regular sequence y contained in I can be extended to one which is maximal with respect to inclusion. The depth of I on M is the number

depth_{R}(I, M) = sup{r ∈ ℕ | I contains an M-regular sequence of length r}

B.1.1. Theorem (Rees [58]). Let I be an ideal of R, let L be some finite R-module with rad(ann_{R}(L)) = rad(I), and let y be a maximal M-regular sequence in I. If depth_{R}(I, M) = g, then the following then hold.

(1) y contains g elements.

(2) g = inf {n ∈ ℕ | Ext_{R}^{n}(L, M) ≠ 0}.

(3) Ext_{R}^{g}(R/I, M) ≅ Hom_{R}(R/I, M/yM).

(4) Every sequence of elements of R minimally generating (y) is M-regular.

A useful way to compute the depth of the ideal generated by a subset y = {y_{1}, …, y_{r}} ⊆ R is by using the Koszul complex K(y; M). It is concentrated in degrees 0 through r and may be defined in two steps as follows:

K(y_{1}; R) = 0 ⟶ R ⟶^{y_{1}} R ⟶ 0 and K(y; R) = ⨂_{i=1}^{r} K(y_{i}; R)

The next result is obvious for r = 1; the general case is obtained by induction.

B.1.2. Theorem (Auslander-Buchsbaum [6]). If y = {y_{1}, …, y_{r}}, then

depth_{R}((y), M) = r − sup{n ∈ ℤ | H_{n} ( K(y; R) ⊗_{R} M) ≠ 0}

The number depth_{R}(𝔪, M) is called the depth of M over R, denoted depth_{R} M (the subscript is dropped if M = R). By the convention sup{∅} = −∞ and Nakayama’s Lemma, depth_{R} M = −∞ if and only if M = 0.

B.1.3. Grade. The grade of M over R, denoted grade_{R} M, is defined to be the number grade_{R} M = depth_{R}(ann_{R} M, R) if ann_{R} M = 0, and to be 0 otherwise; we set M_{R}^{†} = Ext_{R}^{g}(M, R), where g = grade_{R} M.

By Theorem (B.1.1.2), if M ≠ then grade_{R} M ≤ proj dim_{R} M. If equality holds, then M is said to be perfect. A P-resolution P → M of length g = proj dim_{R} M then yields Ext_{R}^{n}(M, R) = H^{n} Hom_{R}(P , R) = 0 for n ≠ g. It follows that the R-module M_{R}^{†} = H^{g} Hom_{R}(P , R) is perfect, has proj dim_{R}(M_{R}^{†}) = g, and satisfies M_{R}^{†}_{R}^{†} ≅ M.

B.2. Hierarchy of local rings. Let (R, 𝔪, k) be a local ring.

If there exists an R-regular sequence y such that the residue ring R/(y)

• is a field, then R is called regular.

• is isomorphic to R′/(y′) for some regular ring R′ and some R′-regular sequence y′, then R is called complete intersection.

• has finite length and Hom_{R}(k, R/(y)) ≅ k, then R is called Gorenstein.

• has finite length, then R is called Cohen-Macaulay. These properties can be introduced in several equivalent ways. Our choice of definitions is tailored to the needs of the survey.

B.2.1. Remark. The following implications hold:

regular ⟹ complete intersection ⟹ Gorenstein ⟹ Cohen-Macaulay

To verify the first implication set R′ = R and y′ = ∅. For the middle one, note the isomorphisms Hom_{R}(k, R/(y)) ≅ Hom_{R′}(k, R′/(y′)) ≅ Hom_{R′}(k, k) ≅ k, where the second one is given by Theorem (B.1.1.3) because the maximal ideal 𝔪′ of R′ is generated by an R′-regular sequence. The last implication holds tautologically.

Local rings often appear as homomorphic images of regular local rings. In algebraic geometry, these are localizations of polynomial rings over a field. In analytic geometry, these are rings of convergent power series (usually over ℂ or ℝ). Theorem (B.4.3) yields a third source of such presentations. It is therefore important to be able to recognize when a residue ring of a regular ring belongs to one of the basic classes.

B.2.2. Theorem. If (Q, 𝔫, k) is a regular local ring, J ⊂ Q an ideal, x a minimal set of generators of J and R = Q/J, then the following hold.

(1) R is regular if and only if x is linearly independent modulo 𝔫^{2}.

(2) R is complete intersection if and only if x is a regular sequence.

(3) R is Gorenstein if and only if the Q-module R is perfect and R_{Q}^{†} ≅ R.

(4) R is Cohen-Macaulay if and only if the Q-module R is perfect.

B.3. Dimension. The primary measure of the size of M is its (Krull) dimension dim_{R} M, equal to the least d ∈ ℕ for which there is a set b = {b_{1}, …, b_{d}} ⊆ 𝔪, such that the R-module M/bM has finite length. One sets dim R = dim_{R} R. Note that dim R = 0 if and only if R is artinian.

The cornerstone of dimension theory is Krull’s Principal Ideal Theorem. Here is a generalization for modules:

B.3.1. Theorem. If M has finite length and R^{s} → R^{r} → M → 0 is an exact sequence, then dim R = 0 or dim R ≤ s − r + 1.

Inequalities link depth, dimension, and the number edim R = rank_{k}(𝔪/𝔪^{2}), called the embedding dimensionnequalities of R; the last one comes by applying Theorem (B.3.1) to an exact sequence R^{s} → R → k → 0 with s = edim R.

B.3.2. Theorem. depth_{R} M ≤ dim_{R} M ≤ dim R ≤ edim R.

We can now give standard definitions of two of the concepts defined in (B.2); for the other two, cf. Remark (B.4.4) and Theorem (1.4.1).

B.3.3. Remark. R is regular if and only if dim R = edim R.

R is Cohen-Macaulay if and only if dim R = depth R.

B.4. Completion. The 𝔪-adic topology is the linear topology in which a basis of neighborhoods of 0 is given by the family of ideals {𝔪^{n}}_{n⩾0}. The completion R̂, as usual, consists of equivalence classes of Cauchy sequences, and the completion map R → R̂ sends each element to the class of the constant sequence it determines.

B.4.1. Theorem. R̂ is a local ring with maximal ideal 𝔪R̂ and residue field k; its depth, dimension, and embedding dimension are equal to those of R.

The completion map is a faithfully flat homomorphism, and so injective.

The ring R̂ is regular (respectively, complete intersection, Gorenstein, Cohen-Macaulay) if and only R has the corresponding property.

The completion map is bijective if and only if R is complete.

B.4.2. Examples. If R is artinian, then 𝔪 is nilpotent, so the 𝔪-adic topology is discrete, hence R is trivially complete. If R is complete, then so is the ring R[[t]] of formal power series in any finite set of indeterminates t. For every R, the completion of R[t]_{(𝔪,t)} is isomorphic to R̂[[t]].

A local ring is called equicharacteristic if it is an algebra over some field.

B.4.3. Theorem (Cohen [30]). If (R, 𝔪, k) is any local ring, then R̂ ≅ Q/J for some complete regular local ring (Q, 𝔫, k), which can be chosen to be a ring of formal power series over the field k if R is equicharacteristic, and over a complete discrete valuation ring with residue field k otherwise.

Any isomorphism R̂ ≅ Q/J where Q ia a complete regular local ring is called a Cohen presentation of R̂. There always exists a Cohen presentation R̂ ≅ Q/J which is minimal, in the sense that edim Q = edim R.

As an application of Cohen’s Theorem, we give the standard definition of the complete intersection property.

B.4.4. Remark. If R is complete intersection, then in every Cohen presentation R̂ ≅ Q/J the ideal J can be generated by a regular sequence. Conversely, if in some presentation R̂ ≅ Q/J the ideal J is generated by a regular sequence, then R is a complete intersection.

Indeed, by Theorem (B.4.1), R is complete intersection if and only if so is R̂. By Theorem (B.2.2.2), R̂ is complete intersection if and only if J is generated by a regular sequence x = x_{1}, …, x_{c}.

Expositions. Modern accounts of commutative ring theory are built around the concepts discussed above. The books of Matsumura [52], Bruns and Herzog [26], and Eisenbud [35] provide complementary approaches to the material.

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Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

   E-mail address: avramov@math.purdue.edu