MODULES WITH REDUCIBLE COMPLEXITY

PETTER ANDREAS BERGH

Abstract. For a commutative Noetherian local ring we define and study the class of modules having reducible complexity, a class containing all modules of finite complete intersection dimension. Various properties of this class of modules are given, together with results on the vanishing of homology and cohomology.

1. Introduction

The complexity of a finitely generated module over a commutative Noetherian local ring is a measure on a polynomial scale of the growth of the ranks of the free modules in its minimal free resolution. Over a local complete intersection every finitely generated module has finite complexity. In fact, it follows from [Gul, Theorem 2.3] that this property characterizes a local complete intersection, and that it is equivalent to the complexity of the residue field being finite.

In [AGP] a certain finiteness condition was defined under which a module behaves homologically like a module over a complete intersection. Namely, a module M over a commutative Noetherian local ring A has finite complete intersection dimension if there exist local rings R and Q and a diagram A → R ↞ Q of local homomorphisms (called a quasi-deformation of A) such that A → R is faithfully flat, R ↞ Q is surjective with kernel generated by a regular sequence, and pd_{Q}(R ⊗_{A} M) is finite. There is of course a reason behind the choice of terminology; it was shown in [AGP] that a local ring is a complete intersection if and only if all its finitely generated modules have finite complete intersection dimension, and that this is equivalent to the finiteness of the complete intersection dimension of the residue field. Moreover, it was shown that if the projective dimension of a module is finite, then it is equal to the complete intersection dimension of the module.

We shall study a class of modules whose complexity is “reducible” (defined in the next section), a class which contains all modules of finite complete intersection dimension. In particular, we investigate for this class of modules the vanishing of homology and cohomology, and generalize several of the results in [ArY], [ChI],

[Jo1] and [Jo2].

2. reducible complexity

Throughout this paper we let (A, 𝔪, k) be a commutative Noetherian local ring, and we suppose all modules are finitely generated. We fix a finitely generated nonzero A-module M with minimal free resolution

F_{M}: ⋯ → F_{2} → F_{1} → F_{0} → M → 0.

The rank of F_{n}, i.e. the integer dim_{k} Ext_{A}^{n}(M, k), is the n’th Betti number of M, and we denote this by β_{n}(M). The complexity of M, denoted cx M, is defined as

cx M = inf{t ∈ ℕ_{0} | ∃a ∈ ℝ such that β_{n}(M) ≤ an^{t−1} for n ≫ 0},

where ℕ_{0} = ℕ∪{0}. In general the complexity of a module may be infinite, whereas it is zero if and only if the module has finite projective dimension.

We now give the main definition, which is inductive and defines the class of modules having reducible complexity as a subcategory of the category of (finitely generated) A-modules having finite complexity. However, before stating the definition, recall that the n’th syzygy of an A-module X with minimal free resolution

⋯ → P_{2} → P_{1} → P_{0} → X → 0

is the cokernel of P_{n+1} → P_{n} and denoted by Ω_{A}^{n}(X) (note that Ω_{A}^{0}(X) = X), and it is unique up to isomorphism. For an A-module Y and a homogeneous element η ∈ Ext_{A}^{∗}(X, Y), choose a map f_{η}: Ω_{A}^{|η|}(X) → Y representing η, and denote by K_{η} the pushout of this map and the inclusion Ω_{A}^{|η|}(X) ↪ P_{|η|−1}. As parallel maps in a pushout diagram have isomorphic cokernels, we obtain an exact sequence

0 → Y → K_{η} → Ω_{A}^{|η|−1}(X) → 0.

Note that the middle term K_{η} is independent (up to isomorphism) of the map f_{η} chosen to represent η.

Definition 2.1. Let C_{A} denote the category of all A-modules having finite complexity. The subcategory C_{A}^{r} ⊆ C_{A} of modules having reducible complexity is defined inductively as follows:

(i) Every module of finite projective dimension belongs to C_{A}^{r}.

(ii) A module X ∈ C_{A} with cx X > 0 belongs to C_{A}^{r} if there exists a homogeneous element η ∈ Ext_{A}^{∗}(X, X) of positive degree such that cx K_{η} < cx X, depth K_{η} = depth X and K_{η} ∈ C_{A}^{r}. We say that the element η reduces the complexity of M.

Remark. The condition depth K_{η} = depth X is not very strong; suppose for example that depth Ω_{A}^{i}(X) ≤ depth A for all i (this always happens when A is Cohen-Macaulay). Then we must have depth X ≤ depth Ω_{A}^{|η|−1}(X), implying the equality depth K_{η} = depth X.

Note the trivial fact that if every A-module has reducible complexity, then A must be a complete intersection since then by definition every module has finite complexity. The following result shows that the converse is also true, in fact every module of finite complete intersection dimension has reducible complexity. Moreover, if A is Cohen-Macaulay and M has reducible complexity, then so does any syzygy of M.

Proposition 2.2. (i) Every module of finite complete intersection dimension

has reducible complexity. (ii) If A is Cohen-Macaulay and M has reducible complexity, then so does the kernel of any surjective map F ↠ M when F is free. In particular, any syzygy of M has reducible complexity.

(iii) if B is a local ring and A → B a faithfully flat local homomorphism, then if M has reducible complexity, so does the B-module B ⊗_{A} M.

Proof. (i) If M has finite complete intersection dimension, then from [AGP, Proposition 5.2] it follows that the complexity of M is finite. We argue by induction on cx M that M has reducible complexity, the case cx M = 0 following from the definition. Suppose therefore that the complexity of M is nonzero. By [AGP, Proposition 7.2] there exists an eventually surjective chain map of degree −n (where n > 0) on the minimal free resolution of M. This chain map corresponds to an element η of Ext_{A}^{n}(M, M), giving the exact commutative diagram

[figure: 0; Ω_{A}^{n}(M); f_{η}; F_{n−1}; Ω_{A}^{n−1}(M); 0; 0; M; K_{η}; Ω_{A}^{n−1}(M); 0]

of A-modules.

Now consider the exact sequence involving K_{η}. Since M and Ω_{A}^{n−1}(M) both have finite complete intersection dimension, then so does K_{η}. The exact sequence gives rise to a long exact sequence

⋯ → Ext_{A}^{i}(K_{η}, k) → Ext_{A}^{i}(M, k) ⟶^{∂_{η}} Ext_{A}^{i+n}(M, k) → Ext_{A}^{i+1}(K_{η}, k) → ⋯,

where Ext_{A}^{i}(Ω_{A}^{n−1}(M), k) has been identified with Ext_{A}^{i+n−1}(M, k). It follows from [Mac, Theorem III.9.1] that the connecting homomorphism ∂_{η}: Ext_{A}^{i}(M, k) → Ext_{A}^{i+n}(M, k) is then scalar multiplication by (−1)^{i}η (we think of Ext_{A}^{∗}(M, k) as a graded right module over the graded ring Ext_{A}^{∗}(M, M)), and reversing the arguments in the proof of [Ber, Proposition 2.2] shows that this is an injective map for i ≫ 0. Consequently the equality β_{i+1}(K_{η}) = β_{i+n}(M) − β_{i}(M) holds for large values of i. By [AGP, Theorem 5.3] the complexities cx M and cx K_{η} equal the orders of the poles at t = 1 of the Poincaré series ∑ β_{n}(M)t^{n} and ∑ β_{n}(K_{η})t^{n}, respectively, thus cx K_{η} = cx M − 1.

It remains only to show that depth K_{η} = depth M, but this is easy; from [AGP, Theorem 1.4] we have 0 ≤ CI-dim Ω_{A}^{i}(M) = depth A−depth Ω_{A}^{i}(M) for each i ≥ 0, and this implies the inequality depth Ω_{A}^{i}(M) ≤ depth Ω_{A}^{i+1}(M). In particular we have depth M ≤ depth Ω_{A}^{n−1}(M), and therefore depth K_{η} must equal depth M.

(ii) Let L denote the kernel of the surjective map F ↠ M. Again we argue by induction on cx M. If the projective dimension of M is finite, then so is the projective dimension of L, and we are done. Suppose therefore cx M is nonzero, and let η ∈ Ext_{A}^{∗}(M, M) be an element reducing the complexity of M. By the Horseshoe Lemma we have an exact commutative diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & L \ar[d] \ar[r] & K ⊕ Q \ar[d] \ar[r] & Ω_{A}^{|η|−1}(L) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & F \ar[d] \ar[r] & F′ ⊕ Q \ar[d] \ar[r] & F′′ ⊕ Q \ar[d] \ar[r] & 0 \\ 0 \ar[r] & M \ar[d] \ar[r] & K_{η} \ar[d] \ar[r] & Ω_{A}^{|η|−1}(M) \ar[d] \ar[r] & 0 \\  & 0 & 0 & 0}

where F′ and F′′ are free modules, and Q is a free module such that Ω_{A}^{|η|}(M)⊕Q ≃ Ω_{A}^{|η|−1}(L). If M and K_{η} are maximal Cohen-Macaulay, then so are L and K ⊕ Q, and if not then depth L = depth M +1 and depth(K ⊕Q) = depth K_{η} +1. In either case we see that depth L equals depth(K ⊕ Q). Moreover, we have cx(K ⊕ Q) = cx K_{η} < cx M = cx L, and so by induction we are done.

Note that the upper horizontal short exact sequence in the above diagram corresponds to an element θ in Ext_{A}^{|η|}(L, L) reducing the complexity of L. A map f_{θ}: Ω_{A}^{|η|}(L) → L representing θ is obtained by lifting a map f_{η} representing η along the minimal free resolution of M, as in the diagram

\xymatrix{0 \ar[r] & Ω_{A}^{|η|+1}(M) \ar@{.>}[d]^{Ω_{A}(f_{η})} \ar[r] & F_{|η|} \ar@{.>}[d] \ar[r] & Ω_{A}^{|η|}(M) \ar[d]^{f_{η}} \ar[r] & 0 \\ 0 \ar[r] & L \ar[r] & F \ar[r] & M \ar[r] & 0}

In this way we obtain a map Ω_{A}(f_{η}): Ω_{A}^{|η|+1}(M) → L, and adding to Ω_{A}^{|η|+1}(M) a free module Q′ such that Ω_{A}^{|η|+1}(M) ⊕ Q′ ≃ Ω_{A}^{|η|}(L), we obtain a map Ω_{A}^{|η|}(L) → L representing θ.

(iii) If X is any A-module with a minimal free resolution F_{X}, then the complex B ⊗_{A} F_{X} is a minimal free B-resolution of B ⊗_{A} X, giving the equality cx_{A} X = cx_{B}(B ⊗_{A} X). Moreover, if Y is another A-module then from [Mat, Theorem 23.3] we have depth_{A} X − depth_{A} Y = depth_{B}(B ⊗_{A} X) − depth_{B}(B ⊗_{A} Y). Hence if η ∈ Ext_{A}^{∗}(M, M) reduces the complexity of M, the element B ⊗_{A} η ∈ Ext_{B}^{∗}(B ⊗_{A} M, B ⊗_{A} M) reduces the complexity of B ⊗_{A} M. □

Thus the class of modules having finite complete intersection dimension is contained in the class of modules having reducible complexity. However, the following example shows that the inclusion is strict in general; there are a lot of modules having reducible complexity but whose complete intersection dimension is infinite.

Example. Suppose M is periodic of period p ≥ 3 (i.e. Ω_{A}^{p}(M) is isomorphic to M), and that depth M ≤ depth Ω_{A}^{p−1}(M). Then we have an exact sequence

0 → M → F_{p−1} → Ω_{A}^{p−1}(M) → 0,

and we have 0 = cx F_{p−1} = cx M − 1 and depth F_{p−1} = depth M. Therefore M has reducible complexity, and it cannot be of finite complete intersection dimension since then by [AGP, Theorem 7.3] the period would have been two.

An example of such a module was given in [GaP, Section3]; let (A, 𝔪, k) be the commutative local finite dimensional k-algebra k[x_{1}, x_{2}, x_{3}, x_{4}]/ 𝔞, where 𝔞 is the ideal generated by the quadratic forms

x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}, x_{3}x_{4}, x_{1}x_{4} + x_{2}x_{4}, αx_{1}x_{3} + x_{2}x_{3}

for a nonzero element α ∈ k. The complex

⋯ → A^{2} ⟶^{d_{2}} A^{2} ⟶^{d_{1}} A^{2} ⟶^{d_{0}} M → 0,

where the maps are given by the matrices

d_{n} = ( \begin{matrix}x_{1} & α^{n}x_{3} + x_{4} \\ 0 & x_{2} \end{matrix} ) ,

is a minimal free resolution of the module M := Im d_{0}, and so if α has finite order p we see that M is periodic of period p.

It is worth mentioning that there do exist examples of modules over Gorenstein rings whose complete intersection dimension is not finite but which have reducible complexity (see [GaP, Proposition 3.1] for an example similar to that above).

Now let X and Y be arbitrary A-modules, and θ_{1} ∈ Ext_{A}^{∗}(X, X) and θ_{2} ∈ Ext_{A}^{∗}(X, Y) two homogeneous elements. The following lemma, motivated by [EHSST, Lemma 7.2], links K_{θ_{1}} and K_{θ_{2}} to K_{θ_{2}θ_{1}}, and will be a key ingredient in several of the forthcoming results.

Lemma 2.3. If θ_{1} ∈ Ext_{A}^{∗}(X, X) and θ_{2} ∈ Ext_{A}^{∗}(X, Y) are two homogeneous elements, then there exists an exact sequence

0 → Ω_{A}^{|θ_{2}|}(K_{θ_{1}}) → K_{θ_{2}θ_{1}} ⊕ F → K_{θ_{2}} → 0

of A-modules, where F is free.

Proof. Denote |θ_{i}| by n_{i} for i = 1, 2. The element θ_{1} gives rise to an exact commutative diagram

[figure: 0; Ω_{A}^{n_{1}}(X); f_{θ_{1}}; i; Q_{n_{1}−1}; h; π; Ω_{A}^{n_{1}−1}(X); 0; 0; X; j; K_{θ_{1}}; Ω_{A}^{n_{1}−1}(X); 0]

where Q_{n} denotes the n’th module in the minimal free resolution of X. Letting Q ⟶^{g} X be a surjection, where Q is free, we can modify the diagram and obtain

[figure: 0; 0; Ker(h, jg); Ker(h, jg); 0; Ω_{A}^{n_{1}}(X) ⊕ Q; (f_{θ_{1}} ,g); _{( \begin{matrix}i & 0 \\ 0 & 1 \end{matrix} )}; Q_{n_{1}−1} ⊕ Q; (h,jg); (π,0); Ω_{A}^{n_{1}−1}(X); 0; 0; X; j; K_{θ_{1}}; Ω_{A}^{n_{1}−1}(X); 0; 0; 0]

Since Ker(h, jg) is isomorphic to Ω_{A}^{1}(K_{θ_{1}}) ⊕ Q′ for some free module Q′, the left vertical exact sequence yields an exact sequence

0 → Ω_{A}^{1}(K_{θ_{1}}) ⊕ Q′ → Ω_{A}^{n_{1}}(X) ⊕ Q ⟶^{(f_{θ_{1}} ,g)} X → 0

on which we can apply the Horseshoe Lemma and obtain an exact sequence

μ: 0 → Ω_{A}^{n_{2}}(K_{θ_{1}}) → Ω_{A}^{n_{1}+n_{2}−1}(X) ⊕ F ⟶^{(Ω_{A}^{n_{2}−1}(f_{θ_{1}} ),s)} Ω_{A}^{n_{2}−1}(X) → 0,

where F is free and F ⟶^{s} Ω_{A}^{n_{2}−1}(X) is a map.

The definition of cohomological products and the pushout properties of K_{θ_{2}θ_{1}} give a commutative diagram

[figure: 0; Y; K_{θ_{2}θ_{1}}; w; t; Ω_{A}^{n_{1}+n_{2}−1}(X); Ω_{A}^{n_{2}−1}(f_{θ_{1}} ); 0; 0; Y; K_{θ_{2}}; v; Ω_{A}^{n_{2}−1}(X); 0]

with exact rows. Adding F to K_{θ_{2}θ_{1}} and Ω_{A}^{n_{1}+n_{2}−1}(X) in the right-most square, we obtain the exact commutative diagram

[figure: 0; 0; Ω_{A}^{n_{2}}(K_{θ_{1}}); Ω_{A}^{n_{2}}(K_{θ_{1}}); 0; Y; K_{θ_{2}θ_{1}} ⊕ F; _{( \begin{matrix}w & 0 \\ 0 & 1 \end{matrix} )}; (t,r); Ω_{A}^{n_{1}+n_{2}−1}(X) ⊕ F; (Ω_{A}^{n_{2}−1}(f_{θ_{1}} ),s); 0; 0; Y; K_{θ_{2}}; v; Ω_{A}^{n_{2}−1}(X); 0; 0; 0]

in which the right vertical exact sequence is μ and F ⟶^{r} K_{θ_{2}} is a map with the property that s = vr. The left vertical exact sequence is of the form we are seeking. □

We end this section with two results on modules over complete intersections. Recall that a maximal Cohen-Macaulay (or “MCM” from now on) approximation of an A-module X is an exact sequence

0 → Y_{X} → C_{X} → X → 0

where C_{X} is MCM and Y_{X} has finite injective dimension, and that a hull of finite injective dimension of X is an exact sequence

0 → X → Y^{X} → C^{X} → 0

where C^{X} is MCM and Y^{X} has finite injective dimension. These notions were introduced in [AuB], where it was shown that every finitely generated module over a commutative Noetherian ring admitting a dualizing module has a MCM approximation and a hull of finite injective dimension. The following result provides a simple proof of the complete intersection case, using a technique similar to the proof of the main result in [Bak] and the fact that over a complete intersection every module has reducible complexity.

Proposition 2.4. Suppose A is a complete intersection.

(i) If η ∈ Ext_{A}^{|η|}(M, M) reduces the complexity of M, then so does η^{t} for t ≥ 1. (ii) Every A-module has a MCM approximation and a hull of finite injective

dimension.

Proof. (i) Using Lemma 2.3 it is easily proved by induction on t that cx K_{η^{t}} ≤ cx K_{η}.

(ii) Fix an exact sequence

0 → Y → C → M → 0

where C is MCM (the minimal free cover of M, for example). If the complexity of Y is nonzero, let η ∈ Ext_{A}^{|η|}(Y, Y) be an element reducing it, and choose an integer t ≥ 1 with the property that Ω_{A}^{t|η|−1}(Y) is MCM. The element η^{t} is given by the exact sequence

0 → Y → K_{η^{t}} → Ω_{A}^{t|η|−1}(Y) → 0,

and by (i) it also reduces the complexity of Y. From the pushout diagram

[figure: 0; 0; 0; Y; K_{η^{t}}; Ω_{A}^{t|η|−1}(Y); 0; 0; C; C′; Ω_{A}^{t|η|−1}(Y); 0; M; M; 0; 0]

we obtain the exact sequence

0 → Y′ → C′ → M → 0

(with Y′ = K_{η^{t}}), where C′ is MCM and cx Y′ < cx Y. Repeating the process we eventually obtain a MCM approximation of M, since over a Gorenstein ring a module has finite injective dimension precisely when it has finite projective dimension.

As for a hull of finite injective dimension, fix an exact sequence

0 → M → Y → C → 0

where C is MCM (obtained for example from a suitable power of an element in Ext_{A}^{∗}(M, M) reducing the complexity of M). If the complexity of Y is nonzero, choose as above an element η ∈ Ext_{A}^{|η|}(Y, Y) reducing it, and let t ≥ 1 be an integer such that Ω_{A}^{t|η|−1}(Y) is MCM. From the pushout diagram

[figure: 0; 0; M; M; 0; Y; K_{η^{t}}; Ω_{A}^{t|η|−1}(Y); 0; 0; C; C′; Ω_{A}^{t|η|−1}(Y); 0; 0; 0]

we obtain the exact sequence

0 → M → Y′ → C′ → 0

(with Y′ = K_{η^{t}}), where C′ is MCM and cx Y′ < cx Y. Repeating the process we eventually obtain a hull of finite injective dimension of M. □

3. Vanishing results

This section investigates the vanishing of cohomology and homology for a module having reducible complexity, and the following is assumed throughout:

Assumption. The module M has reducible complexity, and N is a nonzero Amodule. If cx M > 0, then there exist A-modules K_{1}, …, K_{c} and a set of cohomological elements {η_{i} ∈ Ext_{A}^{|η_{i}|}(K_{i−1}, K_{i−1})}_{i=1}^{c} given by exact sequences

0 → K_{i−1} → K_{i} → Ω_{A}^{|η_{i}|−1}(K_{i−1}) → 0

for i = 1, …, c (where K_{0} = M), satisfying depth K_{i} = depth M, cx K_{i} < cx K_{i−1} and cx K_{c} = 0 (such elements η_{i} must exist by Definition 2.1).

For an A-module N, we define q^{A}(M, N) and p^{A}(M, N) by

q^{A}(M, N) = sup{n | Tor_{n}^{A}(M, N) ≠ 0},
p^{A}(M, N) = sup{n | Ext_{A}^{n}(M, N) ≠ 0}.

The definition of modules having reducible complexity suggests that when proving results about q^{A}(M, N) and p^{A}(M, N), we use induction on the complexity of M.

The first result and its corollary (which considers a conjecture of Auslander and Reiten) consider the vanishing of cohomology, and generalize [ArY, Theorem 4.2 and Theorem 4.3].

Theorem 3.1. The following are equivalent.

(i) There exists an integer t > depth A − depth M such that Ext_{A}^{t+i}(M, N) = 0 for 0 ≤ i ≤ |η_{1}| + ⋯ + |η_{c}| − c.

(ii) p^{A}(M, N) < ∞.

(iii) p^{A}(M, N) = depth A − depth M.

Proof. We only need to show the implication (i) ⇒ (iii), and we do this by induction on cx M. If the projective dimension of M is finite, then by the Auslander-Buchsbaum formula it is equal to depth A−depth M. Since N is finitely generated, we have N ≠ 𝔪 N by Nakayama’s Lemma, hence there exists a nonzero element x ∈ N \ 𝔪 N. The map A → N defined by 1 ↦ x then gives rise to a nonzero element of Ext_{A}^{pd M}(M, N), and therefore p^{A}(M, N) = depth A − depth M.

Now suppose the complexity of M is nonzero, and consider the exact sequence

(†)  0 → M → K_{1} → Ω_{A}^{|η_{1}|−1}(M) → 0.

The vanishing interval for Ext_{A}^{i}(M, N) implies that Ext_{A}^{t+i}(K_{1}, N) = 0 for 0 ≤ i ≤ |η_{2}| + ⋯ + |η_{c}| − (c − 1), and so by induction p^{A}(K_{1}, N) = depth A − depth K_{1}. Since we have equalities p^{A}(M, N) = p^{A}(K_{1}, N) and depth M = depth K_{1}, we are done. □

Corollary 3.2. p^{A}(M, M) = pd M.

Proof. Suppose p^{A}(M, M) < ∞. If the projective dimension of M is not finite, i.e. if cx M > 0, then consider the exact sequence (†) representing η_{1} (from the proof of Theorem 3.1), where K_{1} = K_{η_{1}}. Since η_{1} is nilpotent there is an integer t such that η_{1}^{t} = 0, and therefore cx K_{η_{1}^{t}} = cx M. But using Lemma 2.3 we see that cx K_{η_{1}^{i}} ≤ cx K_{η_{1}} for all i ≥ 1, and since cx K_{η_{1}} < cx M we have reached a contradiction. Therefore the projective dimension of M is finite and equal to depth A − depth M by the Auslander-Buchsbaum formula, and from Theorem 3.1 we see that p^{A}(M, M) = pd M. □

The next result is a homology version of Theorem 3.1, and it is closely related to [Jo1, Theorem 2.1].

Theorem 3.3. The following are equivalent.

(i) There exists an integer t > depth A − depth M such that Tor_{t+i}^{A}(M, N) = 0 for 0 ≤ i ≤ |η_{1}| + ⋯ + |η_{c}| − c.

(ii) q^{A}(M, N) < ∞.

(iii) depth A − depth M − depth N ≤ q^{A}(M, N) ≤ depth A − depth M.

Proof. We only need to show the implication (i) ⇒ (iii), and we do this by induction on cx M. The case pd M < ∞ follows from the Auslander-Buchsbaum formula and [ChI, Remark 8], so suppose therefore cx M > 0, and consider the exact sequence (†) from the proof of Theorem 3.1. Since Tor_{t+i}^{A}(K_{1}, N) = 0 for 0 ≤ i ≤ |η_{2}| + ⋯ + |η_{c}| − (c − 1), we get by induction that the inequalities hold for K_{1} and N. But as in the previous proof we have q^{A}(M, N) = q^{A}(K_{1}, N) and depth M = depth K_{1}, hence the inequalities hold for M and N. □

The following result contains half of [ChI, Theorem 3] (and a version of the first half of [ArY, Theorem 2.5]) and the main result from [Jo2] for Cohen-Macaulay rings, which says that the integer q^{A}(M, N) can be computed locally. It also establishes the depth formula provided N is maximal Cohen-Macaulay.

Theorem 3.4. (i) If q^{A}(M, N) is finite and depth Tor_{q^{A}(M,N)}^{A}(M, N) = 0,

then

q^{A}(M, N) = depth A − depth M − depth N.

(ii) If A is Cohen-Macaulay and q^{A}(M, N) is finite, then the equality

q^{A}(M, N) = sup{ht 𝔭 − depth M_{𝔭} − depth N_{𝔭} | 𝔭 ∈ Spec A}

holds. (iii) If q^{A}(M, N) = 0 and N is maximal Cohen-Macaulay, then the depth for-

mula holds for M and N, i.e.

depth M + depth N = depth A + depth(M ⊗ N).

Proof. (i) We argue by induction on cx M, the case pd M < ∞ following from [Aus, Theorem 1.2] and the Auslander-Buchsbaum formula. Suppose therefore that the complexity of M is nonzero, and consider the exact sequence (†) from the proof of Theorem 3.1. Since Tor_{q^{A}(M,N)}^{A}(M, N) is a submodule of Tor_{q^{A}(M,N)}^{A}(K_{1}, N), the latter is also of depth zero, hence by induction and the equalities q^{A}(K_{1}, N) = q^{A}(M, N), depth K_{1} = depth M we are done.

(ii) Suppose q^{A}(M, N) is finite, and let 𝔭 ⊆ A be a prime ideal. If M has finite projective dimension, so has M_{𝔭}, and from Theorem 3.3 we get q^{A_{𝔭}}(M_{𝔭}, N_{𝔭}) ≥ ht 𝔭 − depth M_{𝔭} − depth N_{𝔭}. If cx M > 0, consider the exact sequences

0 → K_{i−1} → K_{i} → Ω_{A}^{|η_{i}|−1}(K_{i−1}) → 0

for i = 1, …, c (where K_{0} = M), satisfying depth K_{i} = depth M, cx K_{i} < cx K_{i−1} and cx K_{c} = 0. Localizing at 𝔭, we see that depth(K_{i})_{𝔭} = depth M_{𝔭} (as in the remark following Definition 2.1), and that q^{A_{𝔭}}((K_{i})_{𝔭}, N_{𝔭}) = q^{A_{𝔭}}((K_{i−1})_{𝔭}, N_{𝔭}). As K_{c} has finite projective dimension we get

q^{A_{𝔭}}(M_{𝔭}, N_{𝔭}) = q^{A_{𝔭}}((K_{c})_{𝔭}, N_{𝔭})
≥ ht 𝔭 − depth(K_{c})_{𝔭} − depth N_{𝔭}
= ht 𝔭 − depth M_{𝔭} − depth N_{𝔭},

hence since q^{A}(M, N) ≥ q^{A_{𝔭}}(M_{𝔭}, N_{𝔭}) the inequality

q^{A}(M, N) ≥ sup{ht 𝔭 − depth M_{𝔭} − depth N_{𝔭} | 𝔭 ∈ Spec A}

holds.

For the reverse inequality, choose any associated prime 𝔭 of Tor_{q^{A}(M,N)}^{A}(M, N). Then q^{A}(M, N) = q^{A_{𝔭}}(M_{𝔭}, N_{𝔭}) and depth Tor_{q^{A𝔭} (M_{𝔭},N_{𝔭})}^{A_{𝔭}}(M_{𝔭}, N_{𝔭}) = 0, and a small adjustment of the proof of (i) above gives

q^{A_{𝔭}}(M_{𝔭}, N_{𝔭}) = ht 𝔭 − depth M_{𝔭} − depth N_{𝔭}.

(iii) Again we argue by induction on cx M, where the case pd M < ∞ follows from [Aus, Theorem 1.2]. Suppose cx M > 0, and consider the exact sequence (†). We have q^{A}(K_{1}, N) = 0, hence by induction the depth formula holds for K_{1} and N. Since depth K_{1} = depth M, we only have to prove that the equality depth(M ⊗ N) = depth(K_{1} ⊗ N) holds. For each i ≥ 0 we have an exact sequence

0 → Ω_{A}^{i+1}(M) ⊗ N → F_{i} ⊗ N → Ω_{A}^{i}(M) ⊗ N → 0,

and as N is maximal Cohen-Macaulay we must have that depth (Ω_{A}^{i}(M) ⊗ N) is at most depth (Ω_{A}^{i+1}(M) ⊗ N). In particular the inequality depth(M ⊗ N) ≤

depth (Ω_{A}^{|η_{1}|−1}(M) ⊗ N) holds, and therefore when tensoring the sequence (†) with N we see that depth(M ⊗ N) = depth(K_{1} ⊗ N). □

The next result deals with symmetry in the vanishing of Ext. It was shown in [AvB] that if X and Y are modules over a complete intersection A, then Ext_{A}^{i}(X, Y) = 0 for i ≫ 0 if and only if Ext_{A}^{i}(Y, X) = 0 for i ≫ 0. This was generalized in [HuJ] to a class of local Gorenstein rings named “AB rings”, a class properly containing the class of complete intersections. Another generalization appeared in [Jør], where techniques from the theory of derived categories were used to show that symmetry in the vanishing of Ext holds for modules of finite complete intersection dimension over local Gorenstein rings.

Theorem 3.5. If A is Gorenstein then the implication

p^{A}(N, M) < ∞ ⇒ p^{A}(M, N) < ∞

holds. In particular, symmetry in the vanishing of Ext holds for modules with reducible complexity over a local Gorenstein ring.

Proof. Define the integer w (depending on M) by

w = depth A − depth M + |η_{1}| + ⋯ + |η_{c}| − c.

If for some integer i ≥ 1 we have Tor_{i}^{A}(M, N) = ⋯ = Tor_{i+w}^{A}(M, N) = 0, then Tor_{i}^{A}(M, N) = 0 for all i ≥ 1 by Theorem 3.3. The result now follows from [Jør, Theorem 1.7 and Proposition 2.2]. □

The final result deals with the vanishing of homology for two modules when A is a complete intersection. Namely, in this situation the homology modules are given as the homology modules of two modules of finite projective dimension, due to the fact that every module over a complete intersection has reducible complexity.

Theorem 3.6. Suppose A is a complete intersection, and let X and Y be A-modules such that Tor_{i}^{A}(X, Y) = 0 for i ≫ 0. Then there exist A-modules X′ and Y′, both of finite projective dimension, such that depth X = depth X′, depth Y = depth Y′ and Tor_{i}^{A}(X, Y) ≃ Tor_{i}^{A}(X′, Y′) for i > 0.

Proof. If the complexity of one of X and Y, say X, is nonzero, choose a homogeneous element η ∈ Ext_{A}^{∗}(X, X) reducing the complexity. By Proposition 2.4(i) any power of η also reduces the complexity of X, so choose an integer t such that Tor_{i}^{A}(Ω_{A}^{t|η|−1}(X), Y) = 0 for i > 0. The element η^{t} is given by the short exact sequence

0 → X → K_{η^{t}} → Ω_{A}^{t|η|−1}(X) → 0,

therefore by the choice of t we see that Tor_{i}^{A}(X, Y) and Tor_{i}^{A}(K_{η^{t}}, Y) are isomorphic for i > 0. Since A is Cohen-Macaulay we automatically have depth X = depth K_{η^{t}}, and repeating this process we eventually obtain what we want. □

This result has consequences for the study of the rigidity of Tor over Noetherian local rings. This study was initiated by M. Auslander in his 1961 paper [Aus], in which he proved his famous rigidity theorem; if X and Y are modules over an unramified regular local ring R, and Tor_{n}^{R}(X, Y) = 0 for some n ≥ 1, then Tor_{i}^{R}(X, Y) = 0 for all i ≥ n (recall that a regular local ring (S, 𝔪_{S}) is said to be ramified if it is of characteristic zero while its residue class field has characteristic p > 0 and p is an element of 𝔪_{S}^{2}). In 1966 S. Lichtenbaum extended Auslander’s rigidity theorem to all regular local rings (see [Lic]), and subsequently Peskine and Szpiro conjectured in [PeS] that the theorem holds for all Noetherian local rings provided one of the modules in question has finite projective dimension. A counterexample to the conjecture was provided by Heitmann in [Hei], where a Cohen-Macaulay ring R together with R-modules X and Y were given, for which pd X = 2 and Tor_{1}^{R}(X, Y) = 0, while Tor_{2}^{R}(X, Y) ≠ 0.

However, whether the rigidity of Tor holds for Noetherian local rings provided both modules involved have finite projective dimension is unknown. If this holds over complete intersections, then Theorem 3.6 shows that the conjecture of Peskine and Szpiro also holds for such rings (i.e. rigidity of Tor holds provided one of the modules involved has finite projective dimension). In fact, the theorem shows that if rigidity holds over a complete intersection R provided both modules have finite projective dimension, then rigidity holds over R for all modules X and Y satisfying Tor_{i}^{R}(X, Y) = 0 for i ≫ 0.

4. A generalization

In this final section we discuss a situation which slightly generalizes the concept of reducible complexity. Instead of letting M have reducible complexity as in Definition 2.1, we make the following assumption:

Assumption. The complexity of M is finite, and if it is nonzero then there exist local rings {R_{i}}_{i=1}^{c} such that for each i ∈ {1, …, c} there is a faithfully flat local homomorphism R_{i−1} → R_{i} (where R_{0} = A), an R_{i}-module K_{i}, an integer n_{i} and an exact sequence

0 → R_{i} ⊗_{R_{i−1}} K_{i−1} → K_{i} → Ω_{R_{i}}^{n_{i}}(R_{i} ⊗_{R_{i−1}} K_{i−1}) → 0

(where K_{0} = M) satisfying depth_{R_{i}} K_{i} = depth_{R_{i}}(R_{i} ⊗_{R_{i−1}} K_{i−1}), cx_{R_{i}} K_{i} < cx_{R_{i−1}} K_{i−1} and pd_{R_{c}} K_{c} < ∞.

Of course, if M has reducible complexity then by choosing each R_{i} to be A we see that the assumption is satisfied. Now let S → T be any faithfully flat local homomorphism, and X and Y any (finitely generated) S-modules. If F_{X} is a minimal S-free resolution of X, then the complex T ⊗_{S} F_{X} is a minimal T -free resolution of T ⊗_{S} X, and by [EGA, Proposition (2.5.8)] we have natural isomorphisms

Hom_{T} (T ⊗_{S} F_{X}, T ⊗_{S} Y) ≃ T ⊗_{S} Hom_{S}(F_{X}, Y),
(T ⊗_{S} F_{X}) ⊗_{T} (T ⊗_{S} Y) ≃ T ⊗_{S} (F_{X} ⊗_{S} Y).

Therefore we have isomorphisms

Ext_{T}^{i} (T ⊗_{S} X, T ⊗_{S} Y) ≃ T ⊗_{S} Ext_{S}^{i}(X, Y),
Tor_{i}^{T} (T ⊗_{S} X, T ⊗_{S} Y) ≃ T ⊗_{S} Tor_{i}^{S}(X, Y),

and as T is faithfully S-flat we then get

Ext_{T}^{i} (T ⊗_{S} X, T ⊗_{S} Y) = 0 ⇔ Ext_{S}^{i}(X, Y) = 0,
Tor_{i}^{T} (T ⊗_{S} X, T ⊗_{S} Y) = 0 ⇔ Tor_{i}^{S}(X, Y) = 0.

We then get the equalities

cx_{S} X = cx_{T} (T ⊗_{S} X)
p^{S}(X, Y) = p^{T} (T ⊗_{S} X, T ⊗_{S} Y)
q^{S}(X, Y) = q^{T} (T ⊗_{S} X, T ⊗_{S} Y)
depth_{S} X − depth_{S} Y = depth_{T} (T ⊗_{S} X) − depth_{T} (T ⊗_{S} Y),

where the one involving depth follows from [Mat, Theorem 23.3].

Using the above facts it is easy to see that both Theorem 3.1 and Corollary 3.2 remain true in this new situation, as does Theorem 3.3 if we drop the left inequality in (iii).

Suppose now that M has finite complete intersection dimension. Then [AGP, Proposition 7.2] and an argument similar to the proof of Proposition 2.2(i) show that M satisfies this new assumption and that n_{i} = 1 for each 1 ≤ i ≤ c. Consequently, the vanishing intervals in Theorem 3.1(i) and Theorem 3.3(i) are of length cx_{A} M + 1, as in [AvB, Theorem 4.7] and [Jo1, Theorem 2.1]. Moreover, we obtain [AvB, Theorem 4.2], which says that M is of finite projective dimension if and only if Ext_{A}^{2n}(M, M) = 0 for some n ≥ 1. To see this, note that when cx_{A} M > 0 the extension

0 → R_{1} ⊗_{A} M → K_{1} → Ω_{R_{1}}^{1}(R_{1} ⊗_{A} M) → 0

corresponds to an element θ ∈ Ext_{R_{1}}^{2}(R_{1} ⊗_{A} M, R_{1} ⊗_{A} M). If Ext_{A}^{2n}(M, M) = 0 for some n ≥ 1, then Ext_{R_{1}}^{2n}(R_{1} ⊗_{A} M, R_{1} ⊗_{A} M) also vanishes, hence θ^{2n} = 0. As in the proof of Corollary 3.2 we obtain the contradiction

cx_{A} M = cx_{R_{1}}(R_{1} ⊗_{A} M)
= cx_{R_{1}} K_{θ^{2n}}
≤ cx_{R_{1}} K_{1}
< cx_{A} M,

showing that we cannot have Ext_{A}^{2n}(M, M) = 0 for some n ≥ 1 when M is of positive complexity.

Acknowledgements

I would like to thank Dave Jorgensen and my supervisor Øyvind Solberg for valuable suggestions and comments on this paper.

References

[Aus] M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631-647.

[AuB] M. Auslander, R.-O. Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France 38 (1989), 5-37.

[AvB] L. Avramov, R.-O. Buchweitz, Support varieties and cohomology over complete intersection, Invent. Math. 142 (2000), 285-318.

[AGP] L. Avramov, V. Gasharov, I. Peeva, Complete intersection dimension, Publ. Math. I.H.E.S. 86 (1997), 67-114.

[ArY] T. Araya, Y. Yoshino, Remarks on a depth formula, a grade inequality and a conjecture of Auslander, Comm. Algebra 26 (1998), 3793-3806.

[Bak] Ø. Bakke, The existence of short exact sequences with some of the terms in given subcategories, in Algebras and modules II (Geiranger 1996), CMS Conf. Proc. 24 (1998), 39-45.

[Ber] P.A. Bergh, Complexity and periodicity, Colloq. Math. 104 (2006), no. 2, 169-191.

[ChI] S. Choi, S. Iyengar, On a depth formula for modules over local rings, Comm. Algebra 29 (2001), 3135-3143.

[EGA] A. Grothendieck, Éléments de géométrie algébrique, chapitre IV, seconde partie, I.H.É.S. Publ. Math. 24 (1965).

[EHSST] K. Erdmann, M. Holloway, N. Snashall, Ø. Solberg, R. Taillefer, Support varieties for selfinjective algebras, K-theory 33 (2004), 67-87.

[GaP] V. Gasharov, I. Peeva, Boundedness versus periodicity over commutative local rings, Trans. Amer. Math. Soc. 320 (1990), 569-580.

[Gul] T. Gulliksen, On the deviations of a local ring, Math. Scand. 47 (1980), 5-20.

[Hei] R. Heitmann, A counterexample to the rigidity conjecture for rings, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 94-97.

[HuJ] C. Huneke, D. Jorgensen, Symmetry in the vanishing of Ext over Gorenstein rings, Math. Scand. 93 (2003), no. 2, 161-184.

[Jo1] D. Jorgensen, Complexity and Tor on a complete intersection, J. Algebra 211 (1999), 578-598.

[Jo2] D. Jorgensen, A generalization of the Auslander-Buchsbaum formula, J. Pure Appl. Algebra 144 (1999), 145-155.

[Jør] P. Jørgensen, Symmetry theorems for Ext vanishing, J. Algebra 301 (2006), 224-239.

[Lic] S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220-226.

[Mac] S. Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, 1995.

[Mat] H. Matsumura, Commutative ring theory, Cambridge University Press, 2000.

[PeS] C. Peskine, L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, I.H.É.S. Publ. Math. 42 (1973), 47-119.

Petter Andreas Bergh, Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

   E-mail address: bergh@math.ntnu.no