COMPLETE INTERSECTION DIMENSION

Luchezar L. Avramov, Vesselin N. Gasharov, and Irena V. Peeva

Abstract. A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI-dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions.

Contents

Introduction

1. Homological dimensions

2. Quantum regular sequences

3. Constructions of modules of finite CI-dimension

4. Cohomology operators

5. Complexity

6. Homological redundancy

7. Critical degree and growth of Betti numbers

8. Asymptotes of Betti sequences

9. Obstructions to embeddings into standard resolutions References

Introduction

Let M be a finite module over a commutative noetherian local ring R. There always exists a minimal free resolution F of M over R, which is unique up to isomorphism, and defines the Betti numbers b_{n}^{R}(M) = rank_{R} F_{n}. When F is finite, its structure is very rigid, but little is known in the infinite case. To some extent, this is due to intrinsic numerical difficulties: the Betti sequence of the residue field k of R, may be non-recurrent [2], and has exponential growth, unless R is a complete intersection [6]. Over such an R, the Betti sequence of k is given by a polynomial [36], and the Betti sequence of any M is eventually equal to a quasi-polynomial of period 2, [23]. Thus, the numerical behavior at infinity of free resolutions over complete intersections is not so daunting.

Our main objective is the introduction and study of a class of modules which afford a highly non-trivial, yet tractable, theory of minimal free resolutions. As necessary ingredients for such a study, we develop a conceptual framework for describing infinite resolutions, and new techniques for their analysis. The point of view of asymptotic homological algebra

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). Primary 13D02, 13D05, 13D25, 13D40.

Secondary 13C40, 13H10.

The first author was partly supported by NSF Grant No. DMS-9102951 .

[8], [9], refocuses attention from properties of an individual module to those of its entire sequence of syzygies, and thus detects the emergence of remarkable patterns at infinity.

∗ ∗ ∗

In Section 1 we define a new homological invariant for a finite module M over a noetherian ring P, its complete intersection dimension CI-dim_{P} M. It is finite for all modules over a complete intersection, and the example is paradigmatic, exactly as modules over a regular local ring provide the paradigm of modules of finite projective dimension. There is more to the concept: exactly as the complete intersection property of a ring is intermediate between its being Gorenstein and regular, the new dimension interpolates between the G-dimension of [3] and the classical projective dimension:

G-dim_{P} M ≤ CI-dim_{P} M ≤ pd_{P} M ,

and equalities hold to the left of any finite dimension.

CI-dimension shares many basic properties with other homological dimensions. In particular, it localizes, so it is more flexible than the strictly local virtual projective dimension of [8], which it generalizes. Over a local ring R it satisfies an analog of the Auslander-Buchsbaum and Auslander-Bridger Equalities: if CI-dim_{R} M is finite, then it equals depth R − depth_{R} M. The finiteness of CI-dim_{R} k implies that the ring R is a complete intersection, giving a CI version of Serre’s characterization of regularity and Auslander and Bridger’s characterization of Gorensteinness.

The class of modules of finite CI-dimension contains all modules of finite projective dimension and all modules over a complete intersection. In Section 3 we show that the notion is meaningful in a much wider context, by constructing modules of finite CI-dimension and infinite projective dimension over any ring which has an embedded deformation.

∗ ∗ ∗

An initial estimate of the size of a F is given by the complexity cx_{R} M of M, that is, the smallest integer d such that lim_{n→∞} b_{n}^{R}(M)/n^{d} = 0, cf. [1], [8]. When CI-dim_{R} M is finite, so is cx_{R} M, due essentially to Gulliksen [23]. In Section 5 we establish a uniform bound on complexities: If CI-dim_{R} M < ∞, then

cx_{R} M ≤ edim R − depth R ,

with strict inequality unless R is a complete intersection. Since complexity provides a polynomial scale for infinite projective dimensions, the inequality may be viewed as a version of the fact that depth R is a universal bound on the finite projective dimensions.

In sections 7 and 8 we obtain definitive results on the behavior of the Betti sequence for a module of finite CI-dimension. An important role is played by the discovery of a “marker” which signals a place after which F starts to exhibit asymptotically stable patterns. This critical degree cr deg_{R} M is equal to the projective dimension when pd_{R} M < ∞, but the finiteness of cr deg_{R} M is a much weaker condition, and is implied by CI-dim_{R} M < ∞.

We prove that the Betti sequence of a module of finite CI-dimension is either constant or strictly increasing after cr deg_{R} M steps, that asymptotically there are equalities

b_{n}^{R}(M) = o(n^{d−1})  and  b_{n}^{R}(M) − b_{n−1}^{R}(M) = O(n^{d−2}) with d = cx_{R} M ,

and that the order cannot be improved in the second one. These facts are obtained as consequences of a very precise statement on the form of the Poincaré series of M. They are new even over complete intersections, and provide the last step in a proof that over local rings of small codimension, or linked in few steps to complete intersections, all Betti series are eventually non-decreasing.

∗ ∗ ∗

When R = Q/(x) for some Q-regular sequence x, Shamash [33] and Eisenbud [18] produce a not necessarily minimal standard resolution of M over R, by using higher order structures on a minimal resolution of M over Q. In Section 6 we investigate how tightly a standard resolution approximates the minimal one. Contrary to many expectations, we prove that this approximation is almost always very weak.

Previous work on modules over a ring R with a deformation Q has extensively used Gulliksen’s finiteness theorem [23]: if Ext_{Q}^{∗}(M, N) is finite over R for some R–module N, then Ext_{R}^{∗}(M, N) is finite over a ring R[χ] of cohomology operators. To study these operators we develop in Section 4 a novel type of change-of-rings spectral sequence, and obtain a perfect converse to Gulliksen’s theorem. We also use it in Section 9 to construct obstructions to the validity of a conjecture [18] on the existence of graded R[χ]–module structure on minimal free resolutions, and produce modules for which they do not vanish.

∗ ∗ ∗

This paper brings new tools to the study of resolutions over commutative rings.

Much of our analysis is based on the Yoneda product structure of Ext_{R}^{∗}(M, k) as a graded left Ext^{∗}(k, k)– and right Ext_{R}^{∗}(M, M)–bimodule. In general these algebras are neither finitely generated nor commutative. We expand on a result of Mehta [30] to show that if CI-dim_{R} M < ∞, then Ext_{R}^{∗}(M, k) is finite over the subalgebra of Ext_{R}^{∗}(M, M) generated by central elements of degree two. Furthermore, by [12] it is also finite over the subalgebra of Ext^{∗}(k, k) generated by the degree 2 part of the center of the homotopy Lie algebra π^{∗}(R): this is crucial in obtaining the universal bound on cx_{R} M, as deep results of Félix et al. [20] on the radical of π^{∗}(R) can be applied.

Non-commutative ring theory is used to produce modules of finite CI-dimension. In Section 2 we develop the new concept of a quantum regular sequence of endomorphisms of a module, which generalizes the classical notion of regular sequence. To study such sequences we introduce constructions over quantum symmetric algebras, in particular an extension of Manin’s [28] quantum Koszul complex. The application of quantum techniques in the commutative context is so efficient as to suggest possibilities of further interaction.

1. Homological dimensions

In this section, R is a commutative noetherian ring and M is a finite R–module.

A primary motive for the introduction of a new homological dimension is to describe a class of modules over a local ring with tractable minimal free resolutions. There are two basic ways of changing a local ring R to a local ring R′ by a local homomorphism (that is, one which maps the maximal ideal of R into that of R′) without introducing complications into the homological structure of M. If R ⟶ R′ is a local flat extension, then the structure of a minimal free resolution of M over R is essentially the same as that of M′ = M ⊗_{R} R′ over R′. If R′ ⟶ R is a (codimension c) deformation, by which we mean a surjective local homomorphism with kernel generated by a (length c) regular sequence, then the resolution of M over R can only be simpler than over R.

We describe a notion which incorporates these two classes of maps.

(1.1) Quasi-deformations. A diagram of local homomorphisms R ⟶ R′ ⟵ Q, with R ⟶ R′ a flat extension and R′ ⟵ Q a (codimension c) deformation. When M is an R–module and R ⟶ R′ ⟵ Q is a quasi-deformation, we set M′ = M ⊗_{R} R′.

The next definition describes the modules that have finite projective dimension “up to quasi-deformation.” As we are targeting properties of minimal resolutions, the new homological dimension is introduced locally. We write (R, 𝔪, k) or (R, 𝔪) to denote a local ring R with maximal ideal 𝔪 and residue field k = R/𝔪.

(1.2) Complete intersection dimension. For a module M ≠ 0 over a local ring R, set

CI-dim_{R} M = inf { pd_{Q} M′ − pd_{Q} R′ | R ⟶ R′ ⟵ Q is a quasi-deformation } ,

and complement this by CI-dim_{R} 0 = 0. The CI-dimension (which is shorthand for complete intersection dimension, of a module M over a noetherian ring R is defined to be the number

CI-dim_{R} M = sup { CI-dim_{R_{𝔪}} M_{𝔪} | 𝔪 ∈ Max R } .

The choice of terminology is motivated by the next two theorems. Recall that R is a complete intersection if the defining ideal of some Cohen presentation of the 𝔪-adic completion R̂ as a quotient of a regular ring can be generated by a regular sequence; when this is the case, any Cohen presentation has the corresponding property.

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

If R is a complete intersection, then each R–module M has finite CI-dimension.

If CI-dim_{R} k < ∞, then R is a complete intersection.

This result should be viewed in the context of homological characterizations of other local properties. By the Auslander–Buchsbaum–Serre theorem if R is regular, then pd_{R} M < ∞ for each R–module M, and pd_{R} k < ∞ implies R is regular. A similar description of Gorenstein rings is due to Auslander and Bridger [5, (4.20)]. It uses their generalizeation of the classical projective dimension pd_{R} M, based on properties of the functor M^{∗} = Hom_{R}(M, R) and of the canonical biduality map β_{M}: M ⟶ M^{∗∗}.

More precisely, G-dim_{R} M = 0 means that M is a reflexive (that is, β_{M} is bijective), and Ext^{i}(M, R) = 0 = Ext^{i}(M^{∗}, R) for i ≠ 0. In general, the Gorenstein dimension (or: G-dimension) of M is the infimum G-dim_{R} M of those n for which there exists an exact sequence 0 ⟶ P_{n} ⟶ … ⟶ P_{1} ⟶ P_{0} ⟶ M ⟶ 0 with G-dim_{R} P_{i} = 0 for i ∈ ℤ.

The place of CI-dimension in the hierarchy of homological dimensions is determined by

(1.4) Theorem. For each finite module M over a noetherian ring R there are inequalities

G-dim_{R} M ≤ CI-dim_{R} M ≤ pd_{R} M .

If some of these dimensions is finite, then it is equal those on its left.

If R is local and CI-dim_{R} M < ∞, then CI-dim_{R} M = depth R − depth_{R} M.

The rank of the n’th free module in the (unique up to isomorphism) minimal free resolution of a finite module M over a local ring R is known as the n’th Betti number of M over R. Betti numbers can be computed from the equality b_{n}^{R}(M) = rank_{k} Ext_{R}^{n}(M, k), and are often most conveniently studied through the generating function P_{M}^{R}(t) = ∑_{n=0}^{∞} b_{n}^{R}(M)t^{n}, known as the Poincaré series of M over R. We include a variation on known results.

(1.5) Lemma. For a codimension c quasi-deformation R ⟶ R′ ⟵ Q, an R–module M, and M′ = M ⊗_{R} R′, there are coefficientwise (in)equalities of formal power series

P_{M}^{R}(t) = P_{M′}^{R′}(t) ;
P_{M′}^{R′}(t) ≤ P_{M′}^{Q}(t)(1 − t^{2})^{−c};
P_{M′}^{Q}(t) ≤ P_{M′}^{R′}(t)(1 + t)^{c} .

Proof. For the equality observe that if F is a minimal free resolution of M over R, then F ⊗_{R} R′ is such a resolution of M′ over R′. For the inequalities it suffices to treat the case c = 1. The standard change-of-rings spectral sequence

2E^{pq} = Ext_{R′}^{p}(M′, Ext_{Q}^{q}(R′, ℓ)) ⟹ Ext_{Q}^{p+q}(M′, ℓ) ,

with ℓ denoting the residue field of R′, degenerates to a long exact sequence

… ⟶ Ext_{R′}^{n−2}(M′, ℓ) ⟶ Ext_{R′}^{n}(M′, ℓ) ⟶ Ext_{Q}^{n}(M′, ℓ) ⟶ Ext_{R′}^{n−1}(M′, ℓ) ⟶ ….

For each n ≥ 0 we get b_{n}^{R′}(M′) ≤ ∑_{i⩾0} b_{n−2i}^{Q}(M′) and b_{n}^{Q}(M′) ≤ b_{n}^{R′}(M′) + b_{n−1}^{R′}(M′): these are transcripts of the desired inequalities. □

Proof of Theorem (1.3). In view of the Auslander–Buchsbaum–Serre characterization of regular local rings, the first assertion follows immediately from the definition in (1.2).

Conversely, let R ⟶ R′ ⟵ Q be a quasi-deformation such that the Q–module R′/𝔪R′ = (R/𝔪) ⊗_{R} R′ has finite projective dimension. By (1.5) the sequence b_{n}^{R}(R/𝔪) is bounded by a polynomial in n, hence [24; (2.3)] shows that R is a complete intersection. □

Recall from [8] that a module over a local ring R has finite virtual projective dimension if its completion has finite projective dimension over some deformation of R̂. It is not known whether localizations of a module of finite virtual projective dimension also have this property. However, such a module has finite CI-dimension, and we show next that so do its localizations. Finiteness of CI-dimension is thus not only a more general property, but also a more natural one.

(1.6) Proposition. For any multiplicatively closed subset U of R there is an inequality CI-dim_{U^{−1}R}(U^{−1}M) ≤ CI-dim_{R} M, and furthermore

CI-dim_{R} M = sup { CI-dim_{R_{𝔭}} M_{𝔭} | 𝔭 ∈ Supp_{R} M } .

Proof. It suffices to establish the last formula, so we need to show that CI-dim_{R_{𝔭}} M_{𝔭} ≤ CI-dim_{R} M for R local and 𝔭 ∈ Supp_{R} M. Let R ⟶ R′ ⟵ Q be a quasi-deformation. By faithful flatness, pick in R′ a prime 𝔭′ lying over 𝔭, let 𝔮 be its inverse image in Q, and note that the induced diagram R_{𝔭} ⟶ R′_{𝔭′} ⟵ Q_{𝔮} is a quasi-deformation. The (in)equalities

pd_{Q_{𝔮}} M′_{𝔮} ≤ pd_{Q} M′ and pd_{Q_{𝔮}} R′_{𝔮} = pd_{Q} R′

finish the proof, since M′_{𝔮} ≅ M_{𝔭} ⊗_{R_{𝔭}} R′_{𝔭′} and R′_{𝔮} ≅ R′_{𝔭′} as Q_{𝔮}–modules. □

(1.7) Lemma. If R is a local ring, R ⟶ R′ ⟵ Q is a quasi-deformation, and M is a non-zero R–module, then

pd_{Q} M′ − pd_{Q} R′ = \{ \begin{matrix}depth R − depth_{R} M & if pd_{Q} M′ < ∞; \\ ∞ & if pd_{Q} M′ = ∞ . \end{matrix}

Proof. Note that depth_{R′} M′ − depth_{R} M = depth R′ − depth R = depth(R′/𝔪R′), cf. e.g. [29; (23.3)]. Thus, when pd_{Q} M′ is finite the Auslander–Buchsbaum Equality yields

pd_{Q} M′ − pd_{Q} R′ = (depth Q − depth_{Q} M′) − (depth Q − depth_{Q} R′)
= depth_{Q} R′ − depth_{Q} M′ = depth R′ − depth_{R′} M′
= depth R − depth_{R} M .  □

(1.8) Syzygies. The n’th syzygy of a finite module M over a local ring R is defined uniquely up to isomorphism by Syz_{n}^{R}(M) = Coker ∂_{n+1}, where (F, ∂) is a minimal free resolution of M. The exact sequence of R–modules with ∂_{n}(F_{n}) ⊆ 𝔪F_{n−1} for n ≥ 1

(1.8.1)  0 ⟶ Syz_{n}^{R}(M) ⟶ F_{n−1} ⟶^{∂_{n−1}} F_{n−2} ⟶ … ⟶ F_{1} ⟶^{∂_{1}} F_{0} ⟶ M ⟶ 0 .

shows that when R ⟶ R′ is a local flat extension, then

(1.8.2)  Syz_{n}^{R′}(M′) ≅ Syz_{n}^{R}(M) ⊗_{R} R′ .

(1.9) Lemma. If M ≠ 0 is a finite module over a local ring R, then

(1.9.1)  CI-dim_{R} Syz_{n}^{R}(M) = max { CI-dim_{R} M − n, 0 } for n ≥ 0 .

If furthermore CI-dim_{R} M is finite, then the following also hold:

(1.9.2)  depth_{R} Syz_{n}^{R}(M) = min { depth_{R} M + n, depth R } for 0 ≤ n ≤ pd_{R} M .

Proof. If R ⟶ R′ ⟵ Q is a quasi-deformation, then it is easily seen from (1.8) that CI-dim_{R} Syz_{n}^{R}(M) is infinite if and only if CI-dim_{R} M is. For the rest of the proof we assume that CI-dim_{R} M is finite. In view of (1.7), then CI-dim_{R} M = depth R−depth_{R} M, hence it suffices to prove (1.9.2). Changing notation, we may also assume pd_{Q} M′ < ∞. If x = x_{1}, …, x_{c} is a regular sequence generating Ker(Q ⟶ R′), then (x)M′ = 0 implies

(1.9.3)  pd_{Q} M′ ≥ grade_{Q} M′ ≥ c = pd_{Q} R′ .

Now (1.7) yields depth_{R} M ≤ depth R, so we conclude by counting depths in (1.8.1). □

Proof of Theorem (1.4). Both the projective dimension and the Gorenstein dimensions can be computed locally, in the sense that there are equalities

pd_{R} M = sup { pd_{R_{𝔭}} M_{𝔭} | 𝔭 ∈ Supp_{R} M };
G-dim_{R} M = sup { G-dim_{R_{𝔭}} M_{𝔭} | 𝔭 ∈ Supp_{R} M } :

this is classical for the first dimension and proved in [5; (4.15)] for the second one. As the CI-dimension has the corresponding property by (1.6), it suffices to treat the local case.

If the projective dimension of M over R is finite, then so is its complete intersection dimension, because the constant quasi-deformation R ⟶^{=} R ⟵^{=} R appears in the determination of CI-dim_{R} M. By expressing pd_{R} M from the Auslander-Buchsbaum Equality and CI-dim_{R} M from (1.7), we see that they are actually equal.

Suppose next that n = CI-dim_{R} M is finite. By (1.9) the CI-dimension of Syz_{n}^{R}(M) is zero and its depth is equal to that of R. Thus, after changing notation, we may may assume that CI-dim_{R} M = 0 and depth_{R} M = depth R, and we want to show that G-dim_{R} M = 0.

Choose a codimension c quasi-deformation R ⟶ R′ ⟵ Q with pd_{Q} M′ < ∞. By the Auslander-Buchsbaum Equality we then have pd_{Q} M′ = pd_{Q} R′ = c, and the inequalities (1.9.3) imply that the Q–module M′ is perfect of projective dimension c.

For any R′–module L there are well known change of rings isomorphisms due to Rees:

(1.9.4)  Ext_{R′}^{n}(L, R′) ≅ Ext_{Q}^{n+c}(L, Q) for n ∈ ℤ .

With L = M′ they show that Ext_{R′}^{n}(M′, R′) = 0 for n > 0, and that the Q–module M′^{∗} = Hom_{R′}(M′, R′) is perfect of projective dimension c and naturally isomorphic to Ext_{Q}^{c}(M′, Q). With L = M′^{∗} the Rees formulas yield Ext_{R′}^{n}(M′^{∗}, R′) = 0 for n > 0, and M′^{∗∗} ≅ Ext_{Q}^{c}(Ext_{Q}^{c}(M′, Q), Q). As M′ is perfect, the last module is isomorphic to M′, hence G-dim_{R′} M′ = 0. In view of the faithful flatness of R′ over R, it follows easily from the definition that G-dim_{R′} M′ = 0 implies G-dim_{R} M = 0, as desired.

We have now shown that G-dim_{R} M ≤ CI-dim_{R} M. Thus, if the CI-dimension of M is finite, then so is its G-dimension. By [5; (4.13.b)] we then have G-dim_{R} M = depth R − depth_{R} M, and (1.7) shows that this is precisely CI-dim_{R} M. □

We establish change of module and ring formulas for CI-dimension under certain types of ring homomorphisms. The arguments are based on the following construction.

(1.10) Compositions. Let R ⟶^{ρ′} R′ ⟵^{ρ} Q and Q ⟶^{κ′} Q′ ⟵^{κ} P be quasi-deformations. The lower row of the following diagram

P
↓^{κ}
Q ⟶^{κ′}  Q′
ρ↓  ↓^{ρ′⊗_{Q}Q′}
R ⟶_{ρ′} R′ ⟶_{R′⊗_{Q}κ′} R′ ⊗_{Q} Q′ .

is then a composition of local flat extensions, and its right hand column is a composition of deformations. Thus, the resulting diagram R ⟶ R′ ⊗_{Q} Q′ ⟵ P is a quasi-deformation, which we call the composition of the initial two. This operation is associative and yields a category whose objects are local rings and whose morphisms are quasi-deformations, with the identity morphism of R given by the constant quasi-deformation R ⟶^{=} R ⟵^{=} R.

(1.11) Lemma. If R ⟶ R′ ⟵ Q is a codimension c quasi-deformation and M is a finite R–module, then

CI-dim_{R} M ≤ CI-dim_{R′} M′ ≤ CI-dim_{Q} M′ − c .

When one of the quantities is finite it is equal to those on its left.

Proof. The first inequality is immediate from the observation that R ⟶ R′ ⟵^{=} R′ is a quasi-deformation, and that quasi-deformations form a category. When CI-dim_{R′} M′ is finite, by using (1.7) we get CI-dim_{R′} M′ = CI-dim_{R} M .

If Q ⟶ Q′ ⟵ P is a quasi-deformation, and R ⟶ R′′ = R′⊗_{Q}Q′ ⟵ P is its composition with R ⟶ R′ ⟵ Q, then in view of the isomorphism M ⊗_{R} R′′ ≅ (M ⊗_{R} R′)⊗_{R′} R′′ we get an inequality CI-dim_{R′} M′ ≤ CI-dim_{Q} M′. When the right hand side is finite, (1.7) shows it is equal to depth Q − depth_{Q} M′ = depth R + c − depth_{R} M′ = CI-dim_{R′} M′ + c. □

(1.12) Proposition. Given a sequence y = y_{1}, …, y_{g} of elements of a noetherian ring R, set R̄ = R/(y)R and M̄ = M/(y)M. If y is M–regular, then

(1.12.1)  CI-dim_{R} M̄ ≤ CI-dim_{R} M + g

with equality when y is in the Jacobson radical of R. If y is R–regular, then

(1.12.2) CI-dimR̄ M̄ ≤ CI-dim_{R} M when y is M–regular ; (1.12.3) CI-dimR̄ M ≤ CI-dim_{R} M − g when (y)M = 0 ,

with equalities when y is in the Jacobson radical of R and CI-dim_{R} M is finite.

Proof. In view of (1.6) and an obvious induction, it suffices to prove the claims when (R, 𝔪) is local, g = 1, and y_{1} = y is a non-zero-divisor in 𝔪.

Consider a quasi-deformation R ⟶ R′ ⟵ Q. If y is M–regular, then the exact sequence 0 ⟶ M ⟶^{y} M ⟶ N ⟶ 0 induces an exact sequence of R′–modules 0 ⟶ M′ ⟶^{y} M′ ⟶ N′ ⟶ 0. Thus, N′ ≅ M′/yM′ and pd_{Q} N′ = pd_{Q} M′ + 1, so (1.12.1) follows.

Assume next that y is R–regular. If (y)M = 0, then M̄ = M, and (1.12.3) is obtained by applying (1.11) to the quasi-deformation R̄ ⟶^{=} R̄ ⟵ R. If y is M–regular and CI-dim_{R} M is finite, then the equality in (1.12.2) is a consequence of the other two formulas. □

(1.13) Proposition. Let M be a module over a noetherian ring R.

(1) If φ: R ⟶ S is a faithfully flat homomorphism of rings, then

CI-dim_{R} M ≤ CI-dim_{S} (M ⊗_{R} S)

with equality when CI-dim_{S} (M ⊗_{R} S) is finite.

(2) If 𝔞 ⫋ R is an ideal, R^{∗} is the 𝔞-adic completion of R and M^{∗} is that of M, then

CI-dim_{R^{∗}} M^{∗} ≤ CI-dim_{R} M

with equality when 𝔞 is contained in the Jacobson radical 𝔧(R) of R.

Proof. (1) Pick 𝔮 ∈ Spec S and set 𝔭 = 𝔮∩R. The induced local homomorphism φ_{𝔮}: R_{𝔭} ⟶ S_{𝔮} is flat, the S_{𝔮}–modules M_{𝔭} ⊗_{R_{𝔭}} S_{𝔮} and (M ⊗_{R} S)_{𝔮} are canonically isomorphic, hence (1.11) yields CI-dim_{R_{𝔭}} M_{𝔭} ≤ CI-dim_{S_{𝔮}}(M ⊗_{R} S)_{𝔮}, with equality when the right hand side is finite. As each prime of R is the contraction of one of S, the desired (in)equality follows.

(2) Any maximal ideal of R^{∗} is of the form 𝔪^{∗} = 𝔪R^{∗} for some maximal ideal 𝔪 ⊆ R with 𝔪 ⊇ 𝔞. Let R_{𝔪} ⟶ R′ ⟵ Q be a quasi-deformation and let 𝔟 be the inverse image in Q of 𝔞R′. Ideal-adic completions yield a quasi-deformation R_{𝔪^{∗}}^{∗} ⟶ R′^{∗} ⟵ Q^{∗}. As pd_{Q^{∗}} (M_{𝔪^{∗}}^{∗} ⊗_{R_{𝔪^{∗}}^{∗}} R′^{∗}) = pd_{Q} (M_{𝔪} ⊗_{R_{𝔪}} R′) and pd_{Q^{∗}} R′^{∗} = pd_{Q} R′, we get the inequality. If 𝔞 ⊆ 𝔧(R), then R^{∗} is faithfully flat over R, so (1) gives a converse inequality. □

(1.14) Residue field extensions. Let R ⟶ R′ ⟵ Q be a quasi-deformation, let k′ be the residue field of R′, and let k′ ⊆ ℓ be a field extension. By [15; Appendice] there is a local flat extension Q ⟶ Q′, such that Q′ ⊗_{Q} k′ = ℓ. Composition of the original quasi-defor-mation with Q ⟶ Q′ ⟵^{=} Q′ yields a quasi-deformation R ⟶ R′′ ⟵ Q′, such that ℓ is the residue field of R′′, and pd_{Q}(M ⊗_{R} R′) = pd_{Q′}(M ⊗_{R} R′′) for each R–module M.

It follows that CI-dim_{R} M can be computed by using quasi-deformations in which the local rings Q have infinite, or even algebraically closed, residue fields. In view of (1.13.2), one can further restrict to those quasi-deformations for which the ring Q is also complete.

2. Quantum regular sequences

In this section E is a module over a commutative ring Q. We introduce a notion of E– regular sequence of endomorphisms of E, which extends the classical notion of E–regular sequence of elements of Q, and reduces to it when the endomorphisms are homotheties.

We say a square matrix q = (q_{ij}) of elements of Q is a commutation matrix if q_{ij}q_{ji} = 1 and q_{jj} = 1 for 1 ≤ i, j ≤ c. A family {φ_{1}, …, φ_{c}} of endomorphisms of a Q–module E is said to be quantum commuting if there is a c × c commutation matrix q such that φ_{i}φ_{j} = q_{ij}φ_{j}φ_{i} for all i, j. Powers of elements of a quantum commuting family also form such a family, with a new commutation matrix.

We denote by (φ_{j_{1}}, …, φ_{j_{s}})E the Q–submodule Im(φ_{j_{1}}) + ⋯ + Im(φ_{j_{s}}) of E, and set (φ)E = (φ_{1}, …, φ_{c})E. If the family is quantum commuting, then for arbitrary integers j, j_{1}, …, j_{s} between 1 and c, the endomorphism φ_{j} maps the (φ_{j_{1}}, …, φ_{j_{s}})E into itself. Thus, each φ_{j} induces on any of the modules E/(φ_{j_{1}}, …, φ_{j_{s}})E an endomorphism, which is also denoted by φ_{j}. A sequence φ = φ_{1}, …, φ_{c} of quantum commuting endomorphisms is said to be E–regular, if E ≠ (φ)E and φ_{j} is an injection on E/(φ_{1}, …, φ_{j−1})E for j = 1, …, c. Note that x_{1}, …, x_{c} ∈ Q form a (classical) E–regular sequence precisely when the commuting sequence of endomorphisms x_{1} id_{E}, …, x_{c} id_{E} is E–regular.

(2.1) Proposition. Let n_{1}, …, n_{c} be positive integers. A sequence φ_{1}, …, φ_{c} of quantum commuting endomorphisms is E–regular if and only if so is the sequence φ_{1}^{n_{1}}, …, φ_{c}^{n_{c}}.

Proof. When the endomorphisms commute, [29; (16.1)] establishes the “only if” part, and the argument carries over. As a first step, it shows that if u_{1}, …, u_{c} ∈ E satisfy φ_{1}(u_{1}) + ⋯ + φ_{c}(u_{c}) = 0, then u_{j} ∈ (φ)E for j = 1, …, c. Suppose the sequence φ_{1}^{n_{1}}, …, φ_{c}^{n_{c}} is E–regular and note that (φ_{1}, …, φ_{c})^{n_{1}+···+n_{c}}E ⊆ (φ_{1}^{n_{1}}, …, φ_{c}^{n_{c}})E implies (φ)E ≠ E. To establish the regularity of φ_{1}, …, φ_{c} it suffices to show that φ_{1}, φ_{2}^{n_{2}}, …, φ_{c}^{n_{c}} is E–regular. We assume n_{1} ≥ 2 and argue by induction on c.

When c = 1 we have to show is that φ_{1} is injective, and this follows from the injectivity of φ_{1}^{n_{1}}. Let c ≥ 2 and suppose that

φ_{c}^{n_{c}}(u) = φ_{1}(u_{1}) + φ_{2}^{n_{2}}(u_{2}) + ⋯ + φ_{c−1}^{n_{c−1}}(u_{c−1})

for appropriate u, u_{1}, u_{2}, …, u_{c−1} in E. Applying φ_{1}^{n_{1}−1} to this expression we get

φ_{c}^{n_{c}}(q_{c}φ_{1}^{n_{1}−1}(u)) = φ_{1}^{n_{1}}(u_{1}) + φ_{2}^{n_{2}}(q_{2}φ_{1}^{n_{1}−1}(u_{2})) + ⋯ + φ_{c−1}^{n_{c−1}}(q_{c−1}φ_{1}^{n_{1}−1}(u_{c−1})),

where q_{j} = (q_{1j})^{(n_{1}−1)n_{j}} is a unit in Q for j = 2, …, c. Since φ_{1}^{n_{1}}, …, φ_{c}^{n_{c}} is E–regular, there are v_{1}, …, v_{c−1} ∈ E for which φ_{1}^{n_{1}−1}(u) = φ_{1}^{n_{1}}(v_{1}) + ⋯ + φ_{c−1}^{n_{c−1}}(v_{c−1}), that is:

φ_{1}^{n_{1}−1}(u − φ_{1}(v_{1})) = φ_{2}^{n_{2}}(v_{1}) + ⋯ + φ_{c−1}^{n_{c−1}}(v_{c−1}) .

Since φ_{1}, φ_{2}^{n_{2}}, …, φ_{c−1}^{n_{c−1}} is E–regular by the induction hypothesis, the “only if” part shows φ_{1}^{n_{1}−1}, φ_{2}^{n_{2}}, …, φ_{c−1}^{n_{c−1}} has the same property. Thus, there exist w_{1}, …, w_{c−1} ∈ E such that u − φ_{1}(v_{1}) = φ_{1}^{n_{1}−1}(w_{1}) + ⋯ + φ_{c−1}^{n_{c−1}}(w_{c−1}). Rewriting the last equality as

u = φ_{1}(v_{1} + φ_{1}^{n_{1}−2}(w_{1})) + φ_{2}^{n_{2}}(w_{2}) + ⋯ + φ_{c−1}^{n_{c−1}}(w_{c−1}) ∈ (φ)E ,

we see that the sequence φ_{1}, φ_{2}^{n_{2}}, …, φ_{c}^{n_{c}} is E–regular, as desired. □

(2.2) Quantum symmetric algebras. Let q = (q_{ij}) be a c × c commutation matrix with entries in Q. Following Manin [28], we call the Q–algebra Q_{q}[X] generated by a set X = {X_{1}, …, X_{c}} whose elements are subject only to the relations X_{i}X_{j} = q_{ij}X_{j}X_{i} for 1 ≤ i, j ≤ c, a quantum symmetric Q–algebra (with commutation matrix q). For a c-tuple J = (j_{1}, …, j_{c}) ∈ ℕ^{c}, set |J| = j_{1} + ⋯ + j_{c}. Clearly, the algebra Q_{q}[X] is ℕ^{c}–graded, with component of multidegree J the free Q–module with basis element X^{J} = ∏_{i=1}^{c} X_{i}^{j_{i}}.

We use without further comment the coincidence of the left, right, and two-sided ideals in Q_{q}[X] generated by some set of monomials in the indeterminates. Any such ideal is called a monomial ideal. These are the only ideals homogeneous with respect to the multigrading of Q_{q}[X]. Such an ideal 𝔄 is cofinite—in the sense that the Q–module Q_{q}[X]/𝔄 is finite—precisely when it contains positive powers of all the variables.

The following properties are easily established by tracking multi-degrees.

(2.2.1) The monomial ideals of Q_{q}[X] are generated by finite sets of monomials. As in [17], where monomials in commuting indeterminates are treated, we consider the lattice with respect to sums and intersections of the (cofinite) monomial ideals in Q_{q}[X], and note that it is distributive. It follows that any (cofinite) monomial ideal is the intersection of a finite family of ideals generated by positive powers of (all) the indeterminates.

(2.2.2) The lattice of (cofinite) monomial ideals is closed under colons 𝔄 : 𝔅 = {u ∈ Q_{q}[X] | 𝔅u ⊆ 𝔄}. In particular, setting X_{i}^{n_{i}−s_{i}} = 1 if n_{i} ≤ s_{i}, one has equalities

(X_{1}^{n_{1}}, …, X_{c}^{n_{c}}) : (∏_{i=1}^{c} X_{i}^{s_{i}}) = (X_{1}^{n_{1}−s_{1}}, …, X_{c}^{n_{c}−s_{c}});

(X_{1}^{n_{1}}, …, X_{c}^{n_{c}}) : (X_{1}^{s_{1}}, …, X_{c}^{s_{c}}) = (∏_{i=1}^{c} X_{i}^{n_{i}−s_{i}}) + (X_{1}^{n_{1}}, …, X_{c}^{n_{c}}) .

(2.3) Quantum Koszul complexes. Let φ = {φ_{1}, …, φ_{c}} be a quantum commuting family of endomorphisms of the Q–module E, with commutation matrix q. Consider a graded free Q–module Y with Y_{n} free on a basis {Y_{H} | H ⊆ {1, …, c}, #(H) = n}. Let h_{1}, …, h_{n} be the elements of H listed in increasing order, and define a degree −1 endomorphism ∂ of the graded Q–module Y ⊗_{Q} E by the formula:

(2.3.1)  ∂(Y_{H} ⊗ v) = ∑_{i=1}^{n}(−1)^{i−1}(∏_{j=1}^{i−1} q_{h_{i}h_{j}})Y_{H∖h_{i}} ⊗ φ_{h_{i}}(v) .

A direct computation which uses the quantum commutativity of the sequence φ_{1}, …, φ_{c} shows that ∂^{2} = 0. We denote by K(φ; E) the complex (Y ⊗_{Q} E, ∂), call it the quantum Koszul complex of the family {φ_{1}, …, φ_{c}}, and set H_{∗}(φ; E) = H_{∗} K(φ; E).

(2.3.2) Example. When x = {x_{1}, …, x_{c}} is a set of elements of Q, the quantum Koszul complex K(φ; E) on the family φ = {x_{1} id_{E}, …, x_{c} id_{E}} of commuting endomorphisms of E coincides with the classical Koszul complex K(x; E) = K(x; Q) ⊗_{Q} E.

As in the classical case, vanishing of Koszul homology is related to regularity.

(2.3.3) Proposition. For a quantum commuting family φ = {φ_{1}, …, φ_{c}} of endomorphisms of a Q–module E, there is an isomorphism H_{0}(φ; E) ≅ E/(φ)E.

If the sequence φ is E–regular, then H_{n}(φ; E) = 0 for n ≠ 0.

Proof. The expression for the zeroth homology is clear from the definitions.

If φ is regular, then φ′ = φ_{1}, …, φ_{c−1} is a quantum commuting E–regular sequence. By induction, we may assume H_{n}(φ′; E) = 0 when n ≠ 0. For each H ⊆ {1, …, c} set

α(Y_{H} ⊗ v) = Y_{H} ⊗ v; β(Y_{H} ⊗ v) = \{ \begin{matrix}(−1)^{n−1}Y_{H∖c} ⊗ v & if c ∈ H; \\ 0 & if c ∉ H . \end{matrix}

It is easy to see that 0 ⟶ K(φ′; E) ⟶^{α} K(φ; E) ⟶^{β} ΣK(φ′; E) ⟶ 0 is an exact sequence of complexes of Q–modules. The associated homology exact sequence yields equalities H_{n}(φ; E) = 0 for n ≠ 0, 1 and an exact sequence:

0 ⟶ H_{1}(φ; E) ⟶ E/(φ′)E ⟶^{φ_{c}} E/(φ′)E ⟶ E/(φ)E ⟶ 0 .

As φ is regular, the homomorphism φ_{c} is injective, hence H_{1} K(φ; E) = 0. □

(2.3.4) Example. The sequence X of endomorphisms of Q_{q}[X], induced by left multiplication with X_{1}, …, X_{c}, is quantum commuting and Q_{q}[X]–regular, hence H_{n}(X; Q_{q}[X]) vanishes in degrees n ≠ 0, and equals Q in degree 0 by (2.3.3). Observe that in this case our quantum Koszul complex coincides with that of Manin [28].

Given a quantum commuting family {φ_{1}, …, φ_{c}} ⊂ End_{Q}(E) with commutation matrix q, there is a unique homomorphism of Q–algebras Q_{q}[X] → End_{Q}(E) which maps X_{j} to φ_{j} for 1 ≤ j ≤ c, and so E becomes a left Q_{q}[X]–module. A submodule of E of the form 𝔄E for some monomial ideal 𝔄 in Q_{q}[X] is said to be monomial. It is clear that sums of monomial submodules are monomial. Under additional conditions, intersections and colon submodules (P : 𝔅) = {v ∈ E | 𝔅v ⊂ P } also display a similar stability.

(2.4) Theorem. Let φ = φ_{1}, …, φ_{c} be a quantum commuting sequence of endomorphisms of E with commutation matrix q, and let 𝔄 be a cofinite monomial ideal in Q_{q}[X].

If the sequence φ is E–regular, then Tor_{n}^{Q_{q}[X]}(Q_{q}[X]/𝔄, E) = 0 for n > 0, the natural map 𝔄 ⊗_{Q_{q}[X]} E ⟶ 𝔄E is bijective, and for each cofinite monomial ideal 𝔅 there are equalities (𝔄 ∩ 𝔅)E = (𝔄E) ∩ (𝔅E) and (𝔄E : 𝔅) = (𝔄 : 𝔅)E.

Proof. The first assertion is proved by induction on the rank r of the free Q–module Q_{q}[X]/𝔄. If r = 1 then 𝔄 is the ideal (X_{1}, …, X_{c}). As K(X; Q_{q}[X]) is a complex of free right Q_{q}[X]–modules, by (2.3.4) it provides a free resolution of Q = Q_{q}[X]/(X), hence

Tor_{n}^{Q_{q}[X]}(Q, E) = H_{n}(K(X; Q_{q}[X]) ⊗_{Q_{q}[X]} E) = H_{n}(φ; E) ,

and by (2.3.3) the last group vanishes for n > 0.

When r > 1 there is a sequence Q_{q}[X] = 𝔄_{0} ⊃ ⋯ ⊃ 𝔄_{r} = 𝔄 of monomial ideals such that 𝔄_{i}/𝔄_{i+1} ≅ Q for 0 ≤ i < r. For n > 0 the modules at both ends of the exact sequence

Tor_{n}^{Q_{q}[X]}(Q, E) ⟶ Tor_{n}^{Q_{q}[X]}(Q_{q}[X]/𝔄, E) ⟶ Tor_{n}^{Q_{q}[X]}(Q_{q}[X]/𝔄_{r−1}, E)

are trivial by induction, hence so is the middle one. Using this fact for n = 1 we see that the short exact sequence 0 ⟶ 𝔄 ⟶ Q_{q}[X] ⟶ Q_{q}[X]/𝔄 ⟶ 0 induces a canonical isomorphism 𝔄 ⊗_{Q_{q}[X]} E ⟶ 𝔄E, as desired.

To compute the intersection of monomial submodules, consider the short exact sequence of right Q_{q}[X]–modules 0 ⟶ 𝔄 ∩ 𝔅 ⟶ 𝔄 ⊕ 𝔅 ⟶ 𝔄 + 𝔅 ⟶ 0. It induces a commutative diagram with exact rows and bijective (by the first part of the theorem) vertical maps:

(𝔄 ∩ 𝔅) ⊗_{Q_{q}[X]} E ⟶ (𝔄 ⊗_{Q_{q}[X]} E) ⊕ (𝔅 ⊗_{Q_{q}[X]} E) ⟶ (𝔄 + 𝔅) ⊗_{Q_{q}[X]} E ⟶ 0
≅↓  ≅↓  ≅↓
(𝔄 ∩ 𝔅)E  ⟶^{γ}  𝔄E ⊕ 𝔅E  ⟶^{δ}  (𝔄 + 𝔅)E  ⟶ 0

As (𝔄 + 𝔅)E = 𝔄E + 𝔅E, we see that Ker δ = (𝔄E) ∩ (𝔅E). On the other hand, as Tor_{1}^{Q_{q}[X]}(𝔄 + 𝔅, E) ≅ Tor_{2}^{Q_{q}[X]}(Q_{q}[X]/(𝔄 + 𝔅), E) and the last module vanishes by the first part of the theorem, we see that γ is injective. The exactness of the bottom row now shows Im γ = Ker δ, that is, (𝔄 ∩ 𝔅)E = Ker γ = (𝔄E) ∩ (𝔅E), as desired.

Finally, we compute the colon submodule of a monomial submodule and a monomial ideal, and start with the special case when 𝔄 = (X_{1}^{n_{1}}, …, X_{c}^{n_{c}}) and 𝔅 = (∏_{i=1}^{c} X_{i}^{s_{i}}). In view of (2.2.2) and the obvious inclusion (𝔄 : 𝔅)E ⊆ (𝔄E : 𝔅), we have to show that if (∏_{i=1}^{c} φ_{i}^{s_{i}})(v) ∈ (φ_{1}^{n_{1}}, …, φ_{c}^{n_{c}})E then v is in (φ_{1}^{n_{1}−s_{1}}, …, φ_{c}^{n_{c}−s_{c}}) E, where φ_{i}^{n_{i}−s_{i}} = id_{E} if n_{i} ≤ s_{i}. The assertion is trivial when c = 0, so we assume by induction that it has been established for less than c quantum commuting variables; also, since there is nothing to prove otherwise, we assume that n_{i} > s_{i} for 1 ≤ i ≤ c. The sequence φ_{2}^{n_{2}}, …, φ_{c}^{n_{c}} being regular on E/φ_{1}^{n_{1}}(E) by (2.1), we conclude by the induction hypothesis that φ_{1}^{s_{1}}(v) = φ_{1}^{n_{1}}(w_{1}) + ∑_{i=2}^{c} φ_{i}^{n_{i}−s_{i}}(w_{i}) for appropriate w_{i} ∈ E. The regularity of φ_{1}^{s_{1}}, φ_{2}^{n_{2}−s_{2}}, …, φ_{c}^{n_{c}−s_{c}} implies v − φ_{1}^{n_{1}−s_{1}}(w_{1}) ∈ (φ_{2}^{n_{2}−s_{2}}, …, φ_{c}^{n_{c}−s_{c}}) E, as required.

In general, we know from (2.2) that 𝔄 = ⋂_{a=1}^{s} 𝔄_{a} with each 𝔄_{a} generated by positive powers of the variables X_{i}, and that 𝔅 = (B_{1}, …, B_{t}) for appropriate monomials B_{b}. By the special case of the formula for colons and the formula for intersections, we get

(𝔄E : 𝔅) = ⋂_{a,b}(𝔄_{a}E : B_{b}) = ⋂_{a,b}((𝔄_{a} : B_{b})E) = (⋂_{a,b}(𝔄_{a} : B_{b}))E = (𝔄 : 𝔅)E .

The proof of the theorem is now complete. □

3. Constructions of modules of finite CI-dimension

The simplest type of modules of finite CI-dimension are those with finite projective dimension. It is easily seen that there are rings, over which these are the only ones. For instance, if R is Cohen-Macaulay and its multiplicity e(R) has the minimal possible value, equal to edim R − dim R + 1, then each R–module M has b_{n}^{R}(M) = b_{m}^{R}(M)(edim R)^{n−m} for n ≥ m = depth R−depth_{R} M +1; it follows from (5.3) that CI-dim_{R} M = ∞ whenever e(R) ≥ 3 and pd_{R} M = ∞. This raises the problem to determine which local rings admit modules with finite CI-dimension and infinite projective dimension.

In [16] Buchweitz et al. have produced an interesting class of maximal Cohen–Macaulay modules over a complete hypersurface ring. Their method easily extends to provide modules with periodic of period 2 free resolutions over any quotient of a local ring (R, 𝔪) by a non-zero-divisor in 𝔪^{2}, a fact noted independently by Herzog et al. [25].

In this section we produce a plentiful supply of modules of finite CI-dimension and infinite projective dimension over any quotient of a noetherian ring by a decomposable, in a very weak sense, regular sequence. The input comes from different sources: the technique of quantum regular sequences developed in Section 2; a procedure of Shamash [33] and Eisenbud [18] which yields an R–free resolution starting from a Q–free one; and, to a much larger extent than [16], the rich internal structure of exterior algebras.

(3.1) Theorem. Let Q be a noetherian ring, let 𝔫 = (a_{1}, …, a_{m}) be a proper ideal in Q, and let x = x_{1}, …, x_{c} be a Q–regular sequence contained 𝔫^{2}.

Over R = Q/(x) there exists a sequence of modules {M_{r}^{[c]}}_{r∈ℤ} such that whenever s > r the module M_{s}^{[c]} is an (s − r)’th syzygy of M_{r}^{[c]}, and for each r ∈ ℤ there are equalities pd_{Q}(M_{r}^{[c]}) = c and CI-dim_{R}(M_{r}^{[c]}) = 0. If Q is local, then also

b_{n}^{Q}(M_{0}^{[c]}) = 2^{cm}\binom{c}{n} for n ∈ ℤ , and b_{n}^{R}(M_{r}^{[c]}) = \{ \begin{matrix}2^{cm}\binom{−n−r+c−2}{c−1} & for 0 ≤ n < −r; \\ 2^{cm}\binom{n+r+c−1}{c−1} & for n ≥ max\{0, −r\} . \end{matrix}

The modules above are obtained from an unbounded complex of free R–modules T. We denote by ΣT the shifted complex: (ΣT)_{n} = T_{n−1}, and ∂_{n}^{ΣT} = −∂_{n−1}^{T}.

(3.2) Theorem. Let 𝔞 = (a_{1}, …, a_{m}) and 𝔟 be ideals in a commutative ring Q, and let x = x_{1}, …, x_{c} be a Q–regular sequence contained in 𝔞𝔟. Over the ring R = Q/(x) there exists an exact complex of free modules (T, ∂) such that ∂(T) ⊆ (𝔞 + 𝔟)T (and hence T is not split if 𝔞 + 𝔟 ≠ Q), there is a chain isomorphism Hom_{R}(T, R) ≅ ΣT, and

rank_{R} T_{n} = \{ \begin{matrix}2^{cm}\binom{n+c−1}{c−1} & for n ≥ 0 ; \\ 2^{cm}\binom{−n+c−2}{c−1} & for n < 0 . \end{matrix}

Furthermore, M(a, x) = Im ∂_{0} has a Q–free resolution (K, ∂) with ∂(K) ⊆ (𝔞 + 𝔟)K and

rank_{Q} K_{n} = 2^{cm}\binom{c}{n} for n ∈ ℤ .

We present various stages of the construction of the complex T in a series of steps.

(3.3) The module M(a, x). Write x_{j} = ∑_{r=1}^{m} b_{jr}a_{r} with b_{jr} ∈ 𝔟 for 1 ≤ j ≤ c. Let E be the free module of rank 2^{cm} underlying the (non-graded) exterior algebra ⨁_{h=0}^{cm} ⋀^{h} L on a Q–module L with basis e_{1}, …, e_{cm}. Let ε_{h} be left wedge multiplication by e_{h} on E, let θ_{h} be the Q–linear skew derivation of E with θ_{h}(e_{h}) = 1 and θ_{h}(e_{i}) = 0 for i ≠ h. Set

∂_{j} = ∑_{r=1}^{m} a_{r}θ_{(j−1)m+r} ,  δ_{j} = ∑_{r=1}^{m} b_{jr}ε_{(j−1)m+r} ,

and consider the sequence φ = φ_{1}, …, φ_{c} of endomorphisms of E given by φ_{j} = ∂_{j} + δ_{j}.

Set M(a, x) = E/(φ)E. As a module over Q, it has a free presentation

E^{c} ⟶^{(φ_{1},...,φ_{c})} E ⟶^{π} M(a, x) ⟶ 0 ,

where ψ = (φ_{1}, …, φ_{c}) and π is the canonical projection. The inclusions

(x)E = (x_{1}, …, x_{c})E = (φ_{1}^{2}, …, φ_{c}^{2})E ⊆ (φ_{1}, …, φ_{c})E = (φ)E

show that M(a, x) = E/(φ)E has a natural structure of R–module.

When c = 1 we recover the construction of [16; (2.3)].

(3.4) Lemma. The sequence of endomorphisms φ is quantum commuting, with

φ_{i}φ_{j} = −φ_{j}φ_{i}  φ_{j}^{2} =x_{j} id_{E}  φ_{j}(E) ⊆ (𝔞 + 𝔟)E

for 1 ≤ i, j ≤ c with i ≠ j. If the sequence x is Q–regular, then φ is E–regular.

Proof. The inclusions φ_{j}(E) ⊆ (𝔞 + 𝔟)E are clear from the construction.

The multiplication table for the φ_{j}’s easily follows from the relations

\begin{matrix}∂_{j}∂_{j} = 0 \\ ∂_{i}∂_{j} = −∂_{j}∂_{i} \end{matrix}  \begin{matrix}δ_{j}δ_{j} = 0 \\ δ_{i}δ_{j} = −δ_{j}δ_{i} \end{matrix}  \begin{matrix}∂_{j}δ_{j} + δ_{j}∂_{j} = x_{j} id_{L} \\ ∂_{j}δ_{i} = −δ_{i}∂_{j} \end{matrix}

which hold for 1 ≤ i, j ≤ c with i ≠ j. Indeed, the first column results from the fact that both ∂_{i}∂_{j} + ∂_{j}∂_{i} and ∂_{j}∂_{j} are degree −2 derivations of E which vanish on the algebra generators, hence are identically zero; the second one reflects the strict skew-commutativity of the exterior algebra; the third one follows from the next computation, in which f_{j} = ∑_{r=1}^{m} b_{jr}ε_{(j−1)m+r} and e stands for an arbitrary element of E:

∂_{j}δ_{i}(e) + δ_{i}∂_{j}(e) = ∂_{j}(f_{i} ∧ e) + f_{i} ∧ ∂_{j}(e) = ∂_{j}(f_{i})e = \{ \begin{matrix}x_{j}e & if i = j; \\ 0 & if i ≠ j . \end{matrix}

If x is Q–regular, then the E–regularity of φ follows from (2.1) and (2.3.2). □

(3.5) Grassmann duality. An excellent reference for this is [14; §11], from where we borrow a sign convention: for subsets J and K of I = {1, …, cm}, set ρ_{JK} = (−1)^{p}, with p equal to the number of pairs (j, k) ∈ J × K with j > k. When J = {j_{1}, …, j_{s}}, with elements listed in increasing order, set e_{J} = e_{j_{1}} ∧ ⋯ ∧ e_{j_{s}} ∈ ⋀ L. Note that {e_{J}}_{J⊆I} is a Q–basis of E, and let {e_{K}^{∗}}_{K⊆I} be the dual basis of E^{∗} = Hom_{Q}(E, Q).

The Q–linear homomorphism ω: E ⟶ E^{∗} defined in these bases by

ω(e_{J}) = (−1)^{\binom{s}{2}}ρ_{JK}e_{K}^{∗} , where K = I ∖ J ,

is clearly bijective. Furthermore, it is symmetric or antisymmetric, in the sense that

(3.5.1)  ω = (−1)^{\binom{cm}{2}}ω^{∗}β_{E} ,

where ω^{∗} = Hom_{Q}(ω, Q) and β_{E}: E ⟶ E^{∗∗} is the canonical biduality map. To see it, use

ω(e_{J}) = (−1)^{\binom{s}{2}}ρ_{JK}e_{K}^{∗} ,  ω^{∗}β_{E}(e_{J}) = (−1)^{\binom{cm−s}{2}}ρ_{KJ}e_{K}^{∗} ,

in conjunction with the relation ρ_{JK}ρ_{KJ} = (−1)^{s(cm−s)} and the congruence

\binom{s}{2} + \binom{cm}{2} + \binom{cm − s}{2} + s(cm − s) ≡ 0 mod (2) .

Next we exhibit a beautiful reflexivity of M(a, x). With bars denoting reduction modulo (x) set F = Ē = E/(x)E and Ē^{∗} = F^{∗} = Hom_{R}(F, R) via the canonical isomorphism.

(3.6) Lemma. The homomorphism τ = ω ∏_{j=1}^{c} φ_{j}: E ⟶ E^{∗} induces a homomorphism

α: M(a, x) ⟶ M(a, x)^{∗}

of R–modules, which is bijective if the sequence x is Q–regular.

Proof. For a start, note that for 1 ≤ i, j ≤ c and φ_{j}^{∗} = Hom_{Q}(φ_{j}, Q) there are equalities

(3.6.1)  (φ̄_{j})^{2} = 0 ,  φ̄_{i}φ̄_{j} = −φ̄_{j}φ̄_{i} ,  ω̄ φ̄_{j} = φ̄_{j}^{∗}ω̄ .

The first two are clear from (3.4). For the third, consider θ_{h} from (3.3) and set θ_{j}^{∗} = Hom_{Q}(θ_{j}, Q). If k_{1}, …, k_{t} are the elements of K in listed increasing order, then ωθ_{h}(e_{J}) = 0 = θ_{h}^{∗}ω(e_{J}) for h ∉ J, while for h = j_{u+1} ∈ J with k_{v} < h < k_{v+1} we get

ωθ_{h}(e_{J}) = (−1)^{u+ \binom{s−1}{2}}ρ_{(J∖h)(K∪h)}e_{K∪h}^{∗}
= (−1)^{u+\binom{s}{2}−(s−1)}(−1)^{(s−u−1)+v}ρ_{JK}e_{K∪h}^{∗}
= (−1)^{v+ \binom{s}{2}}ρ_{JK}e_{K∪h}^{∗} = θ_{h}^{∗}ω(e_{J}) .

Thus, ωθ_{j} = θ_{j}^{∗}ω. By a similar computation, ωε_{h}(e_{J}) = ε_{h}^{∗}ω(e_{J}) for ε_{h}^{∗} = Hom_{Q}(ε_{h}, Q).

With homomorphisms ψ: E^{c} ⟶ E and ξ: E ⟶ E^{c} defined by ψ(e_{1}, …, e_{c}) = φ_{1}(e_{1}) + ⋯ + φ_{c}(e_{c}) and ξ(e) = (φ_{1}(e), …, φ_{c}(e)), we now have a diagram of R–modules

(3.6.2)  \begin{matrix}F^{c} & ⟶^{ψ̄} & F & \underrightarrow{∏_{j=1}^{c} φ̄_{j}} & F & ⟶^{ξ̄} & F^{c} \\ _{}_{ω̄^{c}}↓≅ & & ≅↓ω̄  ↘τ̄ & ω̄↓≅ & & ≅↓_{}_{ω̄^{c}} \\ (F^{∗})^{c} & ⟶_{ξ̄^{∗}} & F^{∗} & \overrightarrow{∏_{j=1}^{c} φ̄_{j}^{∗}} & F^{∗} & ⟶_{ψ̄^{∗}} & (F^{∗})^{c} . \end{matrix}

From (3.6.1) we see that the diagram is commutative and that its rows are complexes of R–modules. It follows that τ̄ induces the desired homomorphism

α: M(a, x) = Coker ψ = Coker ψ̄ ⟶ Ker ψ̄^{∗} = M(a, x)^{∗} .

The sequences φ ⊂ End_{R}(E) and φ^{∗} = φ_{1}^{∗}, …, φ_{c}^{∗} ⊂ End_{R}(E^{∗}) quantum commute and the squares of their elements are homotheties by x_{1}, …, x_{c} by (3.4). Thus, φ and φ^{∗} are regular by (2.1), so (2.6.2) has exact rows by (2.4) and (2.2.2), hence α is bijective. □

We study the homological algebra of M(a, x), first over the ring Q.

(3.7) Lemma. If the sequence x is Q–regular, then the quantum Koszul complex K(φ; E) is a complex of free Q–modules which satisfies

H_{0}(φ; E) = M(a, x) and H_{n}(φ; E) = 0 for n > 0
∂(K(φ; E)) ⊆ (𝔞 + 𝔟)K(φ; E) ;
rank_{Q} K_{n} = 2^{cm}\binom{c}{n} for n ∈ ℤ .

In particular, pd_{Q} M(a, x) ≤ c, and equality holds when 𝔞 + 𝔟 ≠ Q.

Proof. By (2.3), we have K(φ; E)^{♮} = Y ⊗_{Q} E where Y is a graded Q–module whose n’th homogeneous component has a basis {y_{H}} indexed by the n–element subsets H of \{1, …, cm\}. Thus, K(φ; E) is a complex of free modules with rank_{Q} K_{n} = 2^{cm}\binom{c}{n} = q_{n}. As φ is E–regular by (3.4), K(φ; E) is a Q–free resolution of M(a, x) by (2.3.3), so pd_{Q} M(a, x) ≤ c. Now (2.3.1) and (3.3) yield ∂(K(φ; E)) ⊆ (𝔞 + 𝔟)K(φ; E), hence

Ext_{Q}^{n}(M(a, x), Q/(𝔞 + 𝔟)) ≅ (Q/(𝔞 + 𝔟))^{q_{n}} ,

which is different from zero when 𝔞 + 𝔟 ≠ Q and 0 ≤ n ≤ c. □

In order to produce an R–free resolution of M(a, x) starting from its Q–free resolution K(φ; E), we use a construction of Shamash and Eisenbud, which we recall below.

(3.8) Higher homotopies. We use the multi-grading convention from (2.2) and abuse notation by writing (0) for (0, …, 0) ∈ ℕ^{c}, and (j) for the c-tuple (0, …, 0, 1, 0, …, 0) in which 1 appears in the j’th place. Let (E, ∂) be a complex of Q–modules. A family σ = {σ^{J} ∈ Hom_{Q}(E, E)_{2|J|−1}}_{J∈ℕ^{c}} is called a system of higher homotopies on E if

σ^{(0)} = ∂;

σ^{(0)}σ^{(j)} + σ^{(j)}σ^{(0)} = x_{j} id_{E} for 1 ≤ j ≤ c;

∑_{J′+J′′=J} σ^{J′}σ^{J′′} = 0 for J ∈ ℕ^{c} with |J| ≥ 2 .

Partly responsible for the name is the following observation: the first two conditions above imply that for j = 1, …, c the endomorphism σ^{(j)} of E is a homotopy between multiplication by x_{j} and the zero map. In particular, (x) annihilates H_{∗} E. Conversely, if M is a module over R = Q/(x), and (E, ∂) is a free resolution of M as a Q–module, then basic homological algebra exhibits a family of homotopies {σ^{(j)} : x_{j} id_{E} ∼ 0_{E}}_{1⩽j⩽c}. If (x) is a proper ideal which contains a non-zero-divisor, then by [18; (8.1)] this family can be extended to a system σ = {σ^{J}}_{J∈ℕ^{c}} of higher homotopies for x.

(3.9) Lemma. The complex K(φ; E) admits a system σ of higher homotopies such that

σ^{(j)}σ^{(j)} = 0 and σ^{(i)}σ^{(j)} = −σ^{(j)}σ^{(i)} for 1 ≤ i ≤ j ≤ c;

σ^{J}(K(φ; E)) ⊆ (𝔞 + 𝔟)K(φ; E) when |J| ≤ 1 , and σ^{J} = 0 when |J| ≥ 2 .

Proof. With the notation of the proof of (3.7), for j = 1, …, c set

σ^{(j)}(y_{H} ⊗ u) = \{ \begin{matrix}y_{H∪j} ⊗ φ_{j}(u) & if j ∉ H; \\ 0 & if j ∈ H . \end{matrix}

A direct computation, which uses (2.3.1) and the anticommutativity (3.4) of the sequence φ, shows that the degree 1 endomorphism σ^{(j)} of K(φ; E) = Y ⊗_{Q} E is a square zero homotopy from x_{j} id to the zero map, and that it anticommutes with σ^{(i)}. By [18; p.56] such a set of homotopies can be extended to a system σ of higher homotopies on K(φ; E) by setting σ^{J} = 0 when |J| ≥ 2. For |J| = 1 we have σ^{J}(K(φ; E)) ⊆ (𝔞 + 𝔟)K(φ; E) by (3.3) and the definition above. The desired inclusion for the σ^{(0)} comes from (3.7). □

(3.10) Standard resolutions. Let Q[χ] = Q[χ_{1}, …, χ_{c}] be a polynomial ring graded by assigning to the indeterminates χ_{1}, … χ_{c} lower degree −2, let σ be a system of higher homotopies on a bounded below complex of Q–modules, E, and set Ẽ{σ} = Hom_{Q}(Q[χ], E).

For γ ∈ Hom_{Q}(Q[χ], E), y ∈ Q[χ], and χ^{J} = χ_{1}^{j_{1}} … χ_{c}^{j_{c}}, all but a finite number of summands in the expression ∑_{J∈ℕ^{c}} σ^{J}γ(χ^{J}y) vanish for degree reasons. Thus,

(3.10.1)  ˜∂ = ∑_{J∈ℕ^{c}} σ^{J} ∘ Hom_{Q}(χ^{J}, E): Ẽ{σ} ⟶ Ẽ{σ}

is a degree −1 homomorphism of graded Q–modules. A direct computation shows that

(3.10.2)  (E{σ}, ∂) = (Ẽ{σ} ⊗_{Q} R, ˜∂ ⊗_{Q} R)

is a DG module over the graded ring R[χ] = Q[χ] ⊗_{Q} R.

If E is a complex of free Q–modules, then E{σ} is one of free R–modules, and

(3.10.3)  Hilb_{E{σ}}^{R}(t) = Hilb_{E}^{Q}(t) Hilb_{Q[χ]}^{Q}(t) = Hilb_{E}^{Q}(t)(1 − t^{2})^{−c} ,

where Hilb_{E}^{Q}(t) denotes the generating function ∑_{n∈ℤ}(rank_{Q} E_{n})t^{n}.

The basic property of this construction is given by [18; (7.2)], which extends [33; §3]:

(3.10.4) There is a natural isomorphism H_{0} E{σ} ≅ M. If E is a Q–free resolution of M, and the sequence x is Q–regular, then E{σ} is an R–free resolution of M. □

A resolution E{σ} as above is called a standard R–free resolution of M constructed from a system of higher homotopies σ on a Q–free resolution E. Now we describe the homological algebra of M(a, x) as an R–module.

(3.11) Lemma. If the sequence x is Q–regular, E = K(φ; E) is the quantum Koszul complex, and {σ} is its system of higher homotopies (3.9), then the complexes of free R– modules F = E{σ} and F^{∗} = Hom_{R}(F, R) satisfy

H_{0} F = M(a, x) ;  H_{0}(F^{∗}) = M(a, x)^{∗};
H_{n} F = 0 for n > 0 ;  H_{n}(F^{∗}) = 0 for n < 0 ;
∂(F) ⊆ (𝔞 + 𝔟)F ;  ∂(F^{∗}) ⊆ (𝔞 + 𝔟)F^{∗};
rank_{R} F_{n} = 2^{cm}\binom{n+c−1}{c−1} for n ≥ 0 ;  rank_{R} F_{n}^{∗} = 2^{cm}\binom{−n+c−1}{c−1} for n ≤ 0 .

In particular, pd_{R} M(a, x) = ∞ when 𝔞 + 𝔟 ≠ Q.

Proof. The preceding proposition computes the homology of F.

It follows that H_{0}(F^{∗}) = M(a, x)^{∗}, and that H_{−n}(F^{∗}) ≅ Ext_{R}^{n}(M(a, x), R) for n > 0. As the Q–regular sequence x of length c annihilates M(a, x), (1.9.4) yields

Ext_{R}^{n}(M(a, x), R) ≅ Ext_{Q}^{n+c}(M(a, x), Q) for n ∈ ℤ .

As pd_{Q} M(a, x) ≤ c by (3.7), these modules vanish when n > 0.

In view of (3.7), the equality of power series (3.10.3) becomes

Hilb_{F}^{R}(t) = 2^{cm}(1 + t)^{c}(1 − t^{2})^{−c} = 2^{cm}(1 − t)^{−c} ,

which is another way to say that r_{n} = rank_{R} F_{n} is equal to 2^{cm}\binom{n+c−1}{c−1} for n ∈ ℤ.

Now (3.10.1), (3.10.2), (3.9) imply ∂(F) ⊆ (𝔞 + 𝔟)F, hence ∂(F^{∗}) ⊆ (𝔞 + 𝔟)F^{∗}. Thus, Ext_{R}^{n}(M(a, x), R/(𝔞 + 𝔟)) ≅ (R/(𝔞 + 𝔟))^{r_{n}}, so pd_{R} M(a, x) = ∞ when 𝔞 + 𝔟 ≠ Q. □

We are finally ready to assemble the information obtained so far for the

Proof of Theorem (3.2). Identify the differential ∂_{1} of the complex (F, ∂) with the homomorphism ψ̄: F^{c} ⟶ F of (3.3). Accordingly, identify the differential ∂′′_{0} of (F, ∂)^{∗} with −ψ̄^{∗}: F^{∗} ⟶ (F^{c})^{∗}. As τ̄∂_{1} = 0 = ∂′′_{0}τ̄ due to the commutativity of (3.6.2), setting

(T, ∂)_{⩾0} = (F, ∂)  ∂_{0} = τ̄  (T, ∂)_{⩽−1} = Σ^{−1}((F, ∂)^{∗})

we obtain a complex of free R–modules (T, ∂).

By inception, it comes along with an exact sequence of complexes

0 ⟶ F ⟶ T ⟶ Σ^{−1}(F^{∗}) ⟶ 0 .

The associated long homology exact sequence degenerates to isomorphisms

H_{n} T ≅ \{ \begin{matrix}H_{n} F & for n ≥ 1 ; \\ H_{n+1} F^{∗} & for n ≤ −2 , \end{matrix}

and an exact sequence 0 ⟶ H_{0} T ⟶ M(a, x) ⟶^{α} M(a, x)^{∗} ⟶ H_{−1} T ⟶ 0 . The exactness of T now follows from (3.6) and (3.11). The latter statement also yields the desired expression for t_{n} = rank_{R} T_{n}, and an inclusion ∂(T) ⊆ (𝔞+𝔟)T. Thus, with N = Q/(𝔞+𝔟) we get H_{n}(T ⊗_{Q} N) ≅ N^{t_{n}} for and n ∈ ℤ, so T is not split if 𝔞 + 𝔟 ≠ Q.

It remains to see that T is self-dual up to shift. We have expressions

∂′_{n} = \{ \begin{matrix}−∂_{n−1} & if n > 1 ; \\ −τ̄ & if n = 1 ; \\ (−1)^{1−n}∂_{1−n}^{∗} & if n < 1 ; \end{matrix}  ∂′′_{n} = \{ \begin{matrix}−∂_{n−1}^{∗∗} & if n > 1 ; \\ τ̄^{∗} & if n = 1 ; \\ (−1)^{1−n}∂_{1−n}^{∗} & if n < 1 , \end{matrix}

for the differentials ∂′ of ΣT and ∂′′ of Hom_{R}(T, R). Using (3.5.1) and (3.6.1) we get

τ̄^{∗}β_{F} = ∏_{j=0}^{c−1} φ̄_{c−j}^{∗}ω̄^{∗}β_{F} = (−1)^{\binom{cm}{2}} ∏_{j=0}^{c−1} φ̄_{c−j}^{∗}ω̄ = (−1)^{\binom{cm}{2}}ω̄ ∏_{j=0}^{c−1} φ̄_{c−j} = (−1)^{\binom{cm}{2}+\binom{c}{2}}τ̄ .

It follows that the maps ζ_{n+1} : T_{n} ⟶ Hom_{R}(T_{−n}, R), defined by

ζ_{n+1} = \{ \begin{matrix}(−1)^{1+\binom{c}{2}+\binom{cm}{2}}β_{T_{n}} & when n > 0 ; \\ id_{T_{n}} & when n ≤ 0 , \end{matrix}

provide a chain isomorphism of complexes ζ: ΣT → Hom_{R}(T, R).

The proof of Theorem (3.2) is now complete. □

Proof of Theorem (3.1). By (1.6) we may assume R is local. The formulas for the Betti numbers of M_{r} = M_{r}^{[c]} follow from (3.2) applied with 𝔞 = 𝔟 = 𝔫. In particular, the Q– module M_{0} has finite projective dimension. This conclusion propagates to all M_{r}, as M_{0} is a syzygy of M_{r} when r < 0, and has M_{r} as a syzygy when r > 0. Since each M_{r} is an infinite syzygy over R, we have depth_{R} M_{r} = depth R, and consequently pd_{Q} M_{r} = pd_{Q} R = c. It follows that CI-dim_{R} M_{r} is finite for all r, and hence zero by (1.9.1). □

4. Cohomology operators

In this section Q denotes a commutative ring, x = x_{1}, …, x_{c} is a Q–regular sequence, ρ: Q ⟶ Q/(x) = R is the canonical projection, and M, N are R-modules.

(4.1) Eisenbud operators. If (F, ∂) is a free resolution of a module M over the ring R = Q/(x), then a pair (F̃, ˜∂) consisting of a graded free Q–module F̃ and its degree −1 endomorphism ˜∂, such that (F, ∂) = (F̃ ⊗_{Q} R, ˜∂ ⊗_{Q} R), is called a lifting of the complex (F, ∂). The relation ∂^{2} = 0 yields an inclusion ˜∂^{2}(F̃) ⊆ (x)F̃, hence for 1 ≤ j ≤ c there are endomorphisms t̃_{j}(Q, x, F̃, ˜∂) ∈ Hom_{Q}^{2}(F̃, F̃), such that ˜∂^{2} = ∑_{j=1}^{c} x_{j}t̃_{j}(Q, x, F̃, ˜∂).

Any lifting as above defines a family of Eisenbud operators

{ t_{j} = t_{j}(Q, x, F) = t̃_{j}(Q, x̃, F, ˜∂) ⊗_{Q} R ∈ Hom_{R}^{2}(F, F)}_{1⩽j⩽c} .

The t_{j}’s are chain maps which are defined uniquely up to homotopy, and they commute up to homotopy, cf. [18; §1]. Thus, for each R–module N, they define a family χ = χ_{1}, …, χ_{c} of commuting cohomology operators χ_{j} = H^{∗}(Hom_{R}(t_{j}, N)). In this way, H^{∗} Hom_{R}(F, N) = Ext_{R}^{∗}(M, N) becomes a graded module over a polynomial algebra S^{∗} = R[χ]. This structure may be computed from any system of Eisenbud operators on any R–free resolution.

In particular, if σ is a system of higher homotopies (3.8) on a Q–free resolution E of M, then Ẽ{σ} = Hom_{Q}(Q[χ], E) with the endomorphism ˜∂ from (3.10.2) is a lifting of the standard resolution (E{σ}, ∂) of M over R. Computing with it, one easily sees that the map t̃_{j}(Q, x, Ẽ{σ}, ˜∂) may be chosen to be multiplication with χ_{j}, cf. [18; (7.2)].

By [12] the action of the χ_{j} coincides with that of the operators studied by Gulliksen, who shows [23; (2.3)] that when the R–module Ext_{Q}^{∗}(M, N) is noetherian, so is the R[χ]– module Ext_{R}^{∗}(M, N). A direct proof for the Eisenbud operators is given in [8; (2.1)], and a partial converse in [8; (3.10)]. Here is the full converse.

(4.2) Theorem. If Ext_{R}^{∗}(M, N) is a noetherian graded R[χ]–module, then Ext_{Q}^{∗}(M, N) is a noetherian graded R–module.

We also characterize the regularity of a family of Eisenbud operators.

(4.3) Theorem. The kernel of the canonical change of rings homomorphism

(4.3.1)  ρ_{MN}^{∗} = Ext_{ρ}^{∗}(M, N): Ext_{R}^{∗}(M, N) ⟶ Ext_{Q}^{∗}(M, N)

contains (χ) Ext_{R}^{∗}(M, N), hence it factors through a homomorphism

(4.3.2)  κ_{MN}^{∗}: \frac{Ext_{R}^{∗}(M, N)}{(χ) Ext_{R}^{∗}(M, N)} ⟶ Ext_{Q}^{∗}(M, N) .

Furthermore, the following conditions are equivalent:

(i) The sequence χ = χ_{1}, …, χ_{c} is Ext_{R}^{∗}(M, N)–regular. (ii) The change of rings homomorphism ρ_{MN}^{∗} is surjective. (iii) The reduced change of rings homomorphism κ_{MN}^{∗} is bijective.

The proofs depend on a very useful change of rings spectral sequence. For group cohomology, a sequence with similar flavor is obtained by Benson and Carlson [13; §4].

(4.4) Theorem. There exists a fourth quadrant spectral sequence

(4.4.1)  {^{r}d_{p}^{q}: ^{r}E_{p}^{q} ⟶ ^{r}E_{p−r}^{q−r+1} }^{r⩾1} ⟹E^{q−p} = Ext_{Q}^{q−p}(M, N)

whose initial terms are expressed in terms of the classical Koszul complex K(χ; −) on the family χ = χ_{1}, …, χ_{c} and its homology H_{∗}(χ; −) as follows:

1E_{p}^{q} = K_{p}(χ; Ext_{R}^{∗}(M, N))^{q} ,  ^{2}E_{p}^{q} = H_{p}(χ; Ext_{R}^{∗}(M, N))^{q} .

The spectral sequence has the following vanishing lines and convergence properties: (4.4.2) ^{r}E_{p}^{q} = 0 when p < 0 or p > c, hence ^{c+1}E_{p}^{q} = ^{c+2}E_{p}^{q} = ⋯ = ^{∞}E_{p}^{q} . (4.4.3) ^{r}E_{p}^{q} = 0 when q < 0, hence there exist vertical edge homomorphisms

rυ_{MN}^{q}: ^{r}E_{0}^{q} ↠ ^{∞}E_{0}^{q} ↪E^{q} = Ext_{Q}^{q}(M, N) ,

and ^{1}υ_{MN}^{q} = ρ_{MN}^{q} and ^{2}υ_{MN}^{q} = κ_{MN}^{q}, the maps from (4.3.1) and (4.3.2). (4.4.4) ^{r}E_{p}^{q} = 0 when 2p > q, hence there exist diagonal edge homomorphisms

δ_{MN}^{p}: Ext_{Q}^{p}(M, N) =E^{p} ↠ ^{∞}E_{p}^{2p} ↪ ^{2}E_{p}^{2p} = H_{p}(χ; Ext_{R}^{∗}(M, N))^{2p} .

We isolate parts of the argument for (4.4) in the form of a lemma and a remark.

(4.5) Lemma. For each R–module N there is a natural isomorphism of DG R[χ]–modules:

(Hom_{R}(E{σ} , N)^{♮}, ∂) ≅ (R[χ] ⊗_{R} Hom_{Q}(E, N)^{♮} , ∑_{J∈ℕ^{c}} χ^{J} ⊗_{R} Hom_{Q}(σ^{J}, N)) .

Proof. The natural isomorphisms of graded modules

Hom_{R}(E{σ}, N)^{♮} ≅ Hom_{Q}(Hom_{Q}(Q[χ], E^{♮}), N)
≅ Hom_{Q}(Hom_{Q}(Q[χ], Q) ⊗_{Q} E^{♮}, N)
≅ Hom_{Q}(Hom_{Q}(Q[χ], Q), Hom_{Q}(E^{♮}, N))
≅ Hom_{R}(Hom_{R}(R[χ], R), Hom_{Q}(E^{♮}, N))
≅ Hom_{R}(Hom_{R}(R[χ], R), R) ⊗_{R} Hom_{Q}(E^{♮}, N)
≅ R[χ] ⊗_{R} Hom_{Q}(E, N)^{♮}

are compatible with the action of R[χ]. The formula for the differential on the right hand side is obtained by transporting that on Hom_{R}(E{σ}, N) via these isomorphisms. □

(4.6) The Koszul DG module. The Koszul complex K(χ; R[χ]) is a complex of graded R[χ]–modules. By means of the standard process of condensation, cf. [27; §X.9], it gives rise to a DG R[χ]–module K = ⨁_{n⩾0} Σ^{−2n}K_{n}(χ; R[χ]), which has a natural filtration by the DG submodules K_{(p)} = ⨁_{n=0}^{p} Σ^{−2n}K_{n}(χ; R[χ]).

Proof of Theorem (4.4). The spectral sequence (4.4.1) is obtained from the filtration of the DG R[χ]–module D = Hom_{R}(E{σ}, N) ⊗_{R[χ]} K induced by {K_{(p)}}_{p⩾0}. The lemma shows that its first page is the Koszul complex on χ_{1}, …, χ_{c}, with the bigrading

1E_{p}^{q} = K_{p}(χ; Ext_{R}^{∗}(M, N))^{q} .

To get the vanishing lines announced in (4.4.2), (4.4.3), and (4.4.4) simply note that the variables χ_{i} have degree −2 and filtration level 1, hence in the bigraded R–module 1E^{♮} = Ext_{R}^{∗}(M, N) ⊗_{R} ⋀^{∗}(R^{c}), the generators of the exterior algebra lie in ^{1}E_{1}^{−2}.

To compute the limit of (4.4.1) consider the filtration bête of Hom_{R}(E{σ}, N): its p’th level consist of the homomorphisms of degree ≤ p. The 0’th page of the spectral sequence \{^{r}\overline{E}\}^{r⩾0} it defines on D is the Koszul complex K(χ; Hom_{R}(E\{σ\}, N)^{♮}) bigraded by

0\overline{E}_{q}^{p} = K_{q}(χ; Hom_{R}(E\{σ\}, N)^{♮})^{p} .

It follows from the lemma that χ_{1}, …, χ_{c} is a regular sequence on Hom_{R}(E{σ}, N)^{♮}, hence the Koszul homology vanishes except in wedge degree zero, where we have

(^{1}\overline{E}_{0}^{p})^{♮} = \frac{Hom_{R}^{p}(E\{σ\}, N)^{♮}}{(χ) Hom_{R}^{p−2}(E\{σ\}, N)^{♮}} ≅ Hom_{Q}^{p}(E, N)^{♮} .

Thus, ^{1}\overline{E} is the complex Hom_{Q}(E, N) concentrated in the line q = 0, hence

H^{∗} D = ^{∞}\overline{E}_{0}^{∗} = ^{2}\overline{E}_{0}^{∗} = H^{∗} Hom_{Q}(E, N) ≅ Ext_{Q}^{∗}(M, N) .

It remains to identify the vertical edge homomorphisms in (4.4.3). Consider the inclusion

C = (χ) Hom_{R}(E{σ}, N) ⊕ ⨁_{n⩾1} (Hom_{R}(E{σ}, N) ⊗_{R[χ]} Σ^{−2n}K_{n}(χ; R[χ])) ⊂ D

of DG R[χ]–modules. The preceding argument shows that H_{∗} C = 0, hence the projection D ⟶ Hom_{Q}(E, N) identifies H^{∗} D with Ext_{Q}^{∗}(M, k). Thus, the vertical edge homomorphism^{r}υ_{MN}^{q} is the map induced in homology by the inclusion ⨁_{p⩽r} Σ^{−2p}K_{p}(χ; R[χ]) ⊆ D.

From the lemma, we get ^{1}υ_{MN}^{∗} = ρ_{MN}^{∗}. Furthermore, as ^{1}υ_{MN}^{∗} is the composition of 1E_{0}^{∗} ↠ with ^{2}υ_{MN}^{∗}: ^{2}E_{0}^{∗} ⟶ E^{∗}, and ^{2}E_{0}^{∗} = Ext_{R}^{∗}(M, k)/(χ) Ext_{R}^{∗}(M, k), we see that ρ_{MN}^{∗} annihilates (χ) Ext_{R}^{∗}(M, k), and that ^{2}υ_{MN}^{∗} is the induced map κ_{MN}^{∗}. □

Proof of Theorem (4.2). For each p the graded R[χ]–module H_{p}(χ; Ext_{R}^{∗}(M, N)) is noetherian and annihilated by (χ), hence is noetherian over R[χ]/(χ) = R. This property is inherited by the columns on each page of the finitely convergent spectral sequence (4.4). Its stable page is the graded object associated to a finite filtration of Ext_{Q}^{∗}(M, N), hence the latter R–module is noetherian. □

Proof of Theorem (4.3). The map κ_{MN}^{∗} has been constructed in the proof of (4.4).

(i) ⇒ (iii). When the sequence χ_{1}, …, χ_{c} is Ext_{R}^{∗}(M, N)–regular ^{2}E_{p}^{∗} = 0 for p ≠ 0 in the spectral sequence (4.4), so (4.4.3) shows that κ_{MN}^{∗} is bijective.

(iii) ⇒ (ii) is obvious.

(ii) ⇒ (i). When c = 1 the page ^{2}E of the spectral sequence (4.4.1) lives in just two columns, hence it reduces to an exact triangle of graded modules:

Ext_{R}^{∗}(M, N) ⟶^{χ} Ext_{R}^{∗}(M, N)
Δ ↖  ↙ ρ_{MN}^{∗}
Ext_{Q}^{∗}(M, N)

in which Δ is a homomorphism of upper degree −1. The desired implication follows.

For the rest of the argument we assume that c is at least 2, and that the implication holds for regular sequences of length less than c.

Set Q′ = Q/(x_{1}, …, x_{c−1}), and let ρ′: Q ⟶ Q′ and ρ′′: Q′ ⟶ R be the canonical projections. Note that if σ is a system of higher homotopies for x on a Q–free resolution E of M then σ′ = {σ^{J} ∈ σ | J = (j_{1}, …, j_{c−1}, 0) ∈ ℕ^{c}} is a system of higher homotopies for x′ = x_{1}, …, x_{c−1} on E. Let χ′ = χ′_{1}, …, χ′_{c−1} be the family of operators on Ext_{Q′}^{∗}(M, N) defined by the standard resolution E{σ′}. The morphisms of complexes of R–modules

Hom_{R}(E{σ}, N) ⟶ Hom_{R}(E{σ′} ⊗_{Q′} R, N) = Hom_{Q′}(E{σ′}, N) ⟶ Hom_{Q}(E, N) .

show that the following diagram commutes:

Ext_{R}^{∗}(M, N) ⟶ \frac{Ext_{R}^{∗}(M, N)}{(χ_{1}, …, χ_{c−1}) Ext_{R}^{∗}(M, N)} ⟶ \frac{Ext_{R}^{∗}(M, N)}{(χ) Ext_{R}^{∗}(M, N)}
ρ_{MN}^{′′∗} ↓  ↓  ↓^{κ_{MN}^{∗}}
Ext_{Q′}^{∗}(M, N) ⟶ \frac{Ext_{Q′}^{∗}(M, N)}{(χ′_{1}, …, χ′_{c−1}) Ext_{Q′}^{∗}(M, N)} ⟶_{κ_{MN}^{′∗}} Ext_{Q}^{∗}(M, N) .

The composition in the lower row is ρ_{MN}^{′∗}. As ρ_{MN}^{′∗}ρ_{MN}^{′′∗} = ρ_{MN}^{∗} is surjective by assumption, it follows that the same holds for ρ_{MN}^{′∗}. We now get from the induction hypothesis that the sequence χ′_{1}, …, χ′_{c−1} is Ext_{Q′}^{∗}(M, N)–regular, and then conclude from the already established implication (i) ⇒ (iii) that κ_{MN}^{′∗} is an isomorphism. Thus, the middle vertical map in the diagram is surjective. It follows by Nakayama that so is ρ_{MN}^{′′∗}. The already established case c = 1 shows that χ_{c} is a non-zero-divisor on Ext_{R}^{∗}(M, N), and that Ext_{R}^{∗}(M, N)/χ_{c} Ext_{R}^{∗}(M, N) ≅ Ext_{Q′}^{∗}(M, N). We now know that the sequence χ_{c}, χ_{1}, …, χ_{c−1} is Ext_{R}^{∗}(M, N)–regular, and then we conclude from Matsumura [29; (16.5)] that the sequence χ = χ_{1}, …, χ_{c} has the same property, as desired. □

(4.7) Yoneda products. The graded R–module Ext_{R}^{∗}(M, N) has a natural structure of left Ext_{R}^{∗}(N, N)– and right Ext_{R}^{∗}(M, M)–bimodule, given by Yoneda products, cf. [27; §III.5]. Mehta [30; (2.3)] shows that there are natural homomorphisms of graded R–algebras Ext_{R}^{∗}(N, N) ⟵ R[χ] ⟶ Ext_{R}^{∗}(M, M) whose images lie in the centers of the corresponding Yoneda algebras. Furthermore, the S^{∗} = R[χ]–module structures on Ext_{R}^{∗}(M, N) induced by either of these homomorphisms coincides with that described in (4.1), cf. [12].

(4.8) Flat extensions. If R ⟶ R′ is a local flat extension, then the canonical homomorphism Ext_{R}^{∗}(M, M) ⊗_{R} R′ ⟶ Ext_{R′}^{∗}(M′, M′) is an isomorphism of graded R′–algebras, and Ext_{R}^{∗}(M, N)⊗_{R} R′ ⟶ Ext_{R′}^{∗}(M′, N′) is an equivariant homomorphism of graded right modules over it. We identify the objects connected by the canonical isomorphisms.

The following two finiteness results are crucial for this investigation.

(4.9) Theorem. If M is a finite module over a noetherian local ring R, and CI-dim_{R} M is finite, then for each finite R–module N the graded right module Ext_{R}^{∗}(M, N) is finite over the R–subalgebra Z^{∗} of Ext_{R}^{∗}(M, M) generated by the central elements in Ext_{R}^{2}(M, M).

If R ⟶ R′ ⟵ Q is a quasi-deformation and pd_{Q} M′ < ∞, then Z^{∗} ⊗_{R} R′ is the graded R′–subalgebra Z′^{∗} of Ext_{R′}^{∗}(M′, M′), generated by the central elements of degree 2.

Proof. By the finiteness theorem [23; (2.3)] or [8; (2.1)], Ext_{R′}^{∗}(M′, N′) and Ext_{R′}^{∗}(M′, M′) are noetherian graded modules over the algebra S^{∗} of cohomology operators of the deformation R′ ⟵ Q. In view of (4.7), we conclude that Ext_{R′}^{∗}(M′, M′) and Ext_{R′}^{∗}(M′, N′) are finite graded Z′^{∗}–modules. In particular, the R′–algebra Ext_{R′}^{∗}(M′, M′) is finitely generated. By faithful flatness so is the R–algebra Ext_{R}^{∗}(M, M). Let μ_{1}, … μ_{n} be a set of homogeneous generators, and let |μ_{j}| denote the degrree of μ_{j}. The map

Ext_{R}^{2}(M, M) ⟶^{α} ⨁_{j=1}^{n} Ext_{R}^{|μ_{j}|+2}(M, M) , α(μ) = ([μ, μ_{1}], …, [μ, μ_{n}]) ,

where [μ, μ_{j}] = μμ_{j} − μ_{j}μ, is a homomorphism of R–modules with Ker α = Z^{2}. As μ_{1} ⊗ 1, …, μ_{n} ⊗ 1 generate the R′–algebra Ext_{R}^{∗}(M, M) ⊗_{R} R′, we get

Z^{2} ⊗_{R} R′ = (Ker α) ⊗_{R} R′ = Ker(α ⊗_{R} R′) = Z′^{2} .

Thus, Z^{∗} ⊗_{R} R′ = Z′^{∗}. Now Ext_{R}^{∗}(M, N) is finite over Z^{∗} by faithfully flat descent. □

When (R, 𝔪, k) is a local ring, it is known from Milnor and Moore [31], André [3], and Sjödin [34], that Ext_{R}^{∗}(k, k) is the universal enveloping algebra of a graded Lie algebra (also known as a Lie superalgebra), the homotopy Lie algebra π^{∗}(R). In this context, we make frequent use of the following result of [12].

(4.10) Theorem. If (R, 𝔪, k) is noetherian local ring and M is a finite R–module such that CI-dim_{R} M < ∞, then graded left module Ext_{R}^{∗}(M, k) is finite over the k–subalgebra P^{∗} of Ext_{R}^{∗}(k, k) generated by the central elements in π^{2}(R). □

5. Complexity

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

One aspect of the asymptotic behavior of the minimal free resolution of M is captured by its complexity, a concept originating in the work of Alperin and Evens [1] on group representations and group cohomology, and adapted to local algebra in [8], [9]. We start this section by recalling some basic facts on this invariant, and proceed to establish a uniform bound on the complexities of all modules of finite CI-dimension over R. We finish by providing new and more direct proofs, based on the work in the preceding section, of some pivotal results of [8].

(5.1) Complexity. The complexity of M over R is the number

cx_{R} M = inf { d ∈ ℕ | there exists γ ∈ ℝ such that b_{n}^{R}(M) ≤ γn^{d−1} for n ≫ 0 } .

We list some properties that hold quite generally, cf. [9; Appendix].

(5.2) Proposition. With R and M as above the following hold.

(1) cx_{R} M = 0 if and only if pd_{R} M < ∞.

(2) If a sequence y ⊂ 𝔪 is both R–regular and M–regular, then cx_{R} M/(y)M = cx_{R} M.

(3) If R ⟶ R′ is a local flat extension and M′ = M ⊗_{R} R′, then cx_{R} M = cx_{R′} M′.

(4) If R ⟵ Q is a codimension c deformation, then cx_{Q} M ≤ cx_{R} M ≤ cx_{Q} M + c.

Proof. (1) is immediate from the definition.

(2) Let K = K(y; R) be the classical Koszul complex, and let F be a minimal free resolution of M over R. Since H_{n}(F ⊗_{R} K) ≅ H_{n}(M ⊗_{R} K) = 0 for n ≠ 0 we see that F ⊗_{R} K is a free resolution of M̄ = M/(y)M over R. It is clearly minimal, hence P R ; M̄(t) = (1 + t)^{g}P_{M}^{R}(t), with g = #y. The desired equality follows.

(3) and (4) follow easily from (1.5). □

A deeper understanding of complexity relies on a study of the Yoneda bimodule structure (4.7) on Ext_{R}^{∗}(M, k). In the case of finite CI-dimension we obtain strong numerical bounds.

(5.3) Theorem. If CI-dim_{R} M < ∞, then cx_{R} M is finite, it is equal to the order of the pole at t = 1 of the Poincaré series P_{M}^{R}(t), and

(1)  cx_{R} M = dim_{Z^{∗}} Ext_{R}^{∗}(M, k) = dim_{P^{∗}} Ext_{R}^{∗}(M, k) .

Furthermore, for each finite R–module N there is an inequality

(2)  dim_{Z^{∗}} Ext_{R}^{∗}(M, N) ≤ dim_{R} N + cx_{R} M .

Proof. (1) We know from (4.9) and (4.10) that Ext_{R}^{∗}(M, k) is a finite module over the graded k–algebras Z^{∗}/𝔪Z^{∗} and P^{∗}, generated over k by a finite number of elements of degree 2. Thus, the desired equalities follow from basic properties of Hilbert functions.

(2) The argument is by induction on n = dim_{R} N, and uses the fact that a short exact sequence of R–modules 0 ⟶ N′ ⟶ N ⟶ N′′ ⟶ 0 induces an exact sequence

Ext_{R}^{∗}(M, N′) ⟶ Ext_{R}^{∗}(M, N) ⟶ Ext_{R}^{∗}(M, N′′) ⟶ Σ^{−1} Ext_{R}^{∗}(M, N′)

of degree zero homomorphisms of graded right Ext_{R}^{∗}(M, M)–modules, hence also one of Z^{∗}–modules. (Note that as we use the homological suspension, Σ^{−1} Ext_{R}^{∗}(M, N′) is the graded module with degree i component equal to Ext_{R}^{i+1}(M, N′).)

If n = 0, then dim_{Z^{∗}} Ext_{R}^{∗}(M, N) ≤ cx_{R} M follows from this exact sequence by induction on the length of M over R. So assume that dim_{R} N = n > 0, and that the statement holds for modules of dimension less than n.

Let N′ = H_{𝔪}^{0}(N) be the largest submodule of finite length in N. As the result is available in dimension zero, and dim_{R} N′′ = dim_{R} N, the exact sequence shows it is sufficient to prove the inequality for N′′. Changing notation, we assume that depth_{R} N is positive and choose an N–regular element x ∈ 𝔪. The cohomology exact sequence induced by the short exact sequence 0 ⟶ N ⟶^{x} N ⟶ N/xN ⟶ 0 shows that Ext_{R}^{∗}(M, N)/x Ext_{R}^{∗}(M, N) is isomorphic to a Z^{∗}–submodule of Ext_{R}^{∗}(M, N/xN). This gives the second inequality in the following computation, where the third one is provided by the induction hypothesis:

dim_{Z^{∗}} Ext_{R}^{∗}(M, N) ≤ dim_{Z^{∗}}(Ext_{R}^{∗}(M, N)/x Ext_{R}^{∗}(M, N)) + 1
≤ dim_{Z^{∗}} Ext_{R}^{∗}(M, N/xN) + 1
≤ dim_{R}(N/xN) + cx_{R} M + 1
= dim_{R} N + cx_{R} M .  □

Ext algebras are almost never commutative and usually very complex, so we note:

(5.4) Corollary. When CI-dim_{R} M < ∞ the graded R–algebra Ext_{R}^{∗}(M, M) is modulefinite over its center, which is a finitely generated graded R–algebra, and whose Krull dimension m satisfies max { dim_{R} M, cx_{R} M } ≤ m ≤ dim_{R} M + cx_{R} M.

Proof. To see that dim_{R} M is a lower bound for the Krull dimension of the center of Ext_{R}^{∗}(M, M), note that Ext_{R}^{∗}(M, M)/ Ext_{R}^{⩾1}(M, M) ≅ Hom_{R}(M, M) is a finite module over this algebra, and its Krull dimension is dim_{R} M. The lower bound cx_{R} M is obtained by remarking that Ext_{R}^{∗}(M, k) is a finite module which has this Krull dimension. □

Clearly, the upper and lower bounds on m coincide if dim_{R} M or cx_{R} M is zero, but

(5.5) Example. The inequality m ≤ dim_{R} M + cx_{R} M may be strict.

Let (S, 𝔭) be a local domain containing ℤ and having residual characteristic p > 0, and let G = ⟨h⟩ be a cyclic group of order q = p^{e} > 1. The group algebra R = S[G] is then a local ring with maximal ideal (𝔭, h − 1)R. When S is considered as an R–module via the isomorphism S ≅ R/(h − 1), the deformation R ⟵ S[Y]_{(𝔭,Y −1)}, h ⟵ Y, shows that CI-dim_{R} S is finite. The sequence σ: 0 ⟶ S ⟶ R ⟶ R ⟶ S ⟶ 0, whose middle map is multiplication by h − 1, is exact. Splicing it with itself we obtain a minimal R–free resolution of S which yields isomorphisms of R–algebras Ext_{R}^{∗}(S, S) = Z^{∗} ≅ S[X]/(qX), with σ ↔ X. Thus, dim Z^{∗} = dim_{R} S < dim_{R} S + 1 = dim_{R} S + cx_{R} S.

The embedding dimension of R, edim R, is the minimal number of generators of 𝔪. Given the latitude of base change allowed by (1.2), it is quite remarkable that there exists an upper bound on the complexities of all R–modules of finite CI-dimension:

(5.6) Theorem. If M has finite CI-dimension, then cx_{R} M ≤ edim R − depth R, and the inequality is strict unless R is a complete intersection.

Proof. Let ζ^{2} denote the (super) center of π^{∗}(R), and set rank_{k} π^{i} = e_{i}. From (4.10) and (5.3.1) we get cx_{R} M ≤ rank_{k} ζ^{2}. It follows from [20; Theorem A] that rank_{k} ζ^{2} ≤ edim R − depth R, and when equality holds e_{i} = 0 for i > 2. In that case, Poincaré– Birkhoff–Witt yields P_{k}^{R}(t) = (1 + t)^{e_{1}}(1 − t^{2})^{−e_{2}}, and we conclude that R is a complete intersection by the criterion of Assmus [4; (2.7)]. □

When pd_{R} M is finite, the Auslander–Buchsbaum Equality implies that 0 ≤ pd_{R} M ≤ depth R, and exactness of the Koszul complexes on R–regular sequences shows that all intermediate values occur. Similarly, (1.4) shows that when CI-dim_{R} M is finite 0 ≤ CI-dim_{R} M ≤ depth R, but we do not know whether these values appear in combination with each complexity allowed by (5.6). Nevertheless, the following example shows that many such combinations do occur: its list will be exhaustive if, as suggested by [9; (4.3)], rank_{k} ζ^{2} is equal to the maximal codimension of an embedded deformation of R̂.

(5.7) Example. If the local ring R has a codimension c deformation Q with edim Q = edim R, then for any pair of integers (d, g) with 0 ≤ d ≤ c and 0 ≤ g ≤ depth R there exists an R–module M with cx_{R} M = d and CI-dim_{R} M = g.

Indeed, given d as above, the original deformation can be factored through an embedded deformation of codimension d, hence it suffices to construct the relevant examples when d = c. Fix an arbitrary integer r, and let M denote the module M_{r}^{[c]} introduced in (3.1). That theorem yields cx_{R} M = c and CI-dim_{R} M = 0, hence depth_{R} M = depth R by (1.4). For g as above choose a length g sequence y ⊂ 𝔪 which is regular on both R and M. Thus, CI-dim_{R} M/(y)M = g by (1.12.1) and cx_{R} M/(y)M = c by (5.2.2).

One of our main techniques for studying a module of finite CI-dimension is to construct a quasi-deformation which reflects a particular aspect of its minimal resolution.

(5.8) Factorizations. Consider a deformation R′ ⟵^{ρ} Q with kernel 𝔠, and set C = (𝔠/𝔠^{2})^{∗}.

A factorization R′ ⟵ Q′ ⟵ Q of ρ into a composition of deformations with kernels 𝔠′ and 𝔡 = 𝔠/𝔠′ yields an epimorphism of free R′–modules 𝔠/𝔠^{2} ⟶ 𝔠/(𝔠′ + 𝔠^{2}) = 𝔡/𝔡^{2}. Thus, (𝔡/𝔡^{2})^{∗} = Hom_{R}(𝔡/𝔡^{2}, R) is a direct summand of C. Conversely, given a decomposition C = D ⊕ E, consider the direct summand D^{⊥} = ⋂_{λ∈D} Ker λ of 𝔠/𝔠^{2}. If a sequence y ⊂ 𝔠 lifts a basis of D^{⊥}, then it is Q–regular, and 𝔡 = 𝔠/(y) is generated by a Q′ = Q/(y)– regular sequence. This provides a factorization R′ ⟵ Q′ ⟵ Q of ρ such that (𝔡/𝔡^{2})^{∗} = D.

Let x_{1}^{∗}, …, x_{c}^{∗} be the basis of (Σ^{2}(𝔠/𝔠^{2}))^{∗} dual to the basis of Σ^{2}(𝔠/𝔠^{2}) provided by the canonical images of Σ^{2}(x_{1}), …, Σ^{2}(x_{c}). Identify the symmetric algebra Sym_{R′}^{∗}(Σ^{−2}C) with the algebra S^{∗} = R[χ] of cohomology operators defined by ρ by an isomorphism under which x_{j}^{∗} corresponds to χ_{j} for 1 ≤ j ≤ c. If η = η_{1}, …, η_{d} is a basis for a direct summand D of C as above, then then R′[η] is identified with a subalgebra of R[χ], and thus Ext_{R}^{∗}(M, N) becomes a graded R′[η]–module for arbitrary R–modules M and N. The functoriality of Eisenbud operators [18; (1.7)] shows that this module structure coincides with the one defined by the deformation R′ ⟵ Q/(y) = Q′ through which ρ factors.

The next result can also be deduced from [8; (3.6)], but the proof of [loc. cit.] is considerably more complex.

(5.9) Theorem. If R ⟶ R′ ⟵ Q is a quasi-deformation with pd_{Q} M′ < ∞, then cx_{R} M ≤ pd_{Q} R′, and the deformation R′ ⟵ Q is embedded when equality holds.

If furthermore the residue field ℓ of R′ is infinite, then R′ ⟵ Q factors as a composition R′ ⟵ Q′ ⟵ Q of deformations with cx_{R} M = pd_{Q′} R′.

Proof. The inequality cx_{R} M ≤ pd_{Q} R′ is immediate from (5.2.4). Assume that equality holds, and that edim R′ < edim Q. We can then generate 𝔠 = Ker(Q ⟶ R′) by a regular sequence whose first element x is not in the square of the maximal ideal of the ring Q. The inequality above and [32; (27.5)] yield cx_{R} M ≤ pd_{Q/(x)} M′ = pd_{Q} M′ − 1 = cx_{R} M − 1, which is absurd.

Set 𝔠 = Ker(Q ⟶ R), and note by (4.10) that Ext_{R′}^{∗}(M′, ℓ) is a finite graded module over S^{∗}, and that its Krull dimension d is equal to cx_{R′} M′ = cx_{R} M. As the residue field of R′ is infinite, we may choose in S^{∗}^{2} a homogeneous system of parameters η for Ext_{R′}^{∗}(M′, ℓ), such that R′η is a direct summand of S^{∗}^{2}. By (5.8) we identify R′[η] with the algebra of cohomology operators defined of a codimension d deformation R′ ⟵ Q′ trough which R′ ⟵ Q factors. As Ext_{R′}^{∗}(M′, ℓ) is finite over R′[η], we get from (4.2) that pd_{Q′} M′ is finite, hence Q′ has the desired properties. □

As an immediate corollary, we obtain:

(5.10) Theorem. If CI-dim_{R} M < ∞, then there exists a quasi-deformation R ⟶ R′ ⟵ Q such that cx_{R} M = pd_{Q} M′. Furthermore, edim R′ = edim Q, and Q may be chosen complete with algebraically closed residue field. □

In the context of CI-dimension it is natural to consider the following variant of the notion of the virtual projective dimension of [8 (3.3)]

qpd_{R} M = inf { pd_{Q}(M ⊗_{R} R′) | R ⟶ R′ ⟵ Q is a quasi-deformation } .

Our last result contains the Auslander–Buchsbaum Equality, and its extension in [8; (3.5)].

(5.11) Theorem. For any R–module M there is an equality

qpd_{R} M = CI-dim_{R} M + cx_{R} M .

If CI-dim_{R} M is finite, then qpd_{R} M = depth R − depth_{R} M + cx_{R} M < ∞.

Furthermore, qpd_{R} M ≤ vpd_{R} M, and equality holds when vpd_{R} M is finite.

Proof. By (5.3), a module of finite CI-dimension has finite complexity, hence CI-dim_{R} M + cx_{R} M and qpd_{R} M are finite simultaneously. Assume this is the case, and choose a quasideformation R ⟶ R′ ⟵ Q with qpd_{R} M = pd_{Q} M′. By (1.14) and (5.2), it is harmless to assume that the residue field of R′ is infinite.

On the one hand, we then have

CI-dim_{R} M + cx_{R} M ≤ CI-dim_{R} M + pd_{Q} R′ = (pd_{Q} M′ − pd_{Q} R′) + pd_{Q} R′ = qpd_{R} M

where the inequality comes from the theorem, and the first equality from (1.7) and (1.3). On the other hand, replacing Q by a deformation Q′ constructed in (5.10), we get

qpd_{R} M ≤ pd_{Q′} M′ = depth Q′ − depth_{Q′} M′
= pd_{Q′} R′ + depth_{Q′} R′ − depth_{Q′} M′
= cx_{R} M + depth R′ − depth_{R′} M′
= cx_{R} M + depth R − depth_{R} M
= cx_{R} M + CI-dim_{R} M

with equalities provided by Auslander–Buchsbaum (twice), (5.10), (1.7), and (1.3).

The inequality qpd_{R} M ≤ vpd_{R} M is clear from the definitions. If the virtual projective dimension is finite, then the preceding argument works for the flat extension R ⟶ R′ = R̃, and shows that vpd = cx_{R} M + CI-dim_{R} M. □

6. Homological redundancy

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

Much of the work in this paper revolves around the perception that for any given quasi-deformation R ⟶ R′ ⟵ Q, the homological properties of the Q–module M′ = M ⊗_{R} R′ provide an “upper bound” for those of the R–module M. A case in point is the coefficientwise inequality of formal power series

(6.0)  P_{M}^{R}(t) ≤ P_{M′}^{Q}(t)(1 − t^{2})^{−c}

recalled from (1.5). On a structural level, the standard resolution E{σ} of Shamash and Eisenbud, recalled in (3.10), provides an approximation to the minimal resolution F′ of M′. The latter is always a direct summand of E{σ}, and when these complexes coincide, then the minimal resolution of M over R is completely determined by data defined over Q. In this section we introduce and study the concept of honological reducibility, which descibes how much structure on F′ can be induced by the procedure above.

Some early results provide handy conditions for equality to hold in (6.0).

(6.1) Example. If (Q, 𝔫) is a deformation of R and M is an R–module with 𝔫 ann_{Q} M ⊇ Ker(Q ⟶ R), then equality holds in (6.0) by [36; Theorem 5] and [33; §3, Corollary 1].

First we explore what can be said in general in the extremal situation.

(6.2) Proposition. Let ℓ denote the residue field of R′, let E{σ} be the standard R′–free resolution (3.10) of M′ = M ⊗_{R} R′ constructed from a system of higher homotopies σ on a minimal Q–free resolution E of M′, and let R′[χ] = R′[χ_{1}, …, χ_{c}] be the algebra of cohomology operators (4.1) defined by R′ ⟵ Q. The following conditions are equivalent.

(i) P_{M}^{R}(t) = P_{M′}^{Q}(t)(1 − t^{2})^{−c}. (ii) σ(E) ⊆ 𝔫E for each σ ∈ σ. (iii) E{σ} is a minimal R–free resolution. (iv) Ext_{R′}^{∗}(M′, ℓ) ≅ ℓ[χ] ⊗_{ℓ} Ext_{Q}^{∗}(M′, ℓ) as graded ℓ[χ]–modules.

(v) χ = χ_{1}, …, χ_{c} is an Ext_{R′}^{∗}(M′, ℓ)–regular sequence. In particular, the validity of (ii) or (iii) does not depend on the choice of σ.

Proof. The equivalence of the first three conditions follows from the expression (3.10.3) for the ranks of free modules in a standard R′–free resolution E{σ}, and from the observation that by (3.10.1) and (3.10.2) the resolution E{σ} is minimal precisely when σ(E) ⊆ 𝔫E for each σ ∈ σ. The implication (iv) ⇒ (v) is obvious.

(iii) ⇒ (iv). Consider the complexes Hom_{R′}(E{σ}, ℓ) and R′[χ] ⊗_{R′} Hom_{Q}(E, ℓ), whose differentials are zero in view of the minimality of the free resolutions E{σ} and E. By (4.5) they are isomorphic as DG R′[χ]–modules, so we get an isomorphism in (iv).

(v) ⇒ (i). Set E^{∗} = Ext_{R′}^{∗}(M′, ℓ)/(χ) Ext_{R′}^{∗}(M′, ℓ). On the one hand, the regularity of the sequence χ_{1}, …, χ_{c} on Ext_{R′}^{∗}(M′, ℓ) produces an equality of formal power series P_{M′}^{R′}(t) = (∑_{i⩾0} rank_{ℓ} E^{i} t^{i})(1 − t^{2})^{−c}. On the other hand, by (4.3) the same assumption yields an isomorphism E^{∗} ≅ Ext_{Q}^{∗}(M′, ℓ), hence ∑_{i⩾0} rank_{ℓ} E^{i} t^{i} = P_{M′}^{Q}(t). □

When the conditions above are fulfilled, the homological algebra of M over R is essentially determined by that of M′ over Q. This motivates the next definition.

(6.3) Homological redundancy. We say that a quasi-deformation R ⟶ R′ ⟵ Q is a homological reduction of the R–module M if the equivalent conditions of (6.2) hold. The homological redundancy of M over R is defined to be the number

hom red_{R} M = sup { c ∈ ℕ | M has a homological reduction of codimension c } .

A module M with hom red_{R} M = 0 is said to be homologically irreducible. Note that such a module is necessarily non-zero, since clearly hom red 0 = ∞.

To study homological redundancy, we introduce the depth of a graded module W over a commutative graded ring A as the supremum of the lengths of A–regular sequences of homogeneous elements of non-zero degree, and denote this number by depth_{A} W. Note that 0 ≤ depth_{A} W ≤ ∞ when W ≠ 0, and depth_{A} 0 = −∞. In the next theorem, which should be compared with (5.3.1), Z^{∗} and P^{∗} are the algebras defined in (4.9) and (4.10).

(6.4) Theorem. If M ≠ 0 is a finite R–module, then 0 ≤ hom red_{R} M ≤ cx_{R} M. If furthermore M has finite CI-dimension, then

hom red_{R} M = depth_{Z^{∗}} Ext_{R}^{∗}(M, k) = depth_{P^{∗}} Ext_{R}^{∗}(M, k) .

A crucial part of the argument for (6.4) is provided by a factorization theorem.

(6.5) Theorem. In a quasi-deformation R ⟶ R′ ⟵^{ρ} Q such that pd_{Q} M′ < ∞ and R′ has an infinite residue field, ρ can be factored as a composition of deformations R′ ⟵ Q′ ⟵ Q, where hom red_{R} M = pd_{Q′} R′ and the Q′–module M′ is homologically irreducible.

After some preparation, the theorems are proved jointly at the end of the section.

(6.6) Lemma. (1) If R ⟶ R′ ⟵ Q and Q ⟶ Q′ ⟵ P are quasi-deformations, then the composition R ⟶ R′ ⊗_{Q} Q′ ⟵ P defined in (1.10) is a homological reduction of M if and only if R ⟶ R′ ⟵ Q is one of M and Q ⟶ Q′ ⟵ P is one of M′.

(2) If R ⟶ R′ ⟵ Q is a homological reduction of M, then edim R′ = edim Q.

Proof. (1) By multiplying each side of (6.0) with the corresponding side of the coefficientwise inequality 1 ⩽ (1 − t^{2})^{− depth R}, we obtain a symmetric relation

P_{M}^{R}(t)(1 − t^{2})^{− depth R} ⩽ P_{M′}^{Q}(t)(1 − t^{2})^{− depth Q}

which becomes an equality precisely when R ⟶ R′ ⟵ Q is a homological reduction of M. Thus, setting M′′ = M′ ⊗_{Q} Q′ = M ⊗_{R} (R′ ⊗_{Q} Q′), we get a concatenation of inequalities

P_{M}^{R}(t)(1 − t^{2})^{− depth R} ⩽ P_{M′}^{Q}(t)(1 − t^{2})^{− depth Q} ⩽ P_{M′′}^{P}(t)(1 − t^{2})^{− depth P}

in which the extreme two terms are equal if and only if all three are.

(2) Suppose that edim R′ < edim Q, and choose a minimal generating set for Ker(Q ⟶ R′) whose first element x is not in the square of the maximal ideal of Q. From (1) we see that R ⟶ R′ ⟵ Q′ = Q/(x) is a homological reduction of M, hence P_{M}^{R}(t) = P_{M′}^{Q′}(t)(1 − t^{2})^{−1}. On the other hand, P_{M}^{R}(t) = P_{M′}^{R′}(t) = P_{M}^{Q′}(t)(1 + t)^{−1} due respectively to (1.5), and to [32; (27.3)] in view of the choice of x. The two expressions for P_{M}^{R}(t) being incompatible, we get a contradiction, whence edim R′ = edim Q. □

The essence of the following remark is that properties invariant under finite descent are also invariant under the more general operation of replacing the original ring by another ring of operators, which acts in a compatible way, and over which the module is noetherian.

(6.7) Bimodules. Let R be a commutative ring, let A and B be graded commutative R– algebras, and let W be a graded A-B–bimodule (over commutative algebras we drop the distinction between left and right actions). Equivalently, W is a graded module over the graded R–algebra T = A⊗_{R}B, and thus it has a canonical structure of faithful module over the graded R–algebra C = T/ ann_{T} W, such that the natural homomorphisms of graded R–algebras α: A ⟶ C and C ⟵ B : β induce the original actions of A and B on W.

Let R be noetherian, the R–algebras A and B finitely generated, and the module W finite over each one of them. In this case C is finitely generated, W is a finite C–module, and the maps α and β are both finite. Indeed, γ: T ⟶ End_{A}(W), γ(a⊗b)(m) = (−1)^{|m||b|}amb, induces an A–linear embedding of C into the noetherian A–module Hom_{A}(W, W). The finiteness of α follows. That of β results by symmetry.

Proof of Theorem (6.4) and Theorem (6.5). The inequality hom red_{R} M ≥ 0 is simply the observation that the constant deformation (1.10) is a homological reduction of any M. For the other inequality, note that if a quasi-deformation R ⟶ R′ ⟵ Q is a codimension c homological reduction of an R–module M ≠ 0, and M′ = M ⊗_{R} R′, then there are coefficientwise inequalities P_{M}^{R}(t) = P_{M′}^{Q}(t)(1 − t^{2})^{−c} ⩾ (1 − t^{2})^{−c}, that is, have b_{2n}^{R}(M) ≥ \binom{2n+c−1}{c−1} for n ≥ 0. Thus, cx_{R} M ≥ c, and hence cx_{R} M ≥ hom red_{R} M.

For the rest of the argument we assume that CI-dim_{R} M is finite, pick a quasi-deforma-tion R ⟶ R′ ⟵ Q such that pd_{Q} M′ < ∞ and the residue field ℓ of R′ is infinite, and set M′ = M ⊗_{R} R′. Also, we denote by Z^{∗} and Z′^{∗} the subalgebras generated by the degree 2 central elements of Ext_{R}^{2}(M, M) and Ext_{R′}^{2}(M′, M′), respectively, and note that Z^{∗} ⊗_{R} R′ = Z′^{∗} by (4.9). Finally, we set depth_{Z^{∗}}(Ext_{R}^{∗}(M, k)) = g.

If F is a minimal free resolution of M over R, then

Ext_{R}^{∗}(M, k) ⊗_{k} ℓ = H_{∗} Hom_{R}(F, k) ⊗_{k} ℓ ≅ H_{∗} Hom_{R′}(F ⊗_{R} R′, ℓ) = Ext_{R′}^{∗}(M′, ℓ) ,

This provides an identification of (Z^{∗} ⊗_{R} k) ⊗_{k} ℓ with Z′^{∗} ⊗_{R′} ℓ, which is compatible with the isomorphism Ext_{R}^{∗}(M, k) ⊗_{k} ℓ ≅ Ext_{R′}^{∗}(M′, ℓ) of graded right modules over these algebras. As by (4.9) the Z^{∗} ⊗_{R} k–module Ext_{R}^{∗}(M, k) is finite, and the depth of a finite graded module over a finitely generated graded k–algebra is invariant under base change by field extensions of k, we see that depth_{(Z′^{∗}⊗_{R′} ℓ)}(Ext_{R}^{∗}(M, k) ⊗_{k} ℓ) = g. This implies that the depth of Ext_{R′}^{∗}(M′, ℓ) over Z′^{∗} is equal to g, and hence, in view of (4.7), so is its depth over the algebra S^{∗} of cohomology operators of the deformation R′ ⟵ Q.

If the R–module M has a homological reduction of codimension c, then (6.2) shows that c ≤ depth_{S^{∗}}(Ext_{R}^{∗}(M, k) ⊗_{k} ℓ), that is, c ≤ g. As ℓ is infinite, then we can choose in S^{2} a regular sequence η of length g, which generates an R′–direct summand of that free R′–module. By (5.8), there is a factorization R′ ⟵ Q′ ⟵ Q such that the algebra of cohomology operators defined by R′ ⟵ Q′ is equal to R′[η]. From (6.6.1) we see that R ⟶ R′ ⟵ Q′ is a homological reduction of M, hence hom red_{R} M ≥ g.

We now know that hom red_{R} M = g. Assuming the Q′–module M′ has a homological reduction Q′ ⟶ Q′′ ⟵ P of positive codimension b, we see from (6.6.1) that R ⟶ R′′ ⟵ P is a homological reduction of the R–module M. But its codimension g+b is strictly greater than hom red_{R} M, which is absurd. This proves M′ is homologically irreducible over Q′.

Finally, recall that Ext_{R}^{∗}(M, k) is a bimodule over Z^{∗} and P^{∗}, which by (4.9) and (4.10) is finite over each one of these algebras. As depth is invariant by finite descent, we conclude from (6.7) that depth_{Z^{∗}} Ext_{R}^{∗}(M, k) = depth_{P^{∗}} Ext_{R}^{∗}(M, k).

All the assertions of (6.4) and (6.5) have now been established. □

7. Critical degree and growth of Betti numbers

We consider finite modules over a local ring (R, 𝔪, k).

We prove there is a critical degree in the minimal resolution of each module M of finite CI-dimension, after which asymptotically stable patterns develop. In particular, we prove that beyond this degree the Betti sequence is non-decreasing, answering a basic question of [7] in the case of finite CI-dimension—and thus for all modules over complete intersections.

We provide examples to illustrate unstable phenomena at the start of Betti sequences. Futhermore, we show that the syzygy following the critical degree is the simplest one in the entire syzygy sequence: it admits a homological reduction of highest codimension, and thus its properties are determined by a module of lowest possible complexity.

(7.1) Critical degree. The critical degree cr deg_{R} M of an R–module M is the infimum of those s ∈ ℕ for which there exists a chain endomorphism μ of degree q < 0 of the minimal resolution F of M, such that μ_{n+q}: F_{n+q} ⟶ F_{n} is surjective for all n > s. If μ is bijective, then M is said to have period q after s steps.

Clearly, cr deg 0 = −∞, and −1 ≤ cr deg_{R} M ≤ ∞ when M ≠ 0. If M ≠ 0 and pd_{R} M is finite, then cr deg_{R} M = pd_{R} M, hence the critical degree sharpens the measure of inhomogeneity of free resolutions provided by the notion of projective dimension. In the case of finite CI-dimension, we have the following cohomological characterization, in which Z^{∗} and P^{∗} are the graded algebras introduced in (4.9) and (4.10).

(7.2) Proposition. If M ≠ 0 is a finite R–module with CI-dim_{R} M < ∞, then the critical degree of M is finite, say cr deg_{R} M = s, and the following hold.

(1) There are equalities

s = sup{r ∈ ℕ | depth_{P^{∗}} Ext_{R}^{⩾r}(M, k) = 0} = sup{r ∈ ℕ | depth_{Z^{∗}} Ext_{R}^{⩾r}(M, k) = 0} .

(2) There is a codimension 1 quasi-deformation R ⟶ R′ ⟵ Q, such that cr deg_{R′} M′ = s and the Eisenbud operator on the minimal resolution of M′ is surjective in degrees n > s.

Part (0) of the next theorem is well known and is included only for comparison. Part (1), which can also be obtained from [8; (4.4)] and extends [18; (7.2)], shows that modules of finite CI-dimension satisfy a conjecture of Eisenbud [18], which fails in general [21].

(7.3) Theorem. Let M ≠ 0 be a finite R–module with depth R − depth_{R} M = g and cr deg_{R} M = s. When CI-dim_{R} M is finite and n > s, the following three cases occur.

(0) If cx_{R} M = 0 then s = g and b_{n}^{R}(M) = 0.

(1) If cx_{R} M = 1 then s ≤ g and b_{n}^{R}(M) = b for some integer b > 0; furthermore, M has period 2 after s + 1 steps.

(2) If cx_{R} M ≥ 2 then s < ∞ and b_{n}^{R}(M) < b_{n+1}^{R}(M).

The next result shows that cr deg_{R} M determines the homological reducibility of all the syzygies M_{n}. The positivity of hom red_{R} M_{n} for n ≫ 0 contains a result of Eisenbud [18; (8.2)]: high syzygies have a homological reduction of codimension at least 1. Surprisingly, we also show that M_{n} cannot be deformed to any higher codimension when n > cr deg_{R} M.

(7.4) Theorem. If M is a finite R–module of finite CI-dimension, s = cr deg_{R} M, and M_{n} is the n’th syzygy of M, then for n ≥ 0 one has

hom red_{R} M_{n} = \{ \begin{matrix}0 & when n ≤ s; \\ h ≥ 1 & when n = s + 1 ; \\ 1 & when n > s + 1 . \end{matrix}

If h ≥ 2, then Ext_{R}^{⩽s}(M, k) is the largest submodule of finite length of Ext_{R}^{∗}(M, k) both as a left Ext_{R}^{∗}(k, k)–module and as a right Ext_{R}^{∗}(M, M)–module, and the decomposition Ext_{R}^{∗}(M, k) = Ext_{R}^{⩽s}(M, k) ⊕ Ext_{R}^{>s}(M, k) is compatible with both structures.

Before giving arguments, we illustrate unexpected patterns of growth of Betti numbers of modules of finite CI-dimension, based on the modules M_{i} = M_{i}^{[c]} from (3.1), with (Q, 𝔫) is a local ring with edim Q = m, x is a length c ≥ 2 regular sequence in 𝔫^{2}, and R = Q/(x).

The critical degree of a module of complexity d ≤ 1 is bounded by depth R, but

(7.5) Example. There is no bound on the critical degree, valid for all modules of complexity equal to some integer d ≥ 2.

Indeed, fix s ≥ 0 and note that M_{0} is the s + 1’st syzygy of M_{−s−1}. As the graded P^{∗}– module Ext_{R}^{∗}(M_{−s−1}, k) is isomorphic to Ext_{R}^{⩽s}(M_{−s−1}, k) ⊕ k[χ] ⊗_{k} Σ^{−s−1} Ext_{Q}^{∗}(M_{0}, k) by (6.2) and (7.4), it follows from (7.2.1) that cr deg_{R} M_{−s−1} = s.

The first (depth R + 1) Betti numbers contain the information that cx_{R} M = 0, but

(7.6) Example. No finite interval of constant Betti numbers implies that cx_{R} M = 1. Indeed, for c = 2, s ≥ 0, and 0 ≤ n ≤ s we have b_{n}^{R}(M_{0} ⊕ M_{−s−1}) = 4^{m}(s + 2) by (3.1).

The Betti sequence of any module is monotonic after the critical degree, but

(7.7) Example. Strict growth of the Betti sequence does not signal the critical degree.

Take c = 2 and s ≥ 0 and note that M_{0} ⊕ M_{0} ⊕ M_{−s−1} has a strictly increasing Betti sequence by (3.1), and critical degree s by (7.5).

Among the consequences of the finiteness of cr deg_{R} M is the validity of a weak form of a conjecture of Eisenbud [18 p.37] (boundedness implies periodicity), as well as a partial answer to a question raised in [7; p.34] (is the Betti sequence of each finite R–module eventually non-decreasing?) Note also that the next result implies that the non-periodic modules with constant Betti numbers constructed in [21; §3] have infinite critical degree.

(7.8) Theorem. If M is a finite R–module such that cr deg_{R} M = s < ∞ and μ: F ⟶ F is a chain endomorphism which satisfies (7.1), then either cx_{R} M ≤ 1 and M has period q after depth R − depth_{R} M steps, or cx_{R} M > 1 and b_{n}^{R}(M) < b_{n+q}^{R}(M) for n > s.

If furthermore q ≤ 2, then b_{n}^{R}(M) ≤ b_{n+1}^{R}(M) for n > s, with equality when cx_{R} M ≤ 1.

Proof. Let F and μ be as in (7.1), set M_{n} = ∂(F_{n}) for n ∈ ℤ, and let μ̃: F_{q} ⟶ M be the composition of μ_{q}: F_{q} ⟶ F_{0} with the augmentation F_{0} ⟶ M, and identify Ext_{R}^{∗}(M, k) with H^{∗} Hom_{R}(F, k). Consider the pushout diagram with exact rows

F_{q+1}⟶^{∂_{q+1}} F_{q} ⟶^{∂_{q}} F_{q−1} ⟶^{∂_{q−1}} F_{q−2} ⟶^{∂_{q−2}} … ⟶^{∂_{1}} F_{0} ⟶ M ⟶ 0
μ̃↓  ↓  ∥  ∥  ∥
0 ⟶ M ⟶ M(μ) ⟶ F_{q−2} ⟶^{∂_{q−2}} … ⟶^{∂_{1}} F_{0} ⟶ M ⟶ 0

It follows from [27; §III.9] that μ^{n} = Hom_{R}(μ_{n+q}, k) is the iterated connecting homomorphism of the bottom row, which itself is the Yoneda splice of the exact sequences

0 ⟶ M ⟶ M(μ) ⟶ M_{q−1} ⟶ 0 , (7.8.1)

0 ⟶ M_{q−1} ⟶ F_{q−2} ⟶^{∂_{q−2}} … ⟶^{∂_{1}} F_{0} ⟶ M ⟶ 0 . (7.8.2)

Thus, μ^{n} = ±β^{n+1}α^{n}, with α^{n} the connecting homomorphism of (7.8.1) and β^{n+1} the iterated connecting homomorphism of (7.8.2). As β^{n+1} is bijective for all n, the cohomology exact sequence of (7.8.1) takes the form

… ⟶ Ext_{R}^{n}(M(μ), k) ⟶ Ext_{R}^{n}(M, k) ⟶^{μ^{n}} Ext_{R}^{n+q}(M, k) ⟶ Ext_{R}^{n+1}(M(μ), k) ⟶ …,

Since μ^{n} is the k–vector space dual of μ_{n+q}: F_{n+q} ⊗_{R} k ⟶ F_{n} ⊗_{R} k, which by assumption is surjective for n > s. Thus, μ^{n} is injective for n > s, the exact sequence splits, and

(7.8.3)  b_{n+q}^{R}(M) = b_{n}^{R}(M) + b_{n+1}^{R}(M(μ)) .

We see that M has complexity ≤ 1 precisely when r = pd_{R} M(μ) < ∞. Set b_{n} = b_{n}^{R}(M).

If cx_{R} M > 1, then (7.8.3) yields b_{n} < b_{n+q} for n > s. On the other hand, if cx_{R} M ≤ 1, then μ^{n} is an isomorphism for n > r, hence by Nakayama μ_{n+q} is a surjective homomorphism of free R–modules of the same rank, and thus an isomorphism.

As r = depth R − depth_{R} M(μ) ≥ 0, to finish the proof of the first part we show that depth_{R} M ≤ depth_{R} M(μ). Assuming depth_{R} M > depth R, we use (7.8.2) to get depth_{R} M_{q−1} ≥ depth R, and then (7.8.1) implies depth_{R} M(μ) ≥ depth R. As M(μ) has finite projective dimension over R, it is free, so M is itself periodic, and thus an infinite syzygy. This forces depth_{R} M = depth R, a contradiction. Assuming depth_{R} M(μ) < depth_{R} M ≤ depth R, we get depth_{R} M_{q−1} = depth_{R} M(μ) < depth_{R} M from (7.8.1) and depth_{R} M_{q−1} ≥ depth_{R} M from (7.8.2). This is a new contradiction, so we are done.

In the proof of the last assertion we may assume that q = 2. The map μ_{n+2} is surjective for n > s, and thus it induces a surjection M_{n+2} ⟶ M_{n}. Choosing a minimal prime ideal 𝔭 of R and counting lengths over R_{𝔭} in the localizations of (7.8.2) and (7.8.1), we get

(b_{n+1} − b_{n}) length(R_{𝔭}) = length(M_{n+2})_{𝔭} − length(M_{n})_{𝔭} ≥ 0 ,

hence b_{n+1} ≥ b_{n} for n > s. If cx_{R} ≤ 1, then M_{r+1} is periodic of period 2 by (1), hence for n > r we have M_{n+2} ≅ M_{n}, and thus b_{n+1} = b_{n}. □

Proof of Proposition (7.2). The finiteness of cr deg_{R} M = s follows from the equalities in (1), in view of (4.9). As both (1) and (2) are trivial when pd_{R} M is finite, we assume cx_{R} M > 0. We use the notation of the preceding proof.

(1) By [27; §III.9], the homomorphism μ^{n} is, up to sign, given by Yoneda multiplication of Ext_{R}^{n}(M, k) with cls(μ̃) ∈ Ext_{R}^{q}(M, M). Thus, the splitting of the cohomology exact sequence established above shows that cls(μ̃) is a non-zero-divisor on Ext_{R}^{>s}(M, k). The graded subalgebra Z^{∗}[cls(μ̃)] of Ext_{R}^{∗}(M, M) is commutative since Z^{∗} is central, and by (4.9) the graded Z^{∗}[cls(μ̃)]–module Ext_{R}^{>s}(M, k) is finite. Thus, its depth over Z^{∗} coincides with its depth over Z^{∗}[cls(μ̃)], which we have just seen to be positive.

This implies s = cr deg_{R} M ≥ max{r ∈ ℕ | depth_{Z^{∗}} Ext_{R}^{⩾r}(M, k) = 0}. As each element of Ext_{R}^{∗}(M, M), in particular of Z^{∗}, comes from a chain endomorphism of F, the opposite inequality is clear, and thus cr deg_{R} M ≥ max{r ∈ ℕ | depth_{Z^{∗}} Ext_{R}^{⩾r}(M, k) = 0}. On the other hand, by (4.9) amd (4.10) the right Z^{∗}– and left P^{∗}–bimodule Ext_{R}^{⩾r}(M, k) is finite on each side. It follows from the construction in (6.7) that its depth is the same over either algebra.

(2) By (1.14), choose a quasi-deformation R ⟶ R′ ⟵ Q such that 𝔪R′ is the maximal ideal of R′, the residue field ℓ of (Q, 𝔫) is infinite, and pd_{Q} M′ is finite. Note that depth_{S^{∗}} Ext_{R′}^{⩾r}(M′, ℓ) = depth_{Z^{∗}} Ext_{R′}^{⩾r}(M′, ℓ) for each r, so (1) yields cr deg_{R′} M′ = cr deg_{R} M. In view of (4.7), it also shows that

depth_{S^{∗}} Ext_{R′}^{>s}(M′, ℓ) = depth_{Z^{∗}} Ext_{R′}^{>s}(M′, ℓ) > 0 ,

As ℓ is infinite, there is an Ext_{R′}^{>s}(M′, ℓ)–regular element η in S^{2}. Note that η ∉ 𝔫S^{2}, for otherwise μ would annihilate Ext_{R′}^{∗}(M′, ℓ). Thus, R′η is a direct summand of S^{2}. By (5.8) there is a factorization R′ ⟵ Q′ ⟵ Q whose Eisenbud operator t′ induces on Ext_{R′}^{∗}(M′, ℓ) the same action as η. It follows that Hom_{R}(t′_{n+2}, k) is injective for n > s, hence t′_{n+2} is surjective by Nakayama. □

Proof of Theorem (7.3). Let R ⟶ R′ ⟵ Q′ be a quasi-deformation given by (7.2.2), in particular, cr deg_{R′} M′ = cr deg_{R} M = s. As R ⟶ R′ is a flat extension and Syz_{n}^{R′}(M′) ≅ Syz_{n}^{R}(M) ⊗_{R} R′ by (1.8.2), Syz_{n}^{R′}(M′) ≅ Syz_{n+2}^{R′}(M′) implies Syz_{n}^{R}(M) ≅ Syz_{n+2}^{R}(M), cf. [22; (2.5.8)]. Thus, neither the hypotheses nor the conclusions of the theorem change when we replace the R–module M by the R′–module M′. Changing notation, we assume R = Q/(x) for a local ring Q and a non-zero-divisor x, and t = t(Q, x, F) is an Eisenbud operator on a minimal resolution F of M over R, with t_{n+2} surjective when n > s.

By (4.1), there is a lifting of the complex (F, ∂) to a sequence of homomorphisms of free Q–modules (F̃, ˜∂), and a degree −2 endomorphism t̃ of F̃ such that t = t̃⊗_{Q} R. As t_{n+2} = t̃_{n+2} ⊗_{Q} Q/(x) is surjective, so is t̃_{n+2} by Nakayama. Thus, F̃_{n+2} = E ⊕ G with E = Ker t̃_{n+2}, and the restriction θ of t̃_{n+2} to G is an isomorphism with F̃_{n}.

Next we assume that b_{n}^{R}(M) = b_{n+1}^{R}(M) = b ≠ 0 for some n > s, and show that b_{n+2}^{R}(M) = b. Let γ: G ⟶ F̃_{n+1} be the restriction of ˜∂_{n+2}. As ˜∂_{n+1}γ is the restriction to G of ˜∂_{n+1} ˜∂_{n+2} = xt̃_{n+2}, we have ˜∂_{n+1}γ = xθ, and hence

⋀^{b} ˜∂_{n+1} ⋀^{b} γ = ⋀^{b}(˜∂_{n+1}γ) = ⋀^{b}(xθ) = x^{b} ⋀^{b} θ .

Note that G, F̃_{n+1}, and F̃_{n} have rank b and fix isomorphisms of Q with ⋀^{b}(G), ⋀^{b}(F̃_{n+1}), and ⋀^{b}(F̃_{n}). The maps ⋀^{b} ˜∂_{n+1}, ⋀^{b} γ, and ⋀^{b} θ are then given by multiplication with elements of Q, say y, z, and u, respectively. The equality above becomes yz = x^{b}u. As θ is bijective so is ⋀^{b}(θ), hence u is a unit in Q. Since x is a non-zero-divisor in Q, so is y, hence ˜∂_{n+1} is injective. From ˜∂_{n+1} ˜∂_{n+2}(E) = xt̃_{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}) ⊗_{Q} R ≅ Coker ∂_{n+3} = M_{n+2}

we conclude that M_{n+2} is a homomorphic image of the free R–module G ⊗_{Q} R ≅ R^{b}. It follows that b_{n+2}^{R}(M) ≤ b = b_{n}^{R}(M). On the other hand, we know from (7.8) that b_{n+2}^{R}(M) ≥ b_{n+1}^{R}(M) ≥ b_{n}^{R}(M), hence all three numbers are equal to b.

Thus, the sequence {b_{n}^{R}(M)}_{n>s} is either strictly increasing or constant. In the second case the equality rank_{R} F_{n+2} = rank_{R} F_{n} implies E = 0, so the surjective homomorphism t_{n+2} is bijective, and hence the induced homomorphism M_{n+2} ⟶ M_{n} is bijective. □

Proof of Theorem (7.4). By (6.4) and (7.2.1), hom red_{R} M_{n} = depth_{P^{∗}} Ext_{R}^{∗}(M_{n}, k) = depth_{P^{∗}} Ext_{R}^{⩾n}(M, k). This number is 0 precisely for n ≤ s. To see it is 1 for n > s + 1, use the homological characterization of depth and the exact sequence of P^{∗}–modules

0 ⟶ Ext_{R}^{⩾n+1}(M, k) ⟶ Ext_{R}^{⩾n}(M, k) ⟶ Σ^{−n}k^{b} ⟶ 0 ,

where b = b_{n}^{R}(M). Set next h = hom red_{R} M_{s+1}, and consider the exact sequence

0 ⟶ Ext_{R}^{>s}(M, k) ⟶ Ext_{R}^{∗}(M, k) ⟶ Ext_{R}^{⩽s}(M, k) ⟶ 0

of left Ext_{R}^{∗}(k, k) ⊗_{R} Ext_{R}^{∗}(M, M)^{op}–modules. The center of the tensor product contains the image of P^{∗} ⊗_{R} Z^{∗}, over which the graded module Ext_{R}^{>s}(M, k) is finite. Using again the construction of (6.7) and the invariance of depth under finite homomorphisms, we get

h = depth_{P^{∗}} Ext_{R}^{>s}(M, k) = depth_{(P^{∗}⊗Z^{∗})} Ext_{R}^{>s}(M, k) = depth_{Z^{∗}} Ext_{R}^{>s}(M, k) .

Thus, an inequality h ≥ 2 implies both that Ext_{R}^{>s}(M, k) has no non-trivial submodule of finite length, and that the exact sequence above splits. □

8. Asymptotes of Betti sequences

In this section R is a local ring, and M is a finite R–module of finite CI-dimension.

We show that the rational function representing the Poincaré series of M satisfies nontrivial arithmetical properties. As a consequence, we determine the first two terms of the asymptote of the Betti sequence of modules of higher complexity, and show that for big n the gaps {b_{n+1}^{R}(M) − b_{n}^{R}(M)} between consecutive Betti numbers grow essentially like a polynomial of degree d − 2. This may be viewed as a quantitative sharpening of the assertion of (7.3.2), that if cx_{R} M ≥ 2, then the Betti numbers of M are eventually nondecreasing. An application of these results yields a description of all Betti sequences over several classes of local rings which are “close” to complete intersections.

(8.1) Theorem. Let M be a finite module over a local ring R. If CI-dim_{R} M < ∞ (for instance, if R is a complete intersection) and cx_{R} M = d ≥ 2, then for n ≫ 0

b_{n}^{R}(M) = \{ \begin{matrix}b_{+}(n) when n is even ; \\ b_{−}(n) when n is odd , \end{matrix}

where b_{±} ∈ ℚ[t] are polynomials of the form

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

with integers b, c_{±}, e such that either 0 ≤ e ≤ d − 2, c_{+} = c_{−} and b > 0, or e = d − 1 and b > |c_{+} − c_{−}|. In particular, both difference polynomials b_{±}(t + 1) − b_{∓}(t) have degree d − 2 and positive leading coefficient.

(8.2) Corollary. When d ≥ 2 there exist periodic of period 2 functions δ_{i}: ℤ ⟶ ℚ, 0 ≤ i ≤ d − 2, such that δ_{0}(0) > 0, δ_{0}(1) > 0, and for n ≫ 0

b_{n}^{R}(M) − b_{n−1}^{R}(M) = δ_{0}(n)n^{d−2} + ⋯ + δ_{d−2}(n) .  □

From (8.1), or from [8; (4.2)], one has lim_{n→∞} b_{n}^{R}(M)/n^{d−1} ∈ (2^{e}(d − 1)!)^{−1}ℤ, but

(8.3) Example. When c_{+} = c_{−} the sequence (b_{n}^{R}(M) − b_{n−1}^{R}(M))/n^{d−2} converges to a number in (2^{d−1}(d − 2)!)^{−1}ℤ; when c_{+} ≠ c_{−} it diverges; furthermore, both cases occur.

Indeed, let (Q, 𝔫) be a two-dimensional regular local ring, let x = x_{1}, x_{2} be a Q–regular sequence contained in 𝔫^{3}, set (R, 𝔪) = (Q/(x), 𝔫/(x)), and note that

b_{n}^{R}(R/𝔪) = n + 1 for each n ≥ 0 by the result of Tate recalled in (6.2).

b_{n}^{R}(R/𝔪^{2}) = \frac{3}{2}n + 1 for even n ≥ 0 and b_{n}^{R}(R/𝔪^{2}) = \frac{3}{2}n + \frac{3}{2} for odd n ≥ 1 by [10; (2.1)].

The theorem has an interesting application to local rings “close” to complete intersections. Indeed, if R satisfies one of conditions (a) through (d) of the next theorem, then by [11] all finite R–modules have rational Poincaré series with a common denominator. Furthermore, [9] determines the possible denominators and shows that if P_{M}^{R}(t) has radius of convergence ≥ 1, then M has finite virtual projective dimension, hence finite CI-dimension. For the rings in (e) the corresponding analysis is carried out in [26]. Thus, we get

(8.4) Theorem. Let R be a local ring which satisfies one of the following conditions: (a) R is one link from a complete intersection;

(b) R is two links from a complete intersection and R is Gorenstein;

(c) edim R − depth R ≤ 3;

(d) edim R − depth R = 4 and R is Gorenstein;

(e) edim R − depth R = 4 and R is a Cohen–Macaulay almost complete intersection of residual characteristic ≠ 2.

If M is a finite R–module whose Poincaré series P_{M}^{R}(t) has radius of convergence ρ ≥ 1, then cx_{R} M = d ≤ edim R − depth R and there exist polynomials Δ_{1} and Δ_{2}, both of degree d − 2 and with positive leading coefficients, such that

Δ_{1}(n) ≤ b_{n}^{R}(M) − b_{n−1}^{R}(M) ≤ Δ_{2}(n) for n ≫ 0 .

In particular, {b_{n}^{R}(M)}_{n⩾0} is eventually either constant or strictly increasing. □

Remark. Using the information on the denominators of Poincaré series from [9] and [26], Sun [35] has shown that if ρ < 1, then lim_{n→∞} b_{n+1}^{R}(M)/b_{n}^{R}(M) = 1/ρ > 1. Thus, the asymptotic pattern—in particular, the eventual growth—of Betti sequences is now known for all modules over complete intersections, and over the rings described in (8.4).

We establish Theorem (8.1) in an equivalent form, stated in terms of Poincaré series. An immediate consequence of the finiteness result in (4.10) is that the Poincaré series

of an R–module M of finite CI-dimension can be written uniquely in the form

(8.5)  P_{M}^{R}(t) = \frac{p_{M}(t)}{(1 − t)^{d}(1 + t)^{e}} with p_{M} ∈ ℤ[t] such that p_{M}(±1) ≠ 0 .

The next theorem shows that it satisfies non-trivial arithmetical relations.

(8.6) Theorem. If M is a finite R–module with CI-dim_{R} M < ∞ and Poincaré series given by (8.5), then cx_{R} M = d, p_{M}(1) > 0, and one of the following cases occurs.

(0) d = 0: e < 0, or e = 0 with p_{M}(−1) > 0 ; also, deg p_{M} = depth R −depth_{R} M +e.

(1) d = 1: e ≤ 0, and deg p_{M} ≤ depth R − depth_{R} M + e.
(2) d ≥ 2: e < d − 1, or e = d − 1 with p_{M}(1) > |p_{M}(−1)|.

Proof. Basic dimension theory and (5.3.1) yield d = cx_{R} M. Set s = cr deg_{R} M.

(0) Here pd_{R} M < ∞, so P_{M}^{R}(t) is a polynomial of degree depth R − depth_{R} M, hence e ≤ 0 and deg p_{M} = depth R − depth_{R} M + e. As P_{M}^{R}(t) has positive coefficients, we get p_{M}(1) = 2^{e}P_{M}^{R}(1) > 0. If e = 0, then p_{M}(−1) = P_{M}^{R}(−1) is the Euler characteristic of M, which is non-negative. As −1 is not a root of p_{M}, this yields p_{M}(−1) > 0.

(1) By (7.3.1), we have s ≤ depth R−depth_{R} M and ∑_{i>s} b_{i}^{R}(M)t^{i} = bt^{s+1}(1−t)^{−1} ≠ 0.

(2) In view of (1.5) and (7.2.2), after performing a local flat extension of R we may assume that it has a deformation R ⟵ Q such that pd_{Q} M < ∞, and that there exists a degree −2 chain endomorphism μ of the minimal free resolution of M, which is surjective in degrees n > s. Thus, we can switch to the notation of the proof of (7.8), with q = 2.

We prove e ≤ d − 1 by induction on d. For d = 1 this is contained in (1), so let d ≥ 2. The exact sequence (7.8.2) shows that pd_{Q} M_{1} is finite, and then the exact sequence (7.8.1) yields pd_{Q} M(μ) < ∞, hence CI-dim_{R} M(μ) < ∞. Write the Poincaré series

(8.7.1)  P_{M(μ)}^{R}(t) = \frac{p_{M(μ)}(t)}{(1 − t)^{d′}(1 + t)^{e′}}

in irreducible form (8.5). The equalities (7.8.3) translate into an equality of power series P_{M}^{R}(t) = t^{2}P_{M}^{R}(t) + tP_{M(μ)}^{R}(t) + r(t), for some polynomial r ∈ ℤ[t]. Thus, we get

(8.7.2)  P_{M(μ)}^{R}(t) = \frac{p_{M}(t) − (1 − t)^{d−1}(1 + t)^{e−1}r(t)}{t(1 − t)^{d−1}(1 + t)^{e−1}} .

At t = 1 the numerator above is equal to p_{M}(1) ≠ 0, so no factor (1−t) cancels. Comparing (8.7.1) with (8.7.2), we see that d′ = cx_{R} M(μ) = d − 1, hence the induction hypothesis applies to M(μ). For e ≤ 1 the desired inequality e ≤ d − 1 is clear, so let e > 1. The numerator of (8.7.2) at t = 1 is then p_{M}(−1) ≠ 0; by another comparison with (8.7.1):

(8.7.3)  e = e′ + 1 ≤ d′ = d − 1 .

Note that u_{0} = p_{M}(1)/2^{e} and v_{0} = p_{M}(−1)/2^{d} in the prime fractions decomposition

\frac{p_{M}(t)}{(1 − t)^{d}(1 + t)^{e}} = ∑_{i=1}^{d} \frac{u_{d−i}}{(1 − t)^{i}} + ∑_{j=1}^{e} \frac{v_{e−j}}{(1 + t)^{j}} + g(t) .

Thus, there are h_{+} and h_{−} in ℚ[t], of degree at most d − 2 and e − 2, respectively, and

(8.7.4) b_{n}^{R}(M) = \frac{p_{M}(1)}{2^{e}} \binom{n + d − 1}{d − 1}+(−1)^{n} \frac{p_{M}(−1)}{2^{d}} \binom{n + e − 1}{e − 1}+h_{+}(n)+(−1)^{n}h_{−}(n)

for n ≫ 0. As d > e and the Betti sequence is positive, we see that p_{M}(1) > 0.

To finish the proof of (8.6) we assume that e = d−1 ≥ 1 and show by another induction on d that p_{M}(1) > |p_{M}(−1)|. If d = 2, then for n ≫ 0 in (8.7.4) h_{+}(n) = h, a constant, and h_{−}(n) = 0. The formula for the Betti numbers simplifies to

b_{n}^{R}(M) = \frac{p_{M}(1)}{2}n + (−1)^{n} \frac{p_{M}(−1)}{4} + h for n ≫ 0 .

By (7.3.2) the Betti sequence of M is eventually strictly increasing. This means that the function b_{n}^{R}(M) − b_{n−1}^{R}(M) = \frac{1}{2}p_{M}(1) + (−1)^{n} \frac{1}{2}p_{M}(−1) is positive for n ≫ 0. In other words, p_{M}(1) > |p_{M}(−1)|, hence we have a basis for our induction. To perform the induction step, choose d ≥ 3 and note from (8.7.3) that the module M(μ) has e′ = e − 1 = d − 2 = d′ − 1 ≥ 2, hence the induction assumption applies to it. Finally, note the equality

p_{M}(t) = tp_{M(μ)}(t) + (1 − t)^{d−1}(1 + t)^{e−1}r(t)

which results from (8.7.1), (8.7.2) and (8.7.3). It shows that p_{M}(1) = p_{M(μ)}(1) and p_{M}(−1) = −p_{M(μ)}(−1), hence |p_{M}(1)| > |p_{M}(−1)| by induction. □

Proof of Theorem (8.1). The polynomials b_{±} whose existence is asserted by the theorem are obtained by splitting (8.7.4) into two expressions, one for each parity of n. By (8.7.3) we have e ≤ d − 1. If the inequality is strict, then both polynomials b_{+} and b_{−} have the same coefficient of t^{d−2}, hence both difference polynomials have degree d − 2 and leading coefficient b/2^{e}(d − 2)! > 0. If e = d − 1, then the coefficients of t^{d−2} in the difference polynomials are (b + c_{±} − c_{∓})/2^{d−1}(d − 2)!. As the numerator of this fraction is equal to p_{M}(1) ± p_{M}(−1), it is positive by (8.6.2). □

9. Obstructions to embeddings into standard resolutions

Consider a local ring (R, 𝔪, k) with a deformation ρ: Q ⟶ R with kernel generated by a regular sequence x = x_{1}, …, x_{c}, and a finite R–module M. For the standard R–free resolution (3.10) of M, Eisenbud [18; p.37] makes the following

Conjecture. If M is a finite R–module M with pd_{Q} M < ∞, then its minimal R–free resolution is a subcomplex of a standard resolution in such a way that the maps t_{j} may be chosen to be induced by the standard χ_{j}. In particular, the t_{j} may be chosen to commute.

We produce obstructions to its validity, and show that any ring R with an embedded deformation of codimension 2 has modules with non-vanishing obstruction.

(9.1) Theorem. If there exist a family {t_{j}(Q, x, F)}_{1⩽j⩽c} of Eisenbud operators on the minimal R–free resolution F of M, a system σ of higher homotopies on E, and a comparison α: F ⟶ E{σ} of R–free resolutions of M such that αt_{j} = χ_{j}α for 1 ≤ j ≤ c, then the spectral sequence (4.4.1) stops at ^{2}E.

In particular, the canonical homomorphism

κ_{Mk}^{∗}: \frac{Ext_{R}^{∗}(M, k)}{(χ) Ext_{R}^{∗}(M, k)} ⟶ Ext_{Q}^{∗}(M, k)

from (4.3.2) is injective, hence Ext_{R}^{∗}(M, k) is generated over k[χ] in degrees ≤ pd_{Q} M.

The theorem suggests a more precise version of an earlier conjecture, cf. [6; (5.3.1)]:

(9.2) Conjecture. If F has any structure of DG R[χ]–module, then κ_{Mk}^{∗} is injective.

(9.3) Example. When s ≥ 0 and c ≥ 2 no embedding of a minimal resolution of the R–module M_{−s−1} = M_{−s−1}^{[c]} into a standard resolution coming from a codimension c deformation is compatible with the action of the Eisenbud operators.

Indeed, the direct sum decomposition of Ext_{R}^{∗}(M_{−s−1}, k) described in (7.5) is also one of graded k[χ]–modules, cf. (4.7), hence any system of generators contains elements of degree pd_{Q} M_{0} + s + 1. As pd_{Q} M_{−s−1} = pd_{Q} M_{0} = c by (3.1), we conclude by (9.1).

Proof of Theorem (9.1). The hypotheses imply that by setting χ_{j}f = t_{j}(Q, x, F)(f) for f ∈ F and 1 ≤ j ≤ c one obtains on F a structure of DG R[χ]–module such that α: F ⟶ E{σ} becomes a morphism of DG modules over R[χ]. This produces a morphism

Hom_{R}(α, k): Hom_{R}(E{σ}, k) ⟶ Hom_{R}(F, k)

of DG modules. Filtering the Koszul DG module K by wedge degree, we get a morphism

Hom_{R}(α, k) ⊗_{R[χ]} K: Hom_{R}(E{σ}, k) ⊗_{R[χ]} K ⟶ Hom_{R}(F, k) ⊗_{R[χ]} K . of filtered DG modules. The first page of the induced morphism of spectral sequences is the map K(χ; H^{∗} Hom_{R}(α, k)) of Koszul complexes. As α is a comparison of R–free resolutions, H^{∗} Hom_{R}(α, k) is bijective, so the sequences are isomorphic from the page ^{1}E onwards. Due to the minimality of the resolution F, the differential of Hom_{R}(F, k) is trivial, hence the second spectral sequence stops at ^{2}E.

This implies the first spectral sequence, which by construction is the one in (4.4.1) with N = k, also collapses at ^{2}E, and thus its vertical edge homomorphism ^{2}υ_{Mk}^{∗} is injective. By (4.4.3), this map is equal to the reduced change of rings homomorphism κ_{Mk}^{∗}. □

Next we prove that the cohomology over R of the high R–syzygies of each finite R– module M completely determines its cohomology over Q. Thus, the obstructions of (9.1) vanish for such syzygies, leaving open the asymptotic form of Eisenbud’s conjecture stated in [18; p.37] as follows: “In the spirit of this paper, it would be interesting to prove this conjecture just for some truncation of the minimal free resolution.”

(9.4) Theorem. Let χ be a system of cohomology operators defined by a codimension c deformation ρ: Q ⟶ R, and let M be a finite R–module such that pd_{Q} M is finite. When n ≫ 0 the spectral sequence (4.4.1) for the modules M_{n} = Syz_{n}^{R}(M) and k stops at the page ^{2}E, and produces for each p ∈ ℤ a natural exact sequence

0 ⟶ H_{p−1}(χ; Ext_{R}^{∗}(M_{n}, k))^{2p−1} ⟶ Ext_{Q}^{p}(M_{n}, k) ⟶^{δ^{p}} H_{p}(χ; Ext_{R}^{∗}(M_{n}, k))^{2p} ⟶ 0

of k–linear homomorphisms, in which δ^{p} is the edge homomorphism from (4.4.4).

Proof. The graded module Ext_{R}^{∗}(M_{n}, k) = Σ^{n} Ext_{R}^{⩾n}(M, k) over the polynomial ring k[χ] splits as a direct sum of its even and odd parts. For n ≫ 0 the minimal resolution of the even (respectively, odd) submodules has its p’th free module generated purely in degree −2p (respectively, −2p−1): By simple regrading this can be deduced from [19; (1.1)], which shows that if M is a finite graded module over a polynomial ring generated by elements of upper degree 1 then for n ≫ 0 the module M^{⩾n} has a (finite) linear resolution. As

Tor_{p}^{k[χ]}(k, Ext_{R}^{∗}(M_{n}, k))^{q} ≅ H_{p}(χ; Ext_{R}^{∗}(M_{n}, k))^{q} ,

we see that the sequence (4.4.1) for M_{n} and k has ^{2}E_{p}^{q} = 0, unless −1 ≤ 2p + q ≤ 0.

This leaves no space for non-trivial differentials, hence the sequence stops at ^{2}E and produces a two-tier filtration of the limit term Ext_{Q}^{q−p}(M_{n}, k). The desired short exact sequences represent precisely this information. □

References

1. Alperin, J., Evens, L., Representations, resolutions, and Quillen’s dimension theorem, J. Pure Appl. Algebra 22 (1981), 1–9.

2. Anick, D., A counterexample to a conjecture of Serre, Ann. of Math. 115 (1982), 1–33.

3. André, M., Hopf algebras with divided powers, J. Algebra 18 (1971), 19–50.

4. Assmus, E. F., Jr., On the homology of local rings, Ill. J. Math. 3 (1959), 187–199.

5. Auslander, M., Bridger, M., Stable module theory, Mem. Amer. Math. Soc. 94 (1969).

6. Avramov, L. L., Obstructions to the existence of multiplicative structures on minimal free resolutions, Amer. J. Math. 103 (1981), 1–31.

7. Avramov, L. L., Local algebra and rational homotopy, Homotopie algébrique et algèbre locale. (J.-M. Lemaire, J.-C. Thomas, eds.), Astérisque, vol. 113-114, Soc. Math. France, Paris, 1984, pp. 15–43.

8. Avramov, L. L., Modules of finite virtual projective dimension, Invent. math. 96 (1989), 71–101.

9. Avramov, L. L., Homological asymptotics of modules over local rings, Commutative algebra. (M. Hochster, C. Huneke, J. Sally, eds.), MSRI Publ., vol. 15, Springer, New York, 1989, pp. 33–62.

10. Avramov, L. L., Local rings over which all modules have rational Poincaré series, J. Pure Appl. Algebra 91 (1994), 29–48.

11. Avramov, L. L., Kustin A. R., Miller, M., Poincaré series of modules over local rings of small embedding codepth or small linking number, J. Algebra 118 (1988), 162–204.

12. Avramov, L. L., Sun, L.-C., Cohomology operators defined by a deformation, Preprint (1995).

13. Benson, D. J., Carlson, J. F., Projective resolutions and Poincaré duality complexes, Trans. Amer. Math. Soc. 342 (1994), 447–488.

14. Bourbaki, N., Algèbre. III., Nouvelle édition, Hermann, Paris, 1970.

15. Bourbaki, N., Algèbre commutative. IX., Masson, Paris, 1983.

16. Buchweitz, R.-O., Greuel, G.-M., Schreyer, F., Cohen–Macaulay modules on hypersurface singularities. II, Invent. math. 88 (1987), 165–182.

17. Eagon, J. A., Hochster, M., R–sequences and indeterminates, Quart. J. Math. Oxford Ser. (2) 25 (1974), 61–71.

18. Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.

19. Eisenbud, D., Goto, S., Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89–133.

20. Félix, Y., Halperin, S., Jacobsson, C., Löfwall, C. Thomas, J.-C., The radical of the homotopy Lie algebra, Amer. J. Math. 110 (1988), 301–322.

21. Gasharov, V. N., Peeva, I. V., Boundedness versus periodicity over commutative local rings, Trans. Amer. Math. Soc. 320 (1990), 569–580.

22. Grothendieck, A., Éléments de géométrie algébrique. IV_{2}, Publ. Math. IHES 24 (1965).

23. Gulliksen, T. H., A change of rings theorem, with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167–183.

24. Gulliksen, T. H., On the deviations of a local ring, Math. Scand. 47 (1980), 5–20.

25. Herzog, J., Ulrich, B., Backelin, J., Linear maximal Cohen–Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 (1991), 187–202.

26. Kustin, A. R., Palmer, S. M., The Poincaré series of every finitely generated module over a codimension 4 almost complete intersection is a rational function, J. Pure Appl. Algebra 95 (1994), 271–295.

27. MacLane, S., Homology, Grundlehren Math. Wiss., vol. 114, Springer, Berlin, 1963.

28. Manin, Yu. I., Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier (Grenoble) 37 (1987), 191–205.

29. Matsumura, H., Commutative ring theory, Stud. Adv. Math., vol. 8, Univ. Press, Cambridge, 1986.

30. Mehta, V. B., Endomorphisms of complexes and modules over Golod rings, Ph. D. Thesis, Univ. of California, Berkeley, 1976.

31. Milnor, J. W., Moore, J. C., On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264.

32. Nagata, M., Local rings, Wiley, New York, 1962.

33. Shamash, J., The Poincaré series of a local ring, J. Algebra 12 (1969), 453–470.

34. Sjödin, G., Hopf algebras and derivations, J. Algebra 64 (1980), 218–229.

35. Sun, L.-C., Growth of Betti numbers over local rings of small embedding codepth or small linking number, J. Pure Appl. Algebra 96 (1994), 57–71.

COMPLETE INTERSECTION DIMENSION 41

36. Tate, J., Homology of Noetherian rings and of local rings, Ill. J. Math. 1 (1957), 14–27. Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 E-mail: avramov@math.purdue.edu Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254 E-mail: gasharov@binah.cc.brandeis.edu Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254 E-mail: peeva@binah.cc.brandeis.edu