COMPLEXITY AND PERIODICITY

PETTER ANDREAS BERGH

Abstract. Let M be a finitely generated module over an Artin algebra. By considering the lengths of each module in the minimal projective resolution of M, we obtain its Betti sequence. This sequence must be bounded if M is eventually periodic, but the converse fails to hold in general. We give conditions under which this holds, using techniques from Hochschild cohomology. We also provide a result which under certain conditions guarantees the existence of periodic modules. Finally, we study the case when an element in the Hochschild cohomology ring ”generates” the periodicity of a module.

1. Introduction

This paper is devoted to investigating connections between periodicity and complexity for modules over Artin algebras, as was done in [9] for modules over both group rings of finite groups and commutative Noetherian local rings. More specifically, let M be a finitely generated module over an Artin algebra. By considering the lengths of each module in the minimal projective resolution of M, we obtain its Betti sequence. If M is eventually periodic, i.e. if its minimal projective resolution becomes periodic from some step on, then the Betti sequence of M must be bounded. The converse, however, fails to hold in general, that is, it need not be true that M is eventually periodic even though its Betti sequence is bounded. We give conditions under which this holds, using techniques from Hochschild cohomology. In addition we provide a result which under certain conditions guarantees the existence of periodic modules.

One reason for restricting our attention to Artin algebras is that we need to make sure that every finitely generated module has a unique minimal projective resolution. Therefore, throughout this paper, we let k be a commutative Artin ring and Λ an Artin k-algebra with Jacobson radical 𝔯. We fix a finitely generated Λ-module M with a minimal projective resolution

(ℙ, d): ⋯ ⟶^{d_{3}} P_{2} ⟶^{d_{2}} P_{1} ⟶^{d_{1}} P_{0} ⟶^{d_{0}} M → 0,

i.e. Ker d_{i} ⊆ 𝔯 P_{i}. The integers β_{n}(M) = ℓ_{k}(P_{n}) are called the Betti numbers of M, and they are all finite since a module is finitely generated over k whenever it is finitely generated over Λ. Moreover, these integers are well defined since any minimal projective resolution is unique up to isomorphism. Thus, we may associate the infinite sequence

β_{0}(M), β_{1}(M), β_{2}(M), …

to M, and this sequence is called the Betti sequence of M. Over a commutative Noetherian local ring it is customary to define the Betti numbers of a finitely generated module as the ranks of the modules in its minimal free resolution. However, this would not make sense in our setting since projective modules need not be free.

2000 Mathematics Subject Classification. 16E05, 16E40, 16P10, 16P20, 16P90 (primary), 20J06 (secondary).

Key words and phrases. Eventually periodic modules, complexity, Hochschild cohomology.

We say that M is periodic if there is an integer p ≥ 1 such that M is isomorphic to Ω_{Λ}^{p}(M) (the p’th syzygy in the minimal projective resolution ℙ), and the least such integer p is the period of M. Furthermore, M is eventually periodic if one of its syzygies (in the minimal projective resolution) is periodic. Clearly, if M has this last property, then its Betti sequence is bounded. The converse is not true in general. A counterexample was given by R. Schulz in [14, Proposition 4.1], where he considered finite dimensional algebras of the form k⟨x, y⟩/(x^{2}, xy + qyx, y^{2}), for k a field and q ∈ k a nonzero element.

In [9] D. Eisenbud proved that the converse does hold over group rings of finite groups, and that it also holds in the commutative Noetherian local setting when the rings considered are complete intersections. In fact, it was shown that over a hypersurface (that is, a complete intersection of codimension one) any minimal free resolution eventually becomes periodic. In the same paper it was therefore conjectured that over a commutative Noetherian local ring a module having bounded Betti numbers must be periodic.

However, as in the case of Artin algebras, the conjecture fails to hold in general. An example of this was given in [11]. Here V. Gasharov and I. Peeva considered the commutative local finite dimensional k-algebra (R, 𝔪, k) = k[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}]/ 𝔞, where 𝔞 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},
x_{4}^{2} − x_{2}x_{5} + x_{1}x_{5}, αx_{1}x_{3} + x_{2}x_{3}, x_{3}^{2} − x_{2}x_{5} + αx_{1}x_{5}

for a nonzero element α ∈ k having infinite order in the multiplicative group k\{0}. They constructed the free resolution

⋯ ⟶^{d_{3}} R^{2} ⟶^{d_{2}} R^{2} ⟶^{d_{1}} R^{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} )

for n ≥ 0, and M = Im d_{0}. The module M has a constant Betti sequence, but is not eventually periodic.

We now briefly describe the contents of the three main sections of this paper. The first section is devoted to constructing a certain chain endomorphism on the minimal projective resolution of a module, given a finite generation hypothesis similar to the one used in [10]. This chain map eventually becomes surjective, and we use this to prove the main result; whenever the finite generation hypothesis holds, a module has bounded Betti numbers precisely when it is eventually periodic. In some cases (Theorem 2.5) we may also determine what the period is.

In the second section we develop a method for ”reducing” the complexity of a module. More precisely, given a module M satisfying certain conditions (and having finite nonzero complexity), we construct a new module closely related to M and having complexity exactly one less than that of M. Iterating this procedure, we end up with a module having complexity one (i.e. having bounded Betti numbers), and this module must be periodic in view of the main result in the first section.

Finally, in the third section we study the case when the period of an eventually periodic module is ”generated” by an element in the Hochschild cohomology ring. As with many other concepts, the inspiration comes from the group ring case, where one uses the group cohomology ring instead of the Hochschild cohomology ring.

2. Preliminary Results

The existence of eventually periodic modules of infinite projective dimension - and therefore of nonzero periodic modules - is far from obvious in general. In the next section we prove a result which under certain circumstances guarantees the existence of such modules. The proof is based on the main result of this section, which for a module gives a sufficient condition under which having a bounded Betti sequence is equivalent to being periodic.

The following proposition is the key to the main results. It guarantees regular elements for graded modules, provided we go ”far enough” out in the grading.

Proposition 2.1. Let A = ⨁_{i=0}^{∞} A_{i} be a commutative Noetherian graded k-algebra of finite type over k (that is, each A_{i} is a finitely generated k-module), generated as an A_{0}-algebra by homogeneous elements a_{1}, …, a_{r} of positive degrees. If N = ⨁_{i=0}^{∞} N_{i} is a finitely generated graded A-module, then there exists a homogeneous element η ∈ A of positive degree, such that the multiplication map

N_{i} ⟶^{η} N_{i+|η|}

is a k-monomorphism for i ≫ 0. Moreover, we can pick this η such that for some j, the degree of a_{j} divides |η|.

Proof. Consider the graded ideal A^{+} = ⨁_{i=1}^{∞} A_{i} of A and the graded submodule

(0 :_{N} A^{+}) = {x ∈ N | A^{+}x = 0}

of N. Since N is Noetherian, this submodule is finitely generated. Moreover, since it is annihilated by A^{+} it is a finitely generated A_{0}-module (A_{0} = A/A^{+}). Now A_{0}, being finitely generated over k, is Artinian, hence (0 :_{N} A^{+}) has the descending chain condition on submodules. Therefore there exists an integer w such that (0 :_{N} A^{+})_{i} = 0 for i ≥ w.

Consider the graded A-submodule N_{≥w} = ⨁_{i=w}^{∞} N_{i} of N. Since this is a finitely generated module, its set of associated prime ideals is finite and consists of graded ideals each of which is the annihilator of a homogeneous element (see, for example, [6, Lemma 1.5.6]), and whose union is the set of zero-divisors on N_{≥w}. If A^{+} is contained in any of these primes, then A^{+} annihilates a nonzero homogeneous element of N_{≥w}, a contradiction. Therefore, by the graded version of the ”prime avoidance” lemma (see, for example [6, Lemma 1.5.10]), there exists a homogeneous N_{≥w}-regular element η in A^{+}, obviously of positive degree. Since a_{1}, …, a_{r} generate A^{+}, a slight modification of the proof of [6, Lemma 1.5.10] shows that η can be chosen so that the degree of a_{j} divides that of η for some j. □

From now on, we assume that Λ is projective (or, equivalently, flat) as a kmodule. We denote by Λ^{e} its enveloping algebra Λ ⊗_{k} Λ^{op}, and by HH^{∗}(Λ) its Hochschild cohomology ring. Since Λ is projective as a k-module, we have

HH^{∗}(Λ) = ⨁_{i=0}^{∞} Ext_{Λ^{e}}^{i}(Λ, Λ)

with Yoneda product as multiplication. For two Λ-modules X and Y we denote the graded k-module ⨁_{i=0}^{∞} Ext_{Λ}^{i}(X, Y) by Ext_{Λ}^{∗}(X, Y), and this is a left and right HH^{∗}(Λ)-module via the ring homomorphisms

− ⊗_{Λ} Y : HH^{∗}(Λ) → Ext_{Λ}^{∗}(Y, Y)
− ⊗_{Λ} X: HH^{∗}(Λ) → Ext_{Λ}^{∗}(X, X)

followed by Yoneda composition. The left and right scalar multiplications on this module are closely related as follows (see [15, Corollary 1.3]): for homogeneous elements η ∈ HH^{∗}(Λ) and θ ∈ Ext_{Λ}^{∗}(X, Y) we have ηθ = (−1)^{|η||θ|}θη, where |η| and |θ| denote the degrees of these elements. In particular, we see that Ext_{Λ}^{∗}(X, Y) is finitely generated as a left HH^{∗}(Λ)-module if and only if it is finitely generated as a right HH^{∗}(Λ)-module.

In view of the counterexamples provided by Schulz, Gasharov and Peeva, we need to impose some restrictions in order to be able to prove that M is eventually periodic whenever its Betti numbers are bounded. The assumption we introduce is a ”local variant” of those used in [10] to develop the theory of support varieties for Artin algebras, and it enables us to use well known techniques from commutative algebra to obtain our results.

Assumption (Fg). Given the finitely generated Λ-module M, there exists a commutative Noetherian graded subalgebra H = ⨁_{i=0}^{∞} H^{i} of the Hochschild cohomology ring HH^{∗}(Λ), with the property that H^{0} = HH^{0}(Λ) (the center of Λ) and that the module Ext_{Λ}^{∗}(M, Λ/ 𝔯) is finitely generated over H.

Now we apply Proposition 2.1 to H and the graded H-module Ext_{Λ}^{∗}(M, Λ/ 𝔯). This we can do because H, being Noetherian, is generated as an algebra over H^{0} by a finite set of homogeneous elements of positive degrees. We obtain from the homogeneous element granted by Proposition 2.1 a negative degree chain endomorphism {ξ_{0}, ξ_{1}, … }: ℙ → ℙ (where ℙ is the minimal projective resolution of M) which eventually becomes surjective, that is, the map ξ_{i} is an epimorphism for i ≫ 0.

Proposition 2.2. Assume Fg holds. Then there exists an integer n ≥ 1 with a map ξ: Ω_{Λ}^{n}(M) → M and a chain map {ξ_{0}, ξ_{1}, … }: ℙ → ℙ over ξ of degree −n, such that ξ_{i} is surjective for i ≫ 0.

Proof. From Proposition 2.1 we see that there exists an integer w and a homogeneous element η ∈ H^{|η|}, with |η| ≥ 1, such that the multiplication map

Ext_{Λ}^{i}(M, Λ/ 𝔯) ⟶^{η} Ext_{Λ}^{i+|η|}(M, Λ/ 𝔯)

is injective for i ≥ w. Moreover, since (ℙ, d) is a minimal projective resolution we have Im d_{i} ⊆ 𝔯 P_{i−1}, implying that the differential in the complex Hom_{Λ}(ℙ, Λ / 𝔯) is zero. Therefore Ext_{Λ}^{i}(M, Λ / 𝔯) = Hom_{Λ}(P_{i}, Λ / 𝔯). Now consider the element η and the maps it induces. The action on Ext_{Λ}^{i}(M, Λ / 𝔯) is via the map − ⊗_{Λ} M: HH^{∗}(Λ) → Ext_{Λ}^{∗}(M, M), followed by Yoneda composition. Let ξ denote the image of η in Ext_{Λ}^{∗}(M, M). It can be interpreted as a Λ-linear map Ω_{Λ}^{|η|}(M) ⟶^{ξ} M, and so by the Comparison Theorem there exist Λ-linear maps {ξ_{0}, ξ_{1}, ξ_{2}, … } making the diagram

\xymatrix{⋯ \ar[r] & P_{|η|+i+1} \ar[r]^{d_{|η|+i+1}} \ar[d]^{ξ_{i+1}} & P_{|η|+i} \ar[r]^{d_{|η|+i}} \ar[d]^{ξ_{i}} & ⋯ \ar[r] & P_{|η|} \ar[r] \ar[d]^{ξ_{0}} & Ω_{Λ}^{|η|}(M) \ar[r] \ar[d]^{ξ} & 0 \\ ⋯ \ar[r] & P_{i+1} \ar[r]^{d_{i+1}} & P_{i} \ar[r]^{d_{i}} & ⋯ \ar[r] & P_{0} \ar[r]^{d_{0}} & M \ar[r] & 0}

commute. We show ξ_{i} is surjective for i ≥ w.

An element θ ∈ Ext_{Λ}^{i}(M, Λ / 𝔯) can be interpreted as a Λ-linear map P_{i} ⟶^{θ} Λ / 𝔯 (having the property θ ∘ d_{i+1} = 0), and then scalar multiplication by η is given by θη = θ ∘ ξ_{i}: P_{i+|η|} → Λ / 𝔯. Thus the map η · (−): Hom_{Λ}(P_{i}, Λ / 𝔯) → Hom_{Λ}(P_{i+|η|}, Λ / 𝔯) is simply given by

ξ_{i}^{∗}: f ↦ f ∘ ξ_{i},

and we know it is injective for i ≥ w. Applying Hom_{Λ}(−, Λ/ 𝔯) to the exact sequence

P_{|η|+i} ⟶^{ξ_{i}} P_{i} → Coker ξ_{i} → 0

therefore shows that Hom_{Λ}(Coker ξ_{i}, Λ/ 𝔯) = 0 for i ≥ w. Now if X is any finitely generated nonzero Λ-module, then X/ 𝔯 X is nonzero by Nakayama’s Lemma. This factor module is semisimple, and since every simple Λ-module occurs as a direct summand of Λ / 𝔯, there exists a nonzero map X → Λ / 𝔯. This shows Coker ξ_{i} = 0 for i ≥ w. By taking n = |η|, we are done. □

We now prove the main result of this section, which gives sufficient (and necessary) conditions for a module to be eventually periodic. Having Proposition 2.2 at hand, the proof is only a formality, as it reduces to simply comparing lengths in the minimal projective resolution of a module.

Theorem 2.3. If Fg holds, then M has bounded Betti numbers if and only if it is an eventually periodic module.

Proof. From the previous proposition, there exist integers w ≥ 0 and n ≥ 1 such that we have n sequences of surjective maps

⋯ ⟶^{ξ_{w+i+2n}} P_{w+i+2n} ⟶^{ξ_{w+i+n}} P_{w+i+n} ⟶^{ξ_{w+i}} P_{w+i} 0 ≤ i < n.

Considering the lengths of these modules over k, we see that we have non-decreasing sequences β_{w+i}(M) ≤ β_{w+i+n}(M) ≤ ⋯ for 0 ≤ i < n. Now if the Betti numbers of M are bounded, then these sequences must all eventually stabilize. Thus there is an integer t such that ξ_{i} is bijective for i ≥ t, and diagram chasing in the commutative diagram

\xymatrix{P_{n+t+1} \ar[r]^{d_{n+t+1}} \ar[d]^{ξ_{t+1}} & P_{n+t} \ar[r] \ar[d]^{ξ_{t}} & Ω_{Λ}^{n+t}(M) \ar[r] & 0 \\ P_{t+1} \ar[r]^{d_{t+1}} & P_{t} \ar[r] & Ω_{Λ}^{t}(M) \ar[r] & 0}

with exact rows provides an isomorphism ψ: Ω_{Λ}^{n+t}(M) → Ω_{Λ}^{t}(M). □

From this result we obtain some insight into the structure of the sequence of Betti numbers of M. Determining how these sequences grow is a problem which has been studied for a long time in the commutative Noetherian local setting. In [1], L. Avramov asked whether the Betti sequence (b_{0}, b_{1}, b_{2}, …) of a finitely generated module over such a ring is eventually non-decreasing, whereas a somewhat weaker question was asked by M. Ramras in [13]; is it true that either (b_{0}, b_{1}, b_{2}, …) is eventually constant, or lim_{i→∞} b_{i} = ∞? With these questions in mind, we include the following corollary to Proposition 2.2 and Theorem 2.3. The result shows that we can split the ”tail” of the sequence of Betti numbers of M into a finite number of sequences, each of which is either eventually constant or strictly increasing, depending on whether M has bounded Betti numbers or not.

Corollary 2.4. Let the setting be as in the theorem. There exist integers w ≥ 0 and n ≥ 1 such that the n sequences

(β_{w+i+jn}(M))_{j=0}^{∞} 0 ≤ i < n

are non-decreasing. In fact, these sequences are all eventually constant if the Betti numbers of M are bounded, and strictly increasing if not. In particular, if the Betti numbers of M are unbounded, then lim_{i→∞} β_{i}(M) = ∞.

Proof. The n non-decreasing sequences were given in the proof of the theorem. It is clear that if the Betti numbers of M are bounded, then the sequences are all eventually constant. In the case when there is no bound on the Betti numbers, the module M is not eventually periodic, and then ξ_{i} cannot be bijective for any i ≥ w. For if ξ_{t} was bijective for some t ≥ w, then since ξ_{t+1} is surjective we would (as in the proof of theorem) have an isomorphism Ω_{Λ}^{n+t}(M) ≃ Ω_{Λ}^{t}(M). □

Although Theorem 2.3 provides a tool for determining whether or not a module M is eventually periodic, it does not indicate when the minimal projective resolution of M becomes periodic, nor what the period actually is, contrary to the commutative local case. In [9, Theorem 4.1] it is shown that over a commutative local complete intersection A any minimal free resolution whose Betti sequence is bounded becomes periodic of period 2 after at most dim A + 1 steps.

Question. Given the assumptions of Theorem 2.3, does there exist a computable integer s such that the minimal projective resolution of M becomes periodic after at most s steps?

As to determining the period, the degree n of the element η, which we obtain from Proposition 2.1, is of course a candidate, but all that is certain is that the (eventual) period must divide n. However, imposing moderate restrictions on the rings k and H, we get a much stronger result.

Theorem 2.5. Suppose k contains an infinite field, and let H be generated as an algebra over H^{0} by homogeneous elements a_{1}, …, a_{r} of positive degrees. Then if Fg holds and M has bounded Betti numbers, the eventual period of M divides lcm(|a_{1}|, …, |a_{r}|).

In particular, if |a_{i}| = 1 for all i then the eventual period of M is 1, and if |a_{i}| ≤ 2 for all i then the Betti sequence of M is eventually constant.

Proof. From Proposition 2.1 we know there exists an integer w and a homogeneous element η ∈ H, of positive degree, such that η is regular on the H-module E_{≥w} = ⨁_{i=w}^{∞} Ext_{Λ}^{i}(M, Λ / 𝔯). Now let {𝔭_{i}}_{i=1}^{s} be the (finite) set of associated primes of E_{≥w}, and denote lcm(|a_{1}|, …, |a_{r}|) by u. Then a suitable power of each a_{i} belongs to H^{u}, and so if H^{u} ⊆ 𝔭_{j} for some j, then each a_{i} belongs to 𝔭_{j}. This implies that the ideal H^{+} = ⨁_{i=1}^{∞} H^{i} is contained in 𝔭_{j}, contradicting the fact that η, which is an element of H^{+}, is regular on E_{≥w}. Therefore H^{u} ⊈ 𝔭_{i} for 1 ≤ i ≤ s, implying H^{u} ∩ 𝔭_{i} ⊂ H^{u} (strict inclusion). In particular, if k′ is an infinite field contained in k, then H^{u} ∩ 𝔭_{i} is a proper k′-subspace of H^{u}. Since over an infinite field no vector space can be written as a finite union of proper subspaces, we get the (strict) inclusion

(H^{u} ∩ 𝔭_{1}) ∪ ⋯ ∪ (H^{u} ∩ 𝔭_{s}) ⊂ H^{u},

and so H^{u} must contain an element η′ which is not contained in 𝔭_{1} ∪ ⋯ ∪ 𝔭_{s}. This union is the set of all zero-divisors on E_{≥w}, hence η′ is regular on this module. In the proofs of Proposition 2.2 and Theorem 2.3 we may replace η by η′, thus proving the first statement.

Suppose |a_{i}| ≤ 2 for all i. Clearly, if each a_{i} is of degree 1 then the eventual period of M is 1, and the sequence of Betti numbers of M is eventually constant. If one of the generators is of degree 2, then the eventual period is either 1 or 2. If it is 2, then for some integer N ≥ 0 we have Ω_{Λ}^{i}(M) ≃ Ω_{Λ}^{i+2}(M) for i ≥ N. By taking the alternate sum of the k-dimensions in the exact sequence

0 → Ω_{Λ}^{i+2}(M) → P_{i+1} → P_{i} → Ω_{Λ}^{i}(M) → 0,

and recalling that this sum has to be zero, we get β_{i}(M) = β_{i+1}(M) for i ≥ N. Therefore the sequence of Betti numbers of M is eventually constant also in this case. □

The results we have proved in this section (and also the main result in the next section) depend on the assumption that Ext_{Λ}^{∗}(M, Λ / 𝔯) is finitely generated as a module over H, and therefore also as a module over HH^{∗}(Λ). As the action of the Hochschild cohomology ring on this module factors through the rings Ext_{Λ}^{∗}(M, M) and Ext_{Λ}^{∗}(Λ / 𝔯, Λ / 𝔯) via ring homomorphisms, the assumption forces Ext_{Λ}^{∗}(M, Λ / 𝔯) to be finitely generated as a module over both these latter rings. In which situations this happens has been studied in the commutative case. Let (A, 𝔪, k′) be a commutative Noetherian local ring, and N a finitely generated Amodule whose so-called complete intersection dimension (this was first defined in [3]) over A is finite (as happens for example when A is a complete intersection). Then L. Avramov and L.-C. Sun proved in [4] that the graded module Ext_{A}^{∗}(N, k′) is finitely generated over Ext_{A}^{∗}(k′, k′), whereas L. Avramov, V. Gasharov and I. Peeva proved in [3] that the module is also finitely generated over Ext_{A}^{∗}(N, N).

Question. When is Ext_{Λ}^{∗}(M, Λ / 𝔯) finitely generated over (one of) Ext_{Λ}^{∗}(M, M) and Ext_{Λ}^{∗}(Λ / 𝔯, Λ / 𝔯)?

It is not difficult to see that Ext_{Λ}^{∗}(M, Λ / 𝔯) is finitely generated over Ext_{Λ}^{∗}(M, M) whenever M is periodic; suppose Ω_{Λ}^{p}(M) = M and let μ denote the extension

0 → M → P_{p−1} → ⋯ → P_{0} → M → 0.

If n ≥ p, say n = qp + i where 0 ≤ i < p, then an element of Ext_{Λ}^{n}(M, Λ / 𝔯) can be written as fμ^{q} where f ∈ Ext_{Λ}^{i}(M, Λ / 𝔯). Since Ext_{Λ}^{i}(M, Λ / 𝔯) is finitely generated over k for 0 ≤ i < p, the result follows.

As to the finiteness of Ext_{Λ}^{∗}(M, Λ / 𝔯) as an H-module, a criterion for when this always happens was given in [10, Proposition 1.4]. This result is actually much stronger, as it states that Ext_{Λ}^{∗}(X, Y) is finite over H for all finite Λ-modules X and Y if and only if Ext_{Λ}^{∗}(Λ / 𝔯, Λ / 𝔯) is finite over H.

3. Reducing Complexity

We can use Theorem 2.3 to construct eventually periodic modules - and therefore also periodic modules - of infinite projective dimension (the modules having finite projective dimension are not very interesting in the context of eventual periodicity). This is done by considering an (almost) arbitrary module and from it obtain a new module whose minimal projective resolution behaves ”nicer”.

Let X = ⨁_{n=0}^{∞} X_{n} be a graded k-module of finite type. The rate of growth of X, denoted γ(X), is defined as

γ(X) = inf{t ∈ ℕ_{0} | ∃a ∈ ℝ such that ℓ_{k}(X_{n}) ≤ an^{t−1} for n ≫ 0},

and it may be finite or infinite (here ℕ_{0} denotes ℕ∪{0}). Now consider our module M with the minimal projective resolution (P_{i}, d_{i}). The complexity of M, denoted cx_{Λ} M, is defined as the rate of growth of the graded k-module ⨁_{n=0}^{∞} P_{n}, that is

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

Thus the complexity of M indicates how the sequence of Betti numbers behaves with respect to polynomial growth. From the definition we see that M has complexity 0 if and only if it has finite projective dimension, and that it has complexity less than or equal to 1 if and only if its sequence of Betti numbers is bounded. The main result of this section gives the existence of a new module whose complexity is exactly one less than that of M. The proof uses the identity

cx_{Λ} M = γ(Ext_{Λ}^{∗}(M, Λ / 𝔯)),

which provides a method for computing the complexity of a module. This identity follows from the identities (see the paragraphs following [5, Definition 5.3.3])

γ(Ext_{Λ}^{∗}(M, Λ / 𝔯)) = max{γ(Ext_{Λ}^{∗}(M, S)) | S simple Λ-module}
β_{n}(M) = ∑_{S simple} \frac{ℓ_{k}(P_{S})}{ℓ_{k}(Hom_{Λ}(S, S))} · ℓ_{k}(Ext_{Λ}^{n}(M, S)),

where P_{S} denotes the projective cover of the simple module S.

Given a homogeneous element η in HH^{∗}(Λ) of positive degree, we can interpret it as a Λ^{e}-linear map η: Ω_{Λ^{e}}^{|η|}(Λ) → Λ, where Ω_{Λ^{e}}^{i}(Λ) denotes the i’th syzygy in the minimal projective Λ^{e}-resolution of Λ. Let Q_{i} denote the i’th module in this resolution. By taking pushout we obtain the exact commutative diagram

[figure: 0; Ω_{Λ^{e}}^{|η|}(Λ); η; Q_{|η|−1}; Ω_{Λ^{e}}^{|η|−1}(Λ); 0; 0; Λ; K_{η}; Ω_{Λ^{e}}^{|η|−1}(Λ); 0]

of Λ^{e}-modules, whose bottom row we denote by ζ_{η}. Since Ω_{Λ^{e}}^{|η|−1}(Λ) is projective as a right (and left) Λ-module, the exact sequence ζ_{η} splits when considered as a sequence of right (and left) Λ-modules. Applying − ⊗_{Λ} M therefore gives the exact commutative diagram

[figure: 0; Ω_{Λ^{e}}^{|η|}(Λ) ⊗_{Λ} M; η⊗_{Λ}M; Q_{|η|−1} ⊗_{Λ} M; Ω_{Λ^{e}}^{|η|−1}(Λ) ⊗_{Λ} M; 0; 0; M; K_{η} ⊗_{Λ} M; Ω_{Λ^{e}}^{|η|−1}(Λ) ⊗_{Λ} M; 0]

of left Λ-modules, whose bottom row we denote by ζ_{η} ⊗_{Λ} M. Even though ζ_{η} splits when considered as a sequence of left Λ-modules, this is not necessarily the case for the new sequence. In fact, from [10, Proposition 2.2] we see that the sequence splits if and only if η annihilates Ext_{Λ}^{∗}(M, M).

The module K_{η} ⊗_{Λ}M is going to be the one having complexity one less than that of M. However, as the above shows, the element η cannot be chosen arbitrarily, for if ζ_{η} ⊗_{Λ} M splits then the complexity of K_{η} ⊗_{Λ} M equals that of M. To see this, note that in a split short exact sequence the complexity of the middle term equals the maximum of the complexities of the end terms, and that the complexities of the end term modules in ζ_{η} ⊗_{Λ} M are equal since Ω_{Λ^{e}}^{|η|−1}(Λ) ⊗_{Λ} M is a syzygy of M (it does not matter that Ω_{Λ^{e}}^{|η|−1}(Λ) ⊗_{Λ} M in general is not a syzygy in the minimal projective resolution of M, since projectively equivalent modules are of equal complexity). Thus we must pick an η not annihilating Ext_{Λ}^{∗}(M, M).

Let N be any Λ-module. Applying the functor Hom_{Λ}(−, N) to ζ_{η} ⊗_{Λ} M gives the long exact sequence

Hom_{Λ}(M, N) ⟶^{∂_{η}} Ext_{Λ}^{|η|}(M, N) → Ext_{Λ}^{1}(K_{η} ⊗_{Λ} M, N) → Ext_{Λ}^{1}(M, N) ⟶^{∂_{η}}

⋮

⟶^{∂_{η}} Ext_{Λ}^{i+|η|−1}(M, N) → Ext_{Λ}^{i}(K_{η} ⊗_{Λ} M, N) → Ext_{Λ}^{i}(M, N) ⟶^{∂_{η}}

⋮

for Ext (we shall refer to this sequence as ES(M, N, η)), where we have replaced Ext_{Λ}^{i}(Ω_{Λ^{e}}^{|η|−1}(Λ) ⊗_{Λ} M, N) by Ext_{Λ}^{i+|η|−1}(M, N) for i ≥ 1 using dimension shift. By

making these replacements, the new connecting homomorphism Ext_{Λ}^{i}(M, N) ⟶^{∂_{η}} Ext_{Λ}^{i+|η|}(M, N) is just multiplication by (−1)^{i}η, a fact which is vital for the proof of the main theorem. To see this, note that applying Hom_{Λ}(−, N) to the above commutative diagram gives rise to a commutative diagram of long exact sequences in Ext. Tracing the connecting homomorphism ∂_{η} then gives the desired result. Whenever we refer to the exact sequence ES(M, N, η), we shall drop the sign (−1)^{i} in front of the multiplication map induced by η, as it is of no relevance.

Note that if Fg holds, then Ext_{Λ}^{∗}(K_{η} ⊗_{Λ} M, Λ / 𝔯) is a finitely generated H module, regardless of the choice of η. To see this, consider the exact sequence

⨁_{i=1}^{∞} Ext_{Λ}^{i+|η|−1}(M, Λ / 𝔯) → ⨁_{i=1}^{∞} Ext_{Λ}^{i}(K_{η} ⊗_{Λ} M, Λ / 𝔯) → ⨁_{i=1}^{∞} Ext_{Λ}^{i}(M, Λ / 𝔯)

induced by ES(M, Λ / 𝔯, η). Both the end terms are finitely generated over H, hence so is the middle term because H is Noetherian. Since in addition Hom_{Λ}(K_{η} ⊗_{Λ} M, Λ / 𝔯) is finitely generated over k (which sits inside H^{0}), the claim follows.

In addition to giving us an important tool for computing the complexity of a module, the equality cx_{Λ} M = γ(Ext_{Λ}^{∗}(M, Λ / 𝔯)) implies that the modules we work with have finite complexity. For if Fg holds, then the rate of growth of Ext_{Λ}^{∗}(M, Λ / 𝔯) is not more than that of H, since it is a quotient of a finitely generated free Hmodule. Now as in the proof of Proposition 2.2, there is a finite set {a_{1}, …, a_{r}} in H of homogeneous elements of positive degrees, generating H as an algebra over H^{0}. By the Hilbert-Serre Theorem (see [5, Proposition 5.3.1]) and [5, Proposition 5.3.2], we have that γ(H) equals the order of the pole at t = 1 of a certain rational function g(t)/ ∏_{i=1}^{r}(1 − t^{|a_{i}|}), where g ∈ ℤ[t]. Hence the rate of growth of H is not more than r, implying cx_{Λ} M ≤ r.

The aim is now to pick an η such that the rate of growth of Ext_{Λ}^{∗}(K_{η} ⊗_{Λ} M, Λ / 𝔯) is one less than the rate of growth of Ext_{Λ}^{∗}(M, Λ / 𝔯). That such an element exists is a consequence of the following result.

Proposition 3.1. Let A = ⨁_{i=0}^{∞} A_{i} be a commutative Noetherian graded k-algebra of finite type over k, generated as an A_{0}-algebra by homogeneous elements a_{1}, …, a_{r} with |a_{i}| = n_{i} > 0. Let N = ⨁_{i=0}^{∞} N_{i} be a finitely generated graded A-module, and pick a homogeneous element η ∈ A as in Proposition 2.1. Let w ∈ ℕ be an integer such that η: N_{i} → N_{i+|η|} is injective for i ≥ w, define V_{i−w} to be the cokernel of this map, and denote by V the graded k-vector space ⨁_{i=0}^{∞} V_{i}. If γ(N) > 0, then γ(V) = γ(N) − 1.

Proof. Consider the Poincaré series P (N, t) = ∑_{i=0}^{∞} ℓ_{k}(N_{i})t^{i} of N. By the Hilbert-Serre Theorem we have

P (N, t) = \frac{f(t)}{∏_{i=1}^{r}(1 − t^{n_{i}})},

for some f(t) ∈ ℤ[t], and from [5, Proposition 5.3.2] we see that γ(N) equals the order of the pole of P (N, t) at t = 1. By assumption this integer is strictly greater than zero.

Now consider the exact sequences

0 → N_{i} ⟶^{η} N_{i+|η|} → V_{i−w} → 0

for i ≥ w. Taking k-module lengths we get ℓ_{k}(V_{i}) = ℓ_{k}(N_{i+w+|η|}) − ℓ_{k}(N_{i+w}) for i ≥ 0, giving

P (V, t) = ∑_{i=0}^{∞} ℓ_{k}(V_{i})t^{i} = ∑_{i=0}^{∞} ℓ_{k}(N_{i+w+|η|})t^{i} − ∑_{i=0}^{∞} ℓ_{k}(N_{i+w})t^{i}.

Multiplying this equation by t^{w+|η|} gives

t^{w+|η|}P (V, t) = ∑_{i=0}^{∞} ℓ_{k}(N_{i+w+|η|})t^{i+w+|η|} − t^{|η|} ∑_{i=0}^{∞} ℓ_{k}(N_{i+w})t^{i+w}
= (1 − t^{|η|})P (N, t) + g(t),

where g(t) is some polynomial in ℤ[t], and therefore

P (V, t) = \frac{(1 − t^{|η|})P (N, t)}{t^{w+|η|}} + \frac{g(t)}{t^{w+|η|}}
= \frac{(1 − t^{|η|})f(t)}{t^{w+|η|} ∏_{i=1}^{r}(1 − t^{n_{i}})} + \frac{g(t)}{t^{w+|η|}} .

Thus the order of the pole of P (V, t) at t = 1 is one less than that of P (N, t), showing γ(V) = γ(N) − 1. □

We now return to the setting given at the beginning of this section. With Proposition 3.1 at hand, the main theorem is merely a corollary.

Theorem 3.2 (Reducing Complexity). Assume Fg holds and that M does not have finite projective dimension. Then there exists a homogeneous element η ∈ H of positive degree such that cx_{Λ}(K_{η} ⊗_{Λ} M) = cx_{Λ} M − 1.

Proof. We use the previous proposition with A = H and N = Ext_{Λ}^{∗}(M, Λ / 𝔯). There is an integer w ∈ ℕ and a homogeneous element η ∈ H of positive degree such that the multiplication map Ext_{Λ}^{i}(M, Λ / 𝔯) ⟶^{η} Ext_{Λ}^{i+|η|}(M, Λ / 𝔯) is injective for i ≥ w. From the long exact sequence ES(M, Λ / 𝔯, η) we then get short exact sequences

0 → Ext_{Λ}^{i}(M, Λ / 𝔯) ⟶^{η} Ext_{Λ}^{i+|η|}(M, Λ / 𝔯) → Ext_{Λ}^{i+1}(K_{η} ⊗_{Λ} M, Λ / 𝔯) → 0

for i ≥ w. Using Proposition 3.1 we now get

cx_{Λ}(K_{η} ⊗_{Λ} M) = γ(Ext_{Λ}^{∗}(K_{η} ⊗_{Λ} M, Λ / 𝔯))
= γ( ⨁_{i=w+1}^{∞} Ext_{Λ}^{i}(K_{η} ⊗_{Λ} M, Λ / 𝔯))
= γ(Ext_{Λ}^{∗}(M, Λ / 𝔯)) − 1
= cx_{Λ} M − 1.
□

As a corollary we get a result which under certain conditions guarantees the existence of nonzero periodic modules. As mentioned in the beginning of the previous section, the existence of such modules is not obvious. For example, in [12], M. Ramras introduced a nonempty class of commutative Noetherian local rings called BNSI rings (short for ”Betti numbers strictly increase” rings), which are rings for which every non-free module has a strictly increasing sequence of Betti numbers. There exist a lot of finite dimensional algebras which are BNSI rings, for example regular local rings of dimension at least two modulo any positive power of the maximal ideal. Clearly, such rings cannot have nonzero periodic modules.

Corollary 3.3. Suppose Fg holds and that M has infinite projective dimension. Then Λ has a nonzero periodic module.

Proof. Let d denote the complexity of M (by assumption d > 0). If d = 1 then M is eventually periodic by Theorem 2.3, whereas if d > 1 the previous theorem provides homogeneous elements η_{1}, …, η_{d−1} ∈ H having the property

cx_{Λ}(K_{η_{i}} ⊗_{Λ} ⋯ ⊗_{Λ} K_{η_{1}} ⊗_{Λ} M) = d − i

for 1 ≤ i ≤ d − 1. Denote the module K_{η_{d−1}} ⊗_{Λ} ⋯ ⊗_{Λ} K_{η_{1}} ⊗_{Λ} M by X. This module has complexity one, and from the discussion prior to Proposition 3.1 we see that Ext_{Λ}^{∗}(X, Λ / 𝔯) is a finitely generated H-module. Using Theorem 2.3 once more, we get that X is eventually periodic.

Now if we take an eventually periodic Λ-module of infinite projective dimension, one of its syzygies is a nonzero periodic module. □

Remark. (i) The existence of a periodic Λ-module implies the existence of a periodic module having period 1; if M is isomorphic to Ω_{Λ}^{p}(M), where p ≥ 1, then the module ⨁_{i=0}^{p−1} Ω_{Λ}^{i}(M) is periodic of period 1 (here Ω_{Λ}^{0}(M) = M).

(ii) Suppose Λ is a BNSI ring. Then there cannot exist a non-free Λ-module M for which Fg holds, i.e. Ext_{Λ}^{∗}(M, Λ / 𝔯) is not finitely generated over any commutative Noetherian graded subalgebra of HH^{∗}(Λ); if such a module did exist, then the corollary would imply the existence of a nonzero periodic Λ-module.

4. Generating Periodicity

In this section we consider the case when the period of an eventually periodic module is ”detected” by a homogeneous element in the Hochschild cohomology ring. We start by recalling the group ring case.

Assume k is an algebraically closed field and let G be a finite group. A nonzero homogeneous element θ ∈ H^{|θ|}(G, k) = Ext_{kG}^{|θ|}(k, k) can be interpreted as a surjective kG-homomorphism θ: Ω_{kG}^{|θ|}(k) → k, the kernel of which it is customary to denote by L_{θ}. The cohomological variety of L_{θ} is easily computed; from [8, Lemma 2.3] we have that V_{G}(L_{θ}) equals V_{G}(θ), i.e. the set of maximal ideals in the group cohomology ring H^{·}(G, k) containing θ. Now let N be a finitely generated kG-module. If V_{G}(θ) ∩ V_{G}(N) = {0}, then from the above and the equality V_{G}(X ⊗_{k} Y) = V_{G}(X) ∩ V_{G}(Y), which holds for all finitely generated kG-modules X and Y, we get V_{G}(L_{θ} ⊗_{k} N) = {0}. This is equivalent to L_{θ} ⊗_{k} N being projective, and it follows from the proof of [5, Theorem 5.10.4] that N is isomorphic to Ω_{kG}^{|θ|}(N) ⊕ P, where P is a projective module. This gives

Ω_{kG}^{|θ|}(N) ≃ Ω_{kG}^{|θ|}(Ω_{kG}^{|θ|}(N) ⊕ P) ≃ Ω_{kG}^{2|θ|}(N),

showing Ω_{kG}^{|θ|}(N) is periodic and therefore that N is isomorphic to a direct sum of a periodic module and a projective module. If N contains no nonzero projective summand we must have N ≃ Ω_{kG}^{|θ|}(N), with the period of N dividing |θ|, and in this case the element θ is said to generate the periodicity of N.

Returning to the setting given in the previous sections, with k a commutative Artin ring, Λ an Artin k-algebra (assumed to be projective as a k-module) and M a finitely generated Λ-module, let η be a nonzero element in HH^{∗}(Λ) of positive degree. Instead of considering the kernel of the corresponding Λ^{e}-linear map η: Ω_{Λ^{e}}^{|η|}(Λ) → Λ (which is not necessarily surjective), we look at the pushout K_{η} and the tensor module K_{η} ⊗_{Λ} M, as we did in the last section.

Proposition 4.1. If the Λ-module K_{η} ⊗_{Λ} M has finite projective dimension, then M is eventually periodic with period dividing |η|.

Proof. Denote the projective dimension of K_{η} ⊗_{Λ} M by d. From the long exact sequence ES(M, Λ / 𝔯, η) we see that scalar multiplication by η induces kisomorphisms

Ext_{Λ}^{i}(M, Λ / 𝔯) ⟶^{η} Ext_{Λ}^{i+|η|}(M, Λ / 𝔯)

for i > d. Now let ⋯ → P_{1} ⟶^{d_{1}} P_{0} ⟶^{d_{0}} M → 0 be the minimal projective resolution of M, and recall the proof of Proposition 2.2. If we denote by ξ the image of η in Ext_{Λ}^{∗}(M, M), represented by a map Ω_{Λ}^{|η|}(M) ⟶^{ξ} M, then in the commutative diagram (obtained from the Comparison Theorem)

\xymatrix{⋯ \ar[r] & P_{|η|+i+1} \ar[r]^{d_{|η|+i+1}} \ar[d]^{ξ_{i+1}} & P_{|η|+i} \ar[r]^{d_{|η|+i}} \ar[d]^{ξ_{i}} & ⋯ \ar[r] & P_{|η|} \ar[r] \ar[d]^{ξ_{0}} & Ω_{Λ}^{|η|}(M) \ar[r] \ar[d]^{ξ} & 0 \\ ⋯ \ar[r] & P_{i+1} \ar[r]^{d_{i+1}} & P_{i} \ar[r]^{d_{i}} & ⋯ \ar[r] & P_{0} \ar[r]^{d_{0}} & M \ar[r] & 0}

we have that ξ_{i} is surjective for i > d. Applying Hom_{Λ}(−, Λ / 𝔯) to the exact sequences

0 → Ker ξ_{i} → P_{|η|+i} ⟶^{ξ_{i}} P_{i} → 0

(for i > d) then gives Hom_{Λ}(Ker ξ_{i}, Λ / 𝔯) = 0, hence Ker ξ_{i} = 0. This shows ξ_{i} is bijective for i > d, and as in the proof of Theorem 2.3 we see that Ω_{Λ}^{i}(M) ≃ Ω_{Λ}^{i+|η|}(M) for i > d. □

The proposition motivates the following definition of an element generating the periodicity of a module:

Definition 4.2. Let M be a Λ-module of infinite projective dimension. A nonzero homogeneous element η ∈ HH^{∗}(Λ) of positive degree generates the periodicity of M if the Λ-module K_{η} ⊗_{Λ} M has finite projective dimension.

Note that in the special case when K_{η} ⊗_{Λ} M is projective, all but the end terms in the exact sequence

0 → M → K_{η} ⊗_{Λ} M → Q_{|η|−2} ⊗_{Λ} M → ⋯ → Q_{0} ⊗_{Λ} M → M → 0

are projective, hence M is isomorphic to Ω_{Λ}^{|η|}(M) ⊕ P, where P is a projective Λmodule. As in the group ring case above, this implies that Ω_{Λ}^{|η|}(M) ≃ Ω_{Λ}^{2|η|}(M), and therefore M is isomorphic to the direct sum of a periodic module and a projective module. In particular, if M contains no nonzero projective summand then it is periodic.

Also note that, whenever there exists an element η generating the periodicity of M, for every Λ-module N we see from the long exact sequence ES(M, N, η) that multiplication

Ext_{Λ}^{i}(M, N) ⟶^{η} Ext_{Λ}^{i+|η|}(M, N)

is an isomorphism for i ≫ 0. From the proof of the previous proposition we see that the converse is also true, hence the following result.

Proposition 4.3. A nonzero homogeneous element η ∈ HH^{∗}(Λ) generates the periodicity of M if and only if scalar multiplication by η induces k-isomorphisms

Ext_{Λ}^{i}(M, N) ⟶^{η} Ext_{Λ}^{i+|η|}(M, N)

for every Λ-module N and i ≫ 0.

As an immediate corollary we obtain the following finite generation result.

Corollary 4.4. If there exists an element η generating the periodicity of M, then Ext_{Λ}^{∗}(M, N) is a finitely generated HH^{∗}(Λ)-module for every Λ-module N. In particular Ext_{Λ}^{∗}(M, M) and Ext_{Λ}^{∗}(M, Λ / 𝔯) are finitely generated.

Proof. Denoting pd_{Λ}(K_{η} ⊗_{Λ} M) by d we see that the k-bases of Hom_{Λ}(M, N), Ext_{Λ}^{1}(M, N), …, Ext_{Λ}^{|η|+d}(M, N) together form a generating set. □

Remark. We actually obtain a stronger result, namely that for every Λ-module N the k-module Ext_{Λ}^{∗}(M, N) is a finitely generated module over the commutative Noetherian graded subalgebra of HH^{∗}(Λ) generated by HH^{0}(Λ) and η (in particular Fg holds). As the element η is not nilpotent, this subalgebra is actually the polynomial ring in one variable over HH^{0}(Λ) (the center of Λ).

It is natural to ask whether an element generating the periodicity of M always exists when M is a periodic module. The following two examples show that this is not the case. In the first example the algebra is selfinjective, whereas in the second it is commutative local.

Example 4.5. Let k be a field, and consider the 4-dimensional algebra

Λ = k⟨x, y⟩/(x^{2}, xy + qyx, y^{2})

where 0 ≠ q ∈ k is not a root of unity. Let M be any 2-dimensional k-vector space having a basis {u, v}, say. Straightforward computation shows that defining

xu = 0, xv = 0
yu = v, yv = 0

gives a Λ-module structure on M (this module was also studied in [14]). Define a Λ-linear map p: Λ → M by 1 ↦ u. This is a surjective map, and as a vector space Ker p has {x, xy} as a basis. Therefore Ker p is contained in the radical of Λ, showing p is the projective cover of M. Define a k-linear map f: M → Ker p by u ↦ x and v ↦ xy. This is an isomorphism, and direct computation shows it is Λ-linear. Hence M ≃ Ω_{Λ}^{1}(M), and so M is periodic of period 1.

Suppose 0 ≠ η ∈ HH^{|η|}(Λ) is an element generating the periodicity of M. Then from Proposition 4.3 scalar multiplication by η induces isomorphisms Ext_{Λ}^{i}(M, M) ≃ Ext_{Λ}^{i+|η|}(M, M) for i ≫ 0. Therefore multiplication by any power of η also induces isomorphisms. However, from [7] we have that HH^{n}(Λ) = 0 for n ≥ 3, in particular η is nilpotent. Hence Ext_{Λ}^{i}(M, M) = 0 for i ≫ 0, and since for each i > 1 we have

Ext_{Λ}^{i}(M, M) ≃ Ext_{Λ}^{1}(Ω_{Λ}^{i−1}(M), M) ≃ Ext_{Λ}^{1}(M, M),

we get Ext_{Λ}^{1}(M, M) = 0. Therefore M is projective, since it is periodic of period 1, and this is a contradiction.

Example 4.6. Let k be an algebraically closed field of characteristic different from 2, and let k[x_{1}, x_{2}, x_{3}, x_{4}] be the polynomial ring in four variables over k. Denote by R the finite dimensional local 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_{1}x_{2} − x_{3}^{2}, x_{1}x_{3} − x_{2}x_{4},
x_{1}x_{4}, x_{2}^{2} + x_{3}x_{4}, x_{2}x_{3}, x_{4}^{2}.

Define two R-endomorphisms ϕ, ψ: R^{2} → R^{2} by the matrices

ϕ = ( \begin{matrix}x_{1} & x_{3} \\ x_{2} & x_{4} \end{matrix} ) , ψ = ( \begin{matrix}x_{4} & −x_{3} \\ −x_{2} & x_{1} \end{matrix} ) ,

and let M = Im ψ. In [2] it is shown that Im ψ = Ker ϕ and Im ϕ = Ker ψ, hence M is periodic of period 2 and has the minimal free resolution

⋯ → R^{2} ⟶^{ϕ} R^{2} ⟶^{ψ} R^{2} ⟶^{ϕ} R^{2} ⟶^{ψ} M → 0.

As a k-vector space M is 8-dimensional, and the elements

v_{1} = ( \begin{matrix}x_{4} \\ −x_{2} \end{matrix} ) , v_{2} = ( \begin{matrix}−x_{3} \\ x_{1} \end{matrix} ) ,

v_{3} = ( \begin{matrix}0 \\ x_{1}x_{2} \end{matrix} ) , v_{4} = ( \begin{matrix}x_{1}x_{3} \\ 0 \end{matrix} ) , v_{5} = ( \begin{matrix}x_{1}x_{3} \\ x_{3}x_{4} \end{matrix} ) , v_{6} = ( \begin{matrix}x_{3}x_{4} \\ 0 \end{matrix} ) , v_{7} = ( \begin{matrix}−x_{1}x_{2} \\ x_{1}x_{3} \end{matrix} ) , v_{8} = ( \begin{matrix}0 \\ x_{1}x_{3} \end{matrix} )

form a basis. Furthermore, as a graded R-module M consists of two nonzero graded components M_{0} and M_{1}, with {v_{1}, v_{2}} as a basis for M_{0} and {v_{3}, …, v_{8}} as a basis for M_{1}. The elements v_{1}, v_{2} generate M as an R-module.

Denote by μ the extension 0 → M → R^{2} ⟶^{ϕ} R^{2} ⟶^{ψ} M → 0 in Ext_{R}^{2}(M, M). Then for all n ≥ 0 every element of Ext_{R}^{2n}(M, M) is of the form fμ^{n} for some f ∈ Hom_{R}(M, M). Now suppose 0 ≠ η ∈ HH^{|η|}(R) is an element generating the periodicity of M. Then |η| is an even number (since M is of period 2, i.e. Ω_{R}^{1}(M) ≄ M), and so the image of η in Ext_{R}^{∗}(M, M) under the ring homomorphism −⊗_{R} M can be written as fμ^{n}, where f is an R-endomorphism of M and n = |η|/2. Since M = M_{0} ⊕ M_{1} as a graded module, we can write f as h + g, where h and g are homogeneous endomorphisms of degree 0 and 1, respectively (see the paragraphs following [6, Theorem 1.5.8]). Therefore the element fμ^{n} is the sum of two elements hμ^{n} and gμ^{n} of different internal degrees in Ext_{R}^{|η|}(M, M). Since the map − ⊗_{R} M: HH^{|η|}(R) → Ext_{R}^{|η|}(M, M) preserves internal grading we must have that hμ^{n} and gμ^{n} both lie in the image of − ⊗_{R} M.

Suppose h is nonzero. Since the degree of this map is zero, we must have that h(v_{1}) = c_{1}v_{1} + c_{2}v_{2} and h(v_{2}) = c_{3}v_{1} + c_{4}v_{2} for elements c_{i} ∈ k. As v_{6} = x_{3}v_{1} = −x_{4}v_{2} and h is R-linear, direct computation of h(v_{6}) gives

c_{1}v_{6} + c_{2}v_{7} = c_{3}v_{8} + c_{4}v_{6}.

Thus c_{2} = c_{3} = 0 and c_{1} = c_{4}, and we see that h is nothing but a scalar c ∈ k times the identity on M. Therefore the element cμ^{n} belongs to the image of − ⊗_{R} M, and multiplying with the inverse of c we get that μ^{n} also lies in this image. From [15, Corollary 1.3] we see that every element of Im(− ⊗_{R} M) belongs to the graded center of Ext_{R}^{∗}(M, M), and so μ^{n}, which is of even degree, is a central element. We show that this is not the case.

An element of Ext_{R}^{1}(M, M) can be considered as an R-linear map θ: R^{2} → M such that θ ∘ ψ = 0. Define three such elements θ_{1}, θ_{2}, θ_{3} by

θ_{1} : ( \begin{matrix}1 \\ 0 \end{matrix} ) ↦ v_{6}, ( \begin{matrix}0 \\ 1 \end{matrix} ) ↦ 0,
θ_{2} : ( \begin{matrix}1 \\ 0 \end{matrix} ) ↦ 0, ( \begin{matrix}0 \\ 1 \end{matrix} ) ↦ v_{4},
θ_{3} : ( \begin{matrix}1 \\ 0 \end{matrix} ) ↦ 0, ( \begin{matrix}0 \\ 1 \end{matrix} ) ↦ v_{5}.

It is easy to check that these elements are linearly independent, and when they are multiplied on the right by any power of μ they stay linearly independent. Direct computation (lifting maps along the minimal free resolution of M) gives

μθ_{1} = (θ_{1}/2 − θ_{2} + θ_{3}/2)μ
μθ_{2} = θ_{2}μ
μθ_{3} = (−θ_{1}/2 − θ_{2} + 3θ_{3}/2)μ,

and so by induction we get μ^{n}θ_{1} = ([1 − n/2]θ_{1} − nθ_{2} + [n/2]θ_{3})μ^{n} for all n ≥ 1. This shows that μ^{n} is not a central element, implying h must be zero.

As a consequence, the image of η in Ext_{R}^{∗}(M, M) is gμ^{n}, where g is a homogeneous degree one R-endomorphism of M. In particular, since M only lives in two degrees, we must have g^{2} = 0. From Proposition 4.3, the element η induces isomorphisms

Ext_{Λ}^{i}(M, M) ⟶^{η} Ext_{Λ}^{i+|η|}(M, M)

for i ≫ 0, and in particular there is an i > 0 such that η · gμ^{i} ≠ 0 (because gμ^{n} ≠ 0 implies gμ^{i} ≠ 0, since the element μ^{j} is not a zerodivisor for any j). But gμ^{n}, being the image of η, is a central element, giving η · gμ^{i} = (gμ^{n})gμ^{i} = g(gμ^{n})μ^{i} = 0, a contradiction. Therefore there does not exist a nonzero homogeneous element of positive degree in HH^{∗}(R) generating the periodicity of M.

These examples show that an element generating the periodicity of an eventually periodic module does not exist in general. However, when dealing with ”suitably nice” algebras, such an element always exists. Suppose M is an eventually periodic Λ-module satisfying Fg. Then Theorem 3.2 implies the existence of a homogeneous element η ∈ HH^{∗}(Λ) of positive degree with the property that K_{η} ⊗_{Λ} M has finite projective dimension over Λ, and so this η generates the periodicity of M.

The following example shows that it may be difficult to decide whether or not there exists an element generating the periodicity of a module, even when the algebra is commutative local and selfinjective.

Example 4.7. Let k be a field of characteristic different from 2, and let k[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}] be the polynomial ring in five variables over k. Denote by R the finite dimensional local k-algebra k[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}]/ 𝔞, where 𝔞 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_{3}^{2} − x_{4}^{2},
x_{1}x_{3} + x_{2}x_{3}, x_{1}x_{4} + x_{2}x_{4}, x_{3}^{2} − x_{2}x_{5} + x_{1}x_{5}.

It is shown in [11] that R is graded selfinjective with Hilbert series 1 + 5t + 5t^{2} + t^{3}, and that the R-endomorphism d: R^{2} → R^{2} defined by

d = ( \begin{matrix}x_{1} & x_{3} + x_{4} \\ 0 & x_{2} \end{matrix} )

satisfies Im d = Ker d. Letting M = Im d, we have that M is periodic of period 1, with minimal free resolution

⋯ → R^{2} ⟶^{d} R^{2} ⟶^{d} R^{2} ⟶^{d} M → 0.

Denote by μ the extension 0 → M → R^{2} ⟶^{d} M → 0 in Ext_{R}^{1}(M, M). Then for all n ≥ 0 every element of Ext_{R}^{n}(M, M) is of the form fμ^{n} for some endomorphism f of M.

Now suppose 0 ≠ η ∈ HH^{|η|}(R) is an element generating the periodicity of M. Then the image of η in Ext_{R}^{∗}(M, M) can be written as fμ^{|η|}, and as in the previous example this implies that μ^{|η|} belongs to the image of − ⊗_{R} M: HH^{|η|}(R) → Ext_{R}^{|η|}(M, M) (in this case M is graded and ”lives” in three degrees, i.e. M = M_{0} ⊕ M_{1} ⊕ M_{2}). In particular μ^{|η|} belongs to the graded center of Ext_{R}^{∗}(M, M).

An element of Ext_{R}^{i}(M, M) can be considered as an R-linear map θ: R^{2} → M with the property θ∘d = 0. In our case we have that Ext_{R}^{i}(M, M) is 12-dimensional, and the maps

\begin{matrix}θ_{1} : e_{1} ↦ ( \begin{matrix}x_{1} \\ 0 \end{matrix} ) , e_{2} ↦ ( \begin{matrix}x_{3}+x_{4} \\ x_{2} \end{matrix} ) & θ_{7} : e_{1} ↦ 0, e_{2} ↦ ( \begin{matrix}0 \\ x_{2}x_{5} \end{matrix} ) \\ θ_{2} : e_{1} ↦ ( \begin{matrix}x_{1}x_{2} \\ 0 \end{matrix} ) , e_{2} ↦ 0 & θ_{8} : e_{1} ↦ 0, e_{2} ↦ ( \begin{matrix}x_{2}x_{5} \\ −x_{1}x_{3} \end{matrix} ) \\ θ_{3} : e_{1} ↦ ( \begin{matrix}x_{1}x_{5} \\ 0 \end{matrix} ) , e_{2} ↦ 0 & θ_{9} : e_{1} ↦ 0, e_{2} ↦ ( \begin{matrix}x_{2}x_{5} \\ −x_{1}x_{4} \end{matrix} ) \\ θ_{4} : e_{1} ↦ ( \begin{matrix}0 \\ x_{1}x_{2} \end{matrix} ) , e_{2} ↦ 0 & θ_{10} : e_{1} ↦ ( \begin{matrix}x_{1}x_{3}−x_{1}x_{4} \\ 0 \end{matrix} ) , e_{2} ↦ 0 \\ θ_{5} : e_{1} ↦ 0, e_{2} ↦ ( \begin{matrix}0 \\ x_{1}x_{2}x_{5} \end{matrix} ) & θ_{11} : e_{1} ↦ 0, e_{2} ↦ ( \begin{matrix}x_{1}x_{3} \\ 0 \end{matrix} ) \\ θ_{6} : e_{1} ↦ 0, e_{2} ↦ ( \begin{matrix}0 \\ x_{1}x_{2} \end{matrix} ) & θ_{12} : e_{1} ↦ ( \begin{matrix}0 \\ x_{1}x_{4}−x_{1}x_{3} \end{matrix} ) , e_{2} ↦ 0 \end{matrix}

represent a basis (here e_{1} = ( \begin{matrix}1 \\ 0 \end{matrix} ) and e_{2} = ( \begin{matrix}0 \\ 1 \end{matrix} )). Direct computation shows that μθ_{j} = θ_{j}μ for j = 1, … 10, while μθ_{j} = −θ_{j}μ for j = 11, 12, and therefore μ^{n} cannot belong to the graded center of Ext_{R}^{∗}(M, M) when n is odd. However, the element μ^{2} does belong to the graded center, hence it is possible that a homogeneous element of even degree in HH^{|η|}(R) generates the periodicity of M.

We end with a result providing a positive answer to the following natural question: if the product or the sum of two homogeneous elements in the Hochschild cohomology ring generates the periodicity of a module, do the elements themselves (or one of them) generate the periodicity? As a corollary, we obtain a result analogous to [5, Corollary 5.10.6], which states that if the group cohomology ring of a finite group G is finitely generated over a subring generated by homogeneous elements x_{1}, …, x_{t}, and N is a periodc kG-module (where k is an algebraically closed field), then one of the x_{i} generates the periodicity of N.

Proposition 4.8. Let η_{1} ∈ HH^{|η_{1}|}(Λ) and η_{2} ∈ HH^{|η_{2}|}(Λ) be nonzero homogeneous elements of positive degrees.

(i) If 0 ≠ η_{1}η_{2} generates the periodicity of M, then η_{1} and η_{2} both generate the periodicity.

(ii) Suppose Λ is Gorenstein and k is an algebraically closed field. If |η_{1}| = |η_{2}|, and the sum η_{1} + η_{2} generates the periodicity of M, then either η_{1} or η_{2} generates the periodicity.

Proof. (i) From Proposition 4.3 we have that multiplication

Ext_{Λ}^{i}(M, Λ / 𝔯) ⟶^{η_{1}η_{2}} Ext_{Λ}^{i+|η_{1}|+|η_{2}|}(M, Λ / 𝔯)

is a k-isomorphism for i ≫ 0, and since HH^{∗}(Λ) is graded commutative the element η_{2}η_{1} also induces isomorphisms. Therefore the map

Ext_{Λ}^{i}(M, Λ / 𝔯) ⟶^{η_{2}} Ext_{Λ}^{i+|η_{2}|}(M, Λ / 𝔯)

is injective for i ≫ 0, whereas the map

Ext_{Λ}^{i+|η_{1}|}(M, Λ / 𝔯) ⟶^{η_{2}} Ext_{Λ}^{i+|η_{1}|+|η_{2}|}(M, Λ / 𝔯)

is surjective for i ≫ 0. It follows from the long exact sequence ES(M, Λ / 𝔯, η) that Ext_{Λ}^{i+|η_{1}|+1}(K_{η_{2}} ⊗_{Λ} M, Λ / 𝔯) = 0 for i ≫ 0, hence K_{η_{2}} ⊗_{Λ} M has finite projective dimension. Similarly K_{η_{1}} ⊗_{Λ} M has finite projective dimension.

(ii) Denote η_{1} + η_{2} by η, and let H be the commutative Noetherian graded subalgebra of HH^{∗}(Λ) generated by HH^{0}(Λ) and η. As mentioned in the remark following Corollary 4.4, the graded k-vector space Ext_{Λ}^{∗}(M, Λ / 𝔯) is a finitely generated H-module, and consequently Fg holds. We may therefore apply the theory of support varieties from [10].

Recall that if X is a finitely generated Λ-module, then the support variety V_{H}(X) is defined to be the variety in MaxSpec H defined by the annihilator of Ext_{Λ}^{∗}(X, Λ / 𝔯) in H. Denote the annihilator ideal associated to M by 𝔞. Since cx_{Λ} M = 1 and cx_{Λ}(K_{η} ⊗_{Λ} M) = 0, it follows from [10, Proposition 1.1] that the variety V_{H}(M) = V_{H}(𝔞) is one dimensional, whereas V_{H}(K_{η} ⊗_{Λ} M) is zero dimensional. By [10, Proposition 3.3] the latter variety equals V_{H}(η) ∩ V_{H}(𝔞) = V_{H}(⟨η⟩ + 𝔞) (note that in order to apply [10, Proposition 3.3] both Fg and the assumption that Λ is Gorenstein is needed, as a substitute for the stronger finite generation hypothesis used in that paper).

Suppose neither of the inclusions \sqrt{𝔞} ⊆ \sqrt{⟨η_{i}⟩ + 𝔞} is strict for i = 1, 2. Then

\sqrt{𝔞} = \sqrt{⟨η_{1}⟩ + 𝔞} = \sqrt{⟨η_{1}⟩ + \sqrt{𝔞}} = \sqrt{⟨η_{1}⟩ + \sqrt{⟨η_{2}⟩ + 𝔞}} = \sqrt{⟨η_{1}⟩ + ⟨η_{2}⟩ + 𝔞},

and since

\sqrt{𝔞} ⊆ \sqrt{⟨η⟩ + 𝔞} ⊆ \sqrt{⟨η_{1}⟩ + ⟨η_{2}⟩ + 𝔞}

we get the equality \sqrt{𝔞} = \sqrt{⟨η⟩ + 𝔞}. Since the variety defined by any ideal equals that defined by the radical of the ideal, we get V_{H}(M) = V_{H}(K_{η} ⊗_{Λ} M), which is impossible. Therefore \sqrt{𝔞} ⊂ \sqrt{⟨η_{i}⟩ + 𝔞} (strict inclusion) at least for one j ∈ \{1, 2\}, and as k is algebraically closed and H is a finitely generated k-algebra, Hilbert’s Nullstellensatz gives the strict inclusion V_{H}(K_{η_{j}} ⊗_{Λ} M) ⊂ V_{H}(M). The variety V_{H}(K_{η_{j}} ⊗_{Λ} M) is then zero dimensional, and consequently K_{η_{j}} ⊗_{Λ} M has finite projective dimension. □

Corollary 4.9. Suppose Λ is Gorenstein and k is an algebraically closed field, and that HH^{∗}(Λ) is generated over a subalgebra by homogeneous elements x_{1}, …, x_{t}. If M is an eventually periodic module and there exists an element in HH^{∗}(Λ) generating the periodicity, then one of the x_{i} generates the periodicity. In particular, the period of M divides the degree of one of the x_{i}.

Remark. Proposition 4.8 enables us to strengthen Theorem 2.5 in the case when Λ is Gorenstein and k is an algebraically closed field; when Fg holds and H is generated as an algebra over H^{0} by homogeneous elements a_{1}, …, a_{r} of positive degrees, then if M has bounded Betti numbers it is eventually periodic with period dividing one of |a_{i}|.

Acknowledgements

I would like to thank The Department of Mathematics at the University of Nebraksa, Lincoln, for their hospitality during my visit there in the spring of 2004. In particular I thank my host Luchezar Avramov and also Srikanth Iyengar for fruitful discussions and comments. Thanks also to my supervisor Øyvind Solberg and to the Norwegian Research Council for financial support.

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

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