MODULES WITH REDUCIBLE COMPLEXITY, II

PETTER ANDREAS BERGH

Abstract. We continue studying the class of modules having reducible complexity over a local ring. In particular, a method is provided for computing an upper bound of the complexity of such a module, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which all modules have finite complexity.

1. Introduction

The notion of complexity of a module was introduced by Alperin and Evens in [AlE], in order to study modular representations of finite groups. A decade later, in [Av1, Av2], Avramov introduced this concept for finitely generated modules over local rings, as a means to distinguish between modules of infinite projective dimension. The complexity of a module measures the growth of its minimal free resolution. For example, a module has complexity zero if and only if its projective dimension is finite, and complexity one precisely when its minimal free resolution is bounded.

For an arbitrary local ring, not much is known about the modules of finite complexity. For example, a characterization of these modules in terms of cohomology does not exist. Even worse, it is unclear whether every local ring has finite finitistic complexity dimension, that is, whether there is a bound on the complexities of the modules having finite complexity. In other words, the status quo for complexity is entirely different from that of projective dimension; there simply does not exist any analogue of the Auslander-Buchsbaum formula. The only class of rings for which these problems are solved are the complete intersections; over such a ring every module has finite complexity, and the complexity is bounded by the codimension of the ring.

In this paper we continue studying modules with so-called “reducible complexity”, a concept introduced in [Be2]. In particular, we give a method for computing an upper bound of the complexity of such a module. This is done by looking at the vanishing of cohomology with certain test modules. We then specialize to the case when the ring is a complete intersection, and use the theory of support varieties, introduced in [Av1] and [AvB], both to sharpen our results and to obtain new ones.

2. Reducible complexity

Throughout we let (A, 𝔪, k) be a local (meaning also commutative Noetherian) ring, and we suppose all modules are finitely generated. For an A-module M with minimal free resolution

⋯ → F_{2} → F_{1} → F_{0} → M → 0,

2000 Mathematics Subject Classification. 13D02, 13D07, 13H10.

Key words and phrases. Complexity, vanishing of cohomology, complete intersections.

The author acknowledges the support of the Marie Curie Research Training Network through a LieGrits fellowship.

the rank of F_{n}, i.e. the integer dim_{k} Ext_{A}^{n}(M, k), is the nth Betti number of M, and we denote this by β_{n}(M). The ith syzygy of M, denoted Ω_{A}^{i}(M), is the cokernel of the map F_{i+1} → F_{i}, and it is unique up to isomorphism. Note that Ω_{A}^{0}(M) = M and that β_{n} (Ω_{A}^{i}(M)) = β_{n+i}(M) for all i. 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}.

In general the complexity of a module may be infinite, in fact the rings for which all modules have finite complexity are precisely the complete intersections. From the definition we see that the complexity is zero if and only if the module has finite projective dimension, and that the modules of complexity one are those whose minimal free resolutions are bounded, for example the periodic modules. Moreover, the complexity of M equals that of Ω_{A}^{i}(M) for every i ≥ 0.

Let N be an A-module, and consider an element η ∈ Ext_{A}^{t}(M, N). By choosing a map f_{η}: Ω_{A}^{t}(M) → N representing η, we obtain a commutative pushout diagram

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

with exact rows. Note that the module K_{η} is independent, up to isomorphism, of the map f_{η} chosen as a representative for η.

We now recall the definition of modules with reducible complexity. Given Amodules X and Y, we denote the graded A-module ⊕_{i=0}^{∞} Ext_{A}^{i}(X, Y) by Ext_{A}^{∗}(X, Y).

Definition 2.1. Denote by C_{A} the category of all A-modules having finite complexity. The full subcategory C_{A}^{r} ⊆ C_{A} consisting of the 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} of positive complexity 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−1, depth K_{η} = depth X and K_{η} ∈ C_{A}^{r}. In this case, the cohomological element η is said to reduce the complexity of M.

The notion of reducible complexity was introduced in [Be2], where several (co)homology vanishing results for such modules were given (see also [Be4]). These results were previously known to hold for modules over complete intersections, and so it is not surprising that over such rings all modules have reducible complexity. In fact, by [Be2, Proposition 2.2] every module of finite complete intersection dimension has reducible complexity, and given such a module the reducing process decreases the complexity by exactly one. Recall that the module M has finite complete intersection dimension, written CI-dim_{A} M < ∞, if there exist local rings R and Q and a diagram A → R ↞ Q of local homomorphisms such that A → R is faithfully flat, R ↞ Q is surjective with kernel generated by a regular sequence (such a diagram A → R ↞ Q is called a quasi-deformation of A), and pd_{Q}(R⊗_{A}M) is finite. Such modules were first studied in [AGP], and the concept generalizes that of virtual projective dimension defined in [Av1]. As the name suggests, modules having finite complete intersection dimension to a large extent behave homologically like modules over complete intersections. Indeed, over a complete intersection (S, 𝔫) every module has finite complete intersection dimension; the completion Ŝ of S with respect to the 𝔫-adic topology is the residue ring of a regular local ring Q modulo an ideal generated by a regular sequence, and so S → Ŝ ↞ Q is a quasi deformation.

It should be commented on the fact that in the original definition of the notion of reducible complexity in [Be2], the reducing process did not necessarily reduce the complexity by exactly one. Namely, if η ∈ Ext_{A}^{∗}(M, M) is a cohomological element reducing the complexity of the module M, then the requirement was only that cx K_{η} be strictly less than cx M, and not necessarily equal to cx M − 1. However, we are unaware of any example where the complexity actually drops by more than one. On the contrary, there is evidence to suggest that this cannot happen. If CI-dim_{A} M is finite and

0 → M → K → Ω_{A}^{n}(M) → 0

is an exact sequence, then it is not hard to see that there exists one “joint” quasi deformation A → R ↞ Q such that all these three modules have finite projective dimension over Q when forwarded to R. In this situation we can apply [Jor, Theorem 1.3], a result originally stated for complete intersections, and obtain a “principal lifting” (of R) over which the complexities drop by exactly one. By alterating this procedure, we can show that the complexity of K cannot drop by more than one. Note also that this is trivial if cx M ≤ 2.

It should also be noted that the requirement on the depth of the modules involved, i.e. the requirement depth K_{η} = depth X in the definition, is not very restrictive. For example, when the ring A is Cohen-Macaulay, this is always the case (see the remark following [Be2, Definition 2.1]).

As mentioned, every A-module of finite complete intersection dimension belongs to C_{A}^{r}. However, the converse is a priori not true, that is, a module in C_{A}^{r} need not have reducible complexity. Moreover, a module having finite complexity need not have free reducible complexity. An example illustrating this was given in [Be2, Section 2], and the following is an example in which the ring is Gorenstein.

Example. Let (A, 𝔪, k) be the local finite dimensional algebra k[X_{1}, …, X_{5}]/ 𝔞 where 𝔞 ⊂ k[X_{1}, …, X_{5}] is the ideal generated by the quadratic forms

X_{1}^{2}, X_{2}^{2}, X_{5}^{2}, X_{3}X_{4}, X_{3}X_{5}, X_{4}X_{5}, X_{1}X_{4} + X_{2}X_{4}
αX_{1}X_{3} + X_{2}X_{3}, X_{3}^{2} − X_{2}X_{5} + αX_{1}X_{5}, X_{4}^{2} − X_{2}X_{5} + X_{1}X_{5}

for a nonzero element α ∈ k. By [GaP, Proposition 3.1] this ring is Gorenstein, and the complex

⋯ → A^{2} ⟶^{d_{n+1}} A^{2} ⟶^{d_{n}} A^{2} → ⋯

with maps given by the matrices d_{n} = ( \begin{matrix}x_{1} & α^{n}x_{3}+x_{4} \\ 0 & x_{2} \end{matrix} ) is exact. This sequence is therefore a minimal free resolution of the module M := Im d_{0}, hence this module has complexity one. If the order of α in k is infinite, then M cannot have reducible complexity; if M has reducible complexity, then there exists an exact sequence

0 → M → K → Ω_{A}^{n}(M) → 0

in which K has finite projective dimension. As A is selfinjective, the module K must be free, hence M is a periodic module, a contradiction. Moreover, if the order of α is finite but at least 3, then the argument in [Be2, Section 2, example] shows that M has reducible complexity but not finite complete intersection dimension

This example also shows that, in general, a module of finite Gorenstein dimension and finite complexity need not have reducible complexity. Recall that a module X over a local ring R has finite Gorenstein dimension, denoted G-dim_{R} X < ∞, if there exists an exact sequence

0 → G_{t} → ⋯ → G_{0} → X → 0

of R-modules in which the modules G_{i} are reflexive and satisfy Ext_{R}^{j}(G_{i}, R) = 0 = Ext_{R}^{j}(Hom_{R}(G_{i}, R), R) for j ≥ 1. Every module over a Gorenstein ring has finite Gorenstein dimension, in fact this property characterizes Gorenstein rings. Using this concept, Gerko introduced in [Ger] the notion of lower complete intersection dimension; the module X has finite lower complete intersection dimension, written CI_{*}-dim_{R} X < ∞, if it has finite Gorenstein dimension and finite complexity (and in this case CI_{*}-dim_{R} X = G-dim_{R} X). The Gorenstein dimension, lower complete intersection dimension, complete intersection dimension and projective dimension of a module are all related via the inequalities

G-dim_{R} X ≤ CI_{*}-dim_{R} X ≤ CI-dim_{R} X ≤ pd_{R} X.

If one of these dimensions happen to be finite, then it is equal to those to its left. Note that the class of modules having reducible complexity and finite Gorenstein dimension lies properly “between” the class of modules having finite lower complete intersection dimension and the class of modules having finite complete intersection dimension.

Given an A-module M, how do we decide whether it has reducible complexity? The following proposition, which does not give a complete answer, shows that at least we can reduce the complexity of M by one when its Ext-algebra is “nice”. Recall that for every A-module N the graded A-module Ext_{A}^{∗}(M, N) is a right Ext_{A}^{∗}(M, M)-module via the Yoneda product.

Proposition 2.2. Let M be an A-module, and suppose there exists a commutative Noetherian graded subalgebra H of Ext_{A}^{∗}(M, M) over which Ext_{A}^{∗}(M, k) is a finitely generated module. Then there exists a homogeneous element η ∈ Ext_{A}^{∗}(M, M), of positive degree, such that cx K_{η} = cx M − 1.

In order to prove this proposition, we need the following two lemmas, the first of which generalizes [Be1, Proposition 2.1].

Lemma 2.3. Let H be a commutative graded ring, and let X be a Noetherian graded H-module. Then there exists a homogeneous element η ∈ H, of positive degree, such that the multiplication map X_{i} ⟶^{η} X_{i+|η|} is injective for large i. If H is Noetherian and therefore finitely generated over H_{0} by homogeneous elements η_{1}, …, η_{t}, say, then we may choose η such that |η_{i}| divides |η| for some i.

Proof. Denote by H_{+} the positive part ⊕_{i=1}^{∞}H_{i} of H, and consider the graded submodule

L =^{def} {x ∈ X | H_{+}x = 0}

of X. Since X is Noetherian, this submodule is finitely generated, and by construction it is therefore also finitely generated as an H_{0}-module (since H_{0} = H/H_{+}). But H_{0} lives only in one degree, and consequently L_{i} = 0 for large i. The last part of the proof of [Be1, Proposition 2.1] now applies. □

The second lemma we need in order to prove Proposition 2.2 is the following version of the Hilbert-Serre Theorem. Its corollary shows that under certain conditions the Poincaré series of a module is a rational function.

Lemma 2.4 (Hilbert-Serre Theorem). Let K be a field, and let R be a commutative Noetherien graded ring, generated over R_{0} by homogeneous elements r_{1}, …, r_{t}, say. Let V be a graded K-R-bimodule such that dim_{K} V_{i} < ∞ for each i, and such that V is a finitely generated R-module. Then the Poincaré series ∑_{i=0}^{∞}(dim_{K} V_{i})x^{i} of V is of the form

∑_{i=0}^{∞}(dim_{K} V_{i})x^{i} = f(x)/ ∏_{i=1}^{t}(1 − x^{|r_{i}|}),

where f(x) is a polynomial with integer coefficients.

Corollary 2.5. Let M be an A-module, and suppose there exists a commutative Noetherian graded subalgebra H of Ext_{A}^{∗}(M, M) over which Ext_{A}^{∗}(M, k) is a finitely generated module. Let H be generated over H_{0} by homogeneous elements η_{1}, …, η_{t}, say. Then the Poincaré series ∑_{i=0}^{∞} β_{i}(M)x^{i} of M is of the form

∑_{i=0}^{∞} β_{i}(M)x^{i} = f(x)/ ∏_{i=1}^{t}(1 − x^{|η_{i}|}),

where f(x) is a polynomial with integer coefficients. Moreover, the complexity of M is finite and equal to the order of the pole at x = 1 of the Poincaré series.

Proof. The rationality of the Poincaré series follows from the Hilbert-Serre Theorem, since Ext_{A}^{∗}(M, k) is a k-H-bimodule and β_{i}(M) = dim_{k} Ext_{A}^{i}(M, k). The last statement follows from [Ben, Proposition 5.3.2]. □

We are now ready to prove Proposition 2.2.

Proof of Proposition 2.2. Since H is Noetherian, it is generated over H_{0} by homogeneous elements η_{1}, …, η_{t}, say. By Lemma 2.3, there exists a homogeneous element η ∈ Ext_{A}^{∗}(M, M), of positive degree, such that scalar multiplication

Ext_{A}^{i}(M, k) ⟶^{·η} Ext_{A}^{i+|η|}(M, k)

is injective for large i. Moreover, we may assume that the degree of one η_{1}, …, η_{t} divides that of η. Consider the short exact sequence

0 → M → K_{η} → Ω_{A}^{|η|−1}(M) → 0

induced by this element. Applying Hom_{A}(−, k) to this sequence, we obtain a long exact sequence

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

in which the connecting homomorphism Ext_{A}^{i}(M, k) ⟶^{∂_{η}} Ext_{A}^{i+|η|}(M, k) is just scalar multiplication with (−1)^{i}η (cf. [Mac, Theorem III.9.1]). This map is injective for large i, hence there exists an integer i_{0} such that

β_{i+1}(K_{η}) = β_{i+|η|}(M) − β_{i}(M)

for i ≥ i_{0}.

From Corollary 2.5 we know that the the Poincaré series ∑_{i=0}^{∞} β_{i}(M)x^{i} of M is of the form

∑_{i=0}^{∞} β_{i}(M)x^{i} = f(x)/ ∏_{i=1}^{t}(1 − x^{|η_{i}|}),

where f(x) is a polynomial with integer coefficients. Denote this rational function by P (M, x), and denote the Poincaré series ∑_{i=0}^{∞} β_{i}(K_{η})x^{i} of K_{η} by P (K_{η}, x). From the equality

∑_{i=i_{0}}^{∞} β_{i+1}(K_{η})x^{i+1} = ∑_{i=i_{0}}^{∞} β_{i+|η|}(M)x^{i+1} − ∑_{i=i_{0}}^{∞} β_{i}(M)x^{i+1}

we obtain the equality

P (K_{η}, x) = \frac{(1 − x^{|η|})P (M, x)}{x^{|η|−1}} + \frac{g(x)}{x^{|η|−1}} ,

where g(x) is a polynomial with integer coefficients. Now since one of |η_{1}|, …, |η_{t}| divides |η|, we see that the order of the pole of P (K_{η}, x) at x = 1 is exactly one less than that of P (M, x). Then from [Ben, Proposition 5.3.2] we see that cx K_{η} = cx M − 1. □

3. Complexity testing

In this section we introduce a method for computing an upper bound for the complexity of a given module in C_{A}^{r}. We start with a key result from [Be2], showing that the modules in C_{A}^{r} having infinite projective dimension also have higher self extensions.

Proposition 3.1. [Be2, Corollary 3.2] If M belongs to C_{A}^{r}, then

pd M = sup{i| Ext_{A}^{i}(M, M) ≠ 0}.

Thus every module in C_{A}^{r} with infinite projective dimension has higher selfextensions. For a given natural number t, denote by C_{A}^{r}(t) the full subcategory

C_{A}^{r}(t) =^{def} {X ∈ C_{A}^{r}| cx X = t}

of C_{A}^{r} consisting of the modules of complexity t. The following result shows that this subcategory serves as a “complexity test category”, in the sense that if a module has no higher extensions with the modules in C_{A}^{r}(t), then its complexity is strictly less than t.

Theorem 3.2. Let M be a module belonging to C_{A}^{r} and t a natural number. If Ext_{A}^{i}(M, N) = 0 for every N ∈ C_{A}^{r}(t) and i ≫ 0, then cx M < t.

Proof. We show by induction that if cx M ≥ t, then there is a module N ∈ C_{A}^{r}(t) with the property that Ext_{A}^{i}(M, N) does not vanish for all i ≫ 0. If the complexity of M is t, then by Proposition 3.1 we may take N to be M itself, so suppose that cx M > t. Choose a cohomological homogeneous element η ∈ Ext_{A}^{∗}(M, M) of positive degree reducing the complexity. In the corresponding exact sequence

0 → M → K_{η} → Ω_{A}^{|η|−1}(M) → 0,

the module K_{η} also belongs to C_{A}^{r} and has complexity one less than that of M, hence by induction there is a module N ∈ C_{A}^{r}(t) such that Ext_{A}^{i}(K_{η}, N) does not vanish for all i ≫ 0. From the long exact sequence

⋯ → Ext_{A}^{i+|η|−1}(M, N) → Ext_{A}^{i}(K_{η}, N) → Ext_{A}^{i}(M, N) → Ext_{A}^{i+|η|}(M, N) → ⋯

resulting from η, we see that Ext_{A}^{i}(M, N) cannot vanish for all i ≫ 0. □

In particular, we can use the category C_{A}^{r}(1) to decide whether a given module in C_{A}^{r} has finite projective dimension. We record this fact in the following corollary.

Corollary 3.3. A module M ∈ C_{A}^{r} has finite projective dimension if and only if Ext_{A}^{i}(M, N) = 0 for every N ∈ C_{A}^{r}(1) and i ≫ 0.

Remark. Let C_{A}^{ci} denote the category of all A-modules of finite complete intersection dimension, and for each natural number t define the category

C_{A}^{ci}(t) =^{def} {X ∈ C_{A}^{ci}| cx X = t}.

Then Theorem 3.2 and Corollary 3.3 remain true if we replace C_{A}^{r} and C_{A}^{r}(t) by C_{A}^{ci} and C_{A}^{ci}(t), respectively. That is, when the module we are considering has finite complete intersection dimension, then we need only use modules of finite complete intersection dimension as test modules.

When the ring is Gorenstein, then it follows from [Be2, Theorem 3.5] that symmetry holds for the vanishing of cohomology between modules of reducible complexity. We therefore have the following symmetric version of Theorem 3.2.

Corollary 3.4. Suppose A is Gorenstein, let M be a module belonging to C_{A}^{r}, and let t be a natural number. If Ext_{A}^{i}(N, M) = 0 for every N ∈ C_{A}^{r}(t) and i ≫ 0, then cx M < t.

We now turn to the setting in which every A-module has reducible complexity. For the remainder of this section, we assume A is a complete intersection, i.e. the 𝔪-adic completion Â of A is the residue ring of a regular local ring modulo a regular sequence. For such rings, Avramov and Buchweitz introduced in [Av1] and [AvB] a theory of cohomological support varieties, and they showed that this theory is similar to that of the cohomological support varieties for group algebras. As we will implicitly use this powerful theory in the results to come, we recall now the definitions (details can be found in [Av1, Section 1] and [AvB, Section 2]).

Denote by c the codimension of A, that is, the integer dim_{k}(𝔪 / 𝔪^{2}) − dim A, and by χ the sequence χ_{1}, …, χ_{c} consisting of the c commuting Eisenbud operators of cohomological degree two. For every Â-module X there is a homomorphism

Â[χ] ⟶^{ϕ_{X}} Ext∗ ; Â(X, X)

of graded rings, and via this homomorphism Ext∗ ; Â(X, Y) is finitely generated over Â[χ] for every Â-module Y. Denote by H the polynomial ring k[χ], and by E(X, Y) the graded space Ext∗ ; Â(X, Y) ⊗Â k. The above homomorphism ϕ_{X}, together with the canonical isomorphism H ≃ Â[χ] ⊗Â k, induce a homomorphism H → E(X, X) of graded rings, under which E(X, Y) is a finitely generated H-module. Now let M be an A-module, and denote by M̂ its 𝔪-adic completion Â ⊗_{A} M. The support variety V(M) of M is the algebraic set

V(M) =^{def} {α ∈ k̃^{c}|f(α) = 0 for all f ∈ Ann_{H} E(M̂, M̂)},

where k̃ is the algebraic closure of k. Finally, for an ideal 𝔞 ⊆ H we define the variety V_{H}(𝔞) ⊆ k̃^{c} to be the zero set of 𝔞.

As mentioned above, this theory shares many properties with the theory of cohomological support varieties for modules over group algebras of finite groups. For instance, the dimension of the variety of a module equals the complexity of the module, in particular the variety is trivial if and only if the module has finite projective dimension. The following complexity test result relies on [Be3, Corollary 2.3], which says that every homogeneous algebraic subset of k̃^{c} is realizable as the support variety of some A-module.

Proposition 3.5. Let M be an A-module, let η_{1}, …, η_{t} ∈ H be homogeneous elements of positive degrees, and choose an A-module T_{η_{1},...,η_{t}} with the property that V(T_{η_{1},...,η_{t}}) = V_{H}(η_{1}, …, η_{t}). If Ext_{A}^{i}(M, T_{η_{1},...,η_{t}}) = 0 for i ≫ 0, then cx M ≤ t.

Proof. Denote the ideal Ann_{H} E(M̂, M̂) ⊆ H by 𝔞. If Ext_{A}^{i}(M, T_{η_{1},...,η_{t}}) = 0 for i ≫ 0, then from [AvB, Theorem 5.6] we obtain

{0} = V(M) ∩ V(T_{η_{1},...,η_{t}}) = V_{H}(𝔞) ∩ V_{H}(η_{1}, …, η_{t}) = V_{H} (𝔞 +(η_{1}, …, η_{t})) ,

hence the ring H/ (𝔞 +(η_{1}, …, η_{t})) is zero dimensional. But then the dimension of the ring H/ 𝔞 is at most t, i.e. cx M ≤ t. □

We illustrate this last result with an example.

Example. Let k be a field and Q the formal power series ring k⟦x_{1}, …, x_{c}⟧ in c variables. For each 1 ≤ i ≤ c, let n_{i} ≥ 2 be an integer, let 𝔞 ⊆ Q be the ideal generated by the regular sequence x_{1}^{n_{1}}, …, x_{c}^{n_{c}}, and denote by A the complete intersection Q/ 𝔞. For each 1 ≤ i ≤ c we shall construct an A-module whose support variety equals V_{H}(χ_{i}), by adopting the techniques used in [SnS, Section 7] to give an interpretation of the Eisenbud operators.

Consider the exact sequence

0 → 𝔪_{Q} → Q → k → 0

of Q-modules. Applying A⊗_{Q} − to this sequence gives the four term exact sequence

(†)  0 → Tor_{1}^{Q}(A, k) → 𝔪_{Q} / 𝔞 𝔪_{Q} → A → k → 0

of A-modules. Consider the first term in this sequence. By tensoring the exact sequence

0 → 𝔞 → Q → A → 0

over Q with k, we obtain the exact sequence

0 → Tor_{1}^{Q}(A, k) → 𝔞 ⊗_{Q}k ⟶^{g} Q ⊗_{Q} k → A ⊗_{Q} k → 0,

in which the map g must be the zero map since 𝔞 k = 0. This gives isomorphisms

Tor_{1}^{Q}(A, k) ≃ 𝔞 ⊗_{Q}k ≃ 𝔞 ⊗_{Q}(A ⊗_{A} k) ≃ 𝔞 / 𝔞^{2} ⊗_{A}k

of A-modules. Since 𝔞 is generated by a regular sequence of length c, the A-module 𝔞 / 𝔞^{2} is free of rank c, and therefore Tor_{1}^{Q}(A, k) is isomorphic to k^{c}. We may now rewrite the four term exact sequence (†) as

0 → k^{c} ⟶^{f} 𝔪_{Q} / 𝔞 𝔪_{Q} → A → k → 0,

and it is not hard to show that the map f is defined by

(α_{1}, …, α_{c}) ↦ ∑ α_{i}x_{i}^{n_{i}} + 𝔞 𝔪_{Q} .

The image of the Eisenbud operator χ_{j} under the homomorphism Â[χ] ⟶^{ϕ_{k}} Ext∗ ; Â(k, k) is the bottom row in the pushout diagram

[figure: 0; k^{c}; f; π_{j}; 𝔪_{Q} / 𝔞 𝔪_{Q}; A; k; 0; 0; k; K_{χ_{j}}; A; k; 0]

of A-modules, in which the map π_{j} is projection onto the jth summand. The pushout module K_{χ_{j}} can be described explicitly as

K_{χ_{j}} = \frac{k ⊕ 𝔪_{Q} / 𝔞 𝔪_{Q}}{\{(α_{j}, − ∑ α_{i}x_{i}^{n_{i}} + 𝔞 𝔪_{Q}) | (α_{1}, …, α_{c}) ∈ k^{c}\}},

and by [Be3, Theorem 2.2] its support variety is given by V(K_{χ_{j}}) = V(k)∩V_{H}(χ_{j}). But the variety of k is the whole space, hence the equality V(K_{χ_{j}}) = V_{H}(χ_{j}). Thus by Proposition 3.5 the A-module K_{χ_{j}} is a test module for finding modules with bounded projective resolutions; if M is an A-module such that Ext_{A}^{i}(M, K_{χ_{j}}) = 0 for i ≫ 0, then cx M ≤ 1.

Before proving the final result, we need a lemma showing that every maximal Cohen-Macaulay module over a complete intersection has reducible complexity by a cohomological element of degree two. This improves [Be4, Lemma 2.1(i)], which states that such a cohomological element exists after passing to some suitable faithfully flat extension of the ring.

Lemma 3.6. If M is a maximal Cohen-Macaulay A-module of infinite projective dimension, then there exists an element η ∈ Ext_{A}^{2}(M, M) reducing its complexity.

Proof. Since the dimension of V(M) is nonzero, the radical \sqrt{Ann_{H} E(M̂, M̂)} of

Ann_{H} E(M̂, M̂) is properly contained in the graded maximal ideal of H. There-

fore one of the Eisenbud operators, say χ_{j}, is not contained in \sqrt{Ann_{H} E(M̂, M̂)}. We now follow the arguments given prior to [Be3, Corollary 2.3]. Viewing χ_{j} as an element of Â[χ], we can apply the homomorphism ϕM̂ and obtain the element ϕM̂(χ_{j}) ⊗ 1 in Ext2 ; Â(M̂, M̂) ⊗Â k. Now Ext2 ; Â(M̂, M̂) is isomorphic to Ext_{A}^{2}(M, M) ⊗_{A} Â, and there is an isomorphism

Ext_{A}^{2}(M, M) ⊗_{A} k ⟶^{∼} Ext2 ; Â(M̂, M̂) ⊗Â k

mapping an element θ ⊗ 1 ∈ Ext_{A}^{2}(M, M) ⊗_{A} k to θ̂⊗ 1. Therefore there exists an element η ∈ Ext_{A}^{2}(M, M) such that η̂⊗ 1 equals ϕM̂(χ_{j}) ⊗ 1 in Ext2 ; Â(M̂, M̂) ⊗Â k. If the exact sequence

0 → M → K_{η} → Ω_{A}^{1}(M) → 0

corresponds to η, then its completion

0 → M̂ → K̂_{η} → Ω1 ; Â(M̂) → 0

corresponds to η̂, and so from [Be3, Theorem 2.2] we see that

V(K_{η}) = V(M) ∩ V_{H}(χ_{j}).

Since χ_{j} was chosen so that it “cuts down” the variety of M, we must have dim V(K_{η}) = dim V(M) − 1, i.e. cx K_{η} = cx M − 1. □

We have now arrived at the final result, which improves Theorem 3.2 when the ring is a complete intersection. Namely, for such rings it suffices to check the vanishing of finitely many cohomology groups “separated” by an odd number. The number of cohomology groups we need to check depends on the complexity value we are testing. Recall that we have denoted the codimension of the complete intersection A by c.

Theorem 3.7. Let M be an A-module and t ∈ {1, …, c} an integer. If for every A-module N of complexity t there is an odd number q such that

Ext_{A}^{n}(M, N) = Ext_{A}^{n+q}(M, N) = ⋯ = Ext_{A}^{n+(c−t)q}(M, N) = 0

for some even number n > dim A − depth M, then cx M < t.

Proof. Since cx M = cx Ω_{A}^{dim A−depth M}(M), we may without loss of generality assume that M is maximal Cohen-Macaulay and that n > 0. We prove by induction that if cx M ≥ t, then for any odd number q and any even integer n > 0, the groups

Ext_{A}^{n}(M, N), Ext_{A}^{n+q}(M, N), …, Ext_{A}^{n+(cx M−t)q}(M, N)

cannot all vanish for every module N of complexity t. When the complexity of M is t, take N to be M itself. In this case it follows from [AvB, Theorem 4.2] that Ext_{A}^{n}(M, N) is nonzero, because t ≥ 1. Now assume cx M > t, and write q as q = 2s − 1 where s ≥ 1 is an integer. By Lemma 3.6 there is an element η ∈ Ext_{A}^{2}(M, M) reducing the complexity of M, and it follows from [Be2, Proposition 2.4(i)] that the element η^{s} ∈ Ext_{A}^{2s}(M, M) also reduces the complexity. The latter element corresponds to an exact sequence

0 → M → K → Ω_{A}^{q}(M) → 0,

in which the complexity of K is one less than that of M. By induction there exists a module N, of complexity t, such that the groups

Ext_{A}^{n}(K, N), Ext_{A}^{n+q}(K, N), …, Ext_{A}^{n+(cx K−t)q}(K, N)

do not all vanish. Then from the exact sequence we see that the groups

Ext_{A}^{n}(M, N), Ext_{A}^{n+q}(M, N), …, Ext_{A}^{n+(cx M−t)q}(M, N)

cannot possible all vanish. Since the complexity of any A-module is at most c, the proof is complete. □

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Petter Andreas Bergh Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, United Kingdom Present address: Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

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