RELATIVE SUPPORT VARIETIES

PETTER ANDREAS BERGH & ØYVIND SOLBERG

Abstract. We define relative support varieties with respect to some fixed module over a finite dimensional algebra. These varieties share many of the standard properties of classical support varieties. Moreover, when introducing finite generation conditions on cohomology, we show that relative support varieties contain homological information on the modules involved. As an application, we provide a new criterion for a selfinjective algebra to be of wild representation type.

1. Introduction

Support varieties for modules over a given algebra are defined in terms of the maximal ideal spectrum of some commutative graded ring of cohomology operators, operators which act centrally on the cohomology groups of the algebra. For group algebras of finite groups, or, more generally, for finite dimensional cocommutative Hopf algebras, this role is played by the cohomology ring of the algebra (cf. [4], [7], [12], [14]). For commutative local complete intersections, one uses the polynomial ring of Eisenbud operators (cf. [2], [3]). In all these cases, the ring of cohomology operators is Noetherian, and all the cohomology groups of the algebra are finitely generated as modules. Consequently, the theory of support varieties over these rings is very powerful, in that the variety of a module contains a lot of homological information on the module itself.

As shown in [18], for a finite dimensional algebra, the Hochschild cohomology ring, with its maximal ideal spectrum, is a natural candidate as a ring of central cohomology operators. However, in general this ring is not Noetherian, and the cohomology groups of the algebra are not always finitely generated modules. But, as shown in [10], when the Hochschild cohomology ring is Noetherian and all the cohomology groups are finitely generated, then one obtains a support variety theory very much like in the classical cases. It is therefore important to establish which finite dimensional algebras have “nice” Hochschild cohomology rings. For quantum complete intersections, this has been solved (cf. [11], [6]).

In this paper, we define relative support varieties with respect to a fixed module. These are defined in terms of the maximal ideal spectrum of some commutative graded subalgebra of the Ext-algebra of the module. As one would expect, these varieties share many of the same properties of “ordinary” support varieties, such as the standard behavior on exact sequences etc. Moreover, when we introduce finite generation conditions, then the relative support varieties contain homological information on the modules involved, just as in the classical case.

As an application, we provide a new criterion for a finite dimensional selfinjective algebra to be of wild representation type. Namely, we show that if there exists a module whose Ext-algebra is “large” enough, Noetherian and finitely generated as a module over its center, then the algebra is wild. This generalizes Farnsteiner’s theorem, which states that the complexity of every module of a tame block of a finite group scheme is at most two (cf. [13]).

2000 Mathematics Subject Classification. 16E05, 16E30, 16G60.

Key words and phrases. Relative support varieties, complexity, wild algebras.

The first author was supported by NFR Storforsk grant no. 167130.

2. Relative support varieties

Throughout this paper, we let k be a field and Λ a finite dimensional k-algebra. We denote by mod Λ the category of finitely generated left Λ-modules, and we fix a module M ∈ mod Λ whose higher self-extensions do not all vanish. Whenever we deal with Λ-modules, we assume they belong to mod Λ. Finally, for two Λ-modules X and Y, we denote by Ext_{Λ}^{∗}(X, Y) the direct sum ⊕_{i=0}^{∞} Ext_{Λ}^{i}(X, Y).

Consider the Ext-algebra Ext_{Λ}^{∗}(M, M) of M, in which multiplication is given by the Yoneda product. Then for any Λ-module N, the graded k-vector space Ext_{Λ}^{∗}(M, N) is a graded right Ext_{Λ}^{∗}(M, M)-module, whereas Ext_{Λ}^{∗}(N, M) is a graded left Ext_{Λ}^{∗}(M, M)-module. Moreover, a Λ-homomorphism N_{1} ⟶^{f} N_{2} induces homomorphisms

Ext_{Λ}^{∗}(M, N_{1}) ⟶^{f_{∗}} Ext_{Λ}^{∗}(M, N_{2})
Ext_{Λ}^{∗}(N_{2}, M) ⟶^{f^{∗}} Ext_{Λ}^{∗}(N_{1}, M)

of right and left Ext_{Λ}^{∗}(M, M)-modules. The homomorphism f_{∗} is given as follows: given a homogeneous element

η: 0 → N_{1} → X_{n} → ⋯ → X_{1} → M → 0

in Ext_{Λ}^{∗}(M, N_{1}), the element f_{∗}(η) is the lower exact sequence in the diagram

[figure: 0; N_{1}; f; X_{n}; X_{n−1}; ⋯; X_{1}; M; 0; 0; N_{2}; K; X_{n−1}; ⋯; X_{1}; M; 0]

in which the module K is a pushout. Similarly, the homomorphism f^{∗} is induced by pullback along f. It follows immediately that f_{∗} and f^{∗} are well defined homomorphisms of right and left Ext_{Λ}^{∗}(M, M)-modules, respectively.

The relative support varieties are defined with respect to some commutative graded subalgebra of Ext_{Λ}^{∗}(M, M), and therefore we now fix such a subalgebra.

Assumption. Fix a commutative graded subalgebra H ⊆ Ext_{Λ}^{∗}(M, M) such that H_{0} is a local ring.

As mentioned, the relative support varieties to be defined are defined with respect to this graded subalgebra H. The assumption that H_{0} is local is made in order to get a nice characterization of the trivial varieties. This assumption is not very restrictive. For example, when we introduce finiteness assumptions later, then we may actually take H to be a polynomial ring over k, so that H_{0} is just k itself. Moreover, the following result shows that when M is an indecomposable module, then H_{0} is automatically a local ring.

Lemma 2.1. If M is indecomposable, then H_{0} is automatically a local ring.

Proof. Since H_{0} is a finite dimensional commutative k-algebra, the factor algebra H_{0}/ rad H_{0} is a product K_{1} × ⋯ × K_{t} of fields. If t ≥ 2, then this factor algebra contains nontrivial idempotents, and these lift to H_{0}. But H_{0}, being a subalgebra of End_{Λ}(M), cannot contain any nontrivial idempotent, hence t = 1. Therefore the radical of H_{0} is a maximal ideal. □

Since we have assumed that H_{0} is a local ring, the graded ideal rad H_{0} ⊕H_{1} ⊕⋯ is maximal in H (and it is the only maximal graded ideal). We denote this ideal by 𝔪_{gr}(H).

We now define a relative support variety theory for Λ-modules, in which the commutative graded ring H is the coordinate ring. Given a Λ-module N, denote by Ann_{H}^{i} N the annihilator of Ext_{Λ}^{∗}(M, N) in H, and by Ann_{H}^{p} N the annihilator of Ext_{Λ}^{∗}(N, M) in H. As the annihilator of any graded module over any graded ring is graded, the ideals Ann_{H}^{i} N and Ann_{H}^{p} N are graded ideals of H. We define the injective and projective support varieties of N with respect to H as

V_{H}^{i}(N) =^{def} {𝔪 ∈ MaxSpec H | Ann_{H}^{i} N ⊆ 𝔪},
V_{H}^{p}(N) =^{def} {𝔪 ∈ MaxSpec H | Ann_{H}^{p} N ⊆ 𝔪},

respectively, where MaxSpec H denotes the set of maximal ideals of H. Note that Ann_{H}^{i} N and Ann_{H}^{p} N are contained in 𝔪_{gr}(H), hence 𝔪_{gr}(H) is trivially a point in both V_{H}^{i}(N) and V_{H}^{p}(N). We call a variety trivial if it only contains this point.

In the following result we record some elementary facts on relative varieties. Whenever we write V_{H}^{∗}(N) or Ann_{H}^{∗} N and make a statement, it is to be understood that the statement holds in both the injective and projective cases. Furthermore, denote by M^{⊥} the category of all Λ-modules X such that Ext_{Λ}^{n}(M, X) = 0 for n ≫ 0, and by ^{⊥}M the category of all Λ-modules Y such that Ext_{Λ}^{n}(Y, M) = 0 for n ≫ 0.

Proposition 2.2. For Λ-modules N, N_{1}, N_{2}, N_{3}, the following hold:

(i) V_{H}^{∗}(M) = MaxSpec H.

(ii) If N ∈ M^{⊥}, then V_{H}^{i}(N) is trivial. In particular, this holds if the injective

dimension of N is finite. (iii) If N ∈ ^{⊥}M, then V_{H}^{p}(N) is trivial. In particular, this holds if the projective

dimension of N is finite. (iv) For any exact sequence

0 → N_{1} → N_{2} → N_{3} → 0,

the inclusion V_{H}^{∗}(N_{u}) ⊆ V_{H}^{∗}(N_{v}) ∪ V_{H}^{∗}(N_{w}) holds whenever {u, v, w} = {1, 2, 3}.

(v) If N = N_{1} ⊕ N_{2}, then V_{H}^{∗}(N) = V_{H}^{∗}(N_{1}) ∪ V_{H}^{∗}(N_{2}).

Proof. Since H is a subalgebra of Ext_{Λ}^{∗}(M, M), no nonzero element of H can annihilate Ext_{Λ}^{∗}(M, M). Therefore Ann_{H}^{∗} M = 0, and this shows (i).

To prove (ii), note that if Ext_{Λ}^{n}(M, N) = 0 for n ≫ 0 and η is a homogeneous element in H of positive degree, then some power of η belongs to Ann_{H}^{i} N. Moreover, if θ is any element of rad H_{0}, then it is nilpotent, and therefore some power of θ also belongs to Ann_{H}^{i} N. Consequently V_{H}^{i}(N) = {𝔪_{gr}(H)}. This proves (ii), and the proof of (iii) is similar.

As for (iv), we prove only the inclusion V_{H}^{i}(N_{3}) ⊆ V_{H}^{i}(N_{1}) ∪ V_{H}^{i}(N_{2}); the other inclusions are proved analogously. The given short exact sequence induces an exact sequence

Ext_{Λ}^{∗}(M, N_{2}) → Ext_{Λ}^{∗}(M, N_{3}) → Ext_{Λ}^{∗+1}(M, N_{1})

of right Ext_{Λ}^{∗}(M, M)-modules, from which we obtain Ann_{H}^{i} N_{2} · Ann_{H}^{i} N_{1} ⊆ Ann_{H}^{i} N_{3}. The inclusion V_{H}^{i}(N_{3}) ⊆ V_{H}^{i}(N_{1}) ∪ V_{H}^{i}(N_{2}) now follows.

The proof of (v) is straightforward. □

By combining properties (ii), (iii) and (iv) in Proposition 2.2, we see that injective varieties are invariant under cosyzygies, whereas projective varieties are invariant under syzygies. We record these facts in the following slightly more general result, which concludes this section.

Corollary 2.3. Let N be a Λ-module, and let

0 → N_{1} → N_{2} → N_{3} → 0

be an exact sequence in mod Λ.

(i) If N_{2} ∈ M^{⊥}, then V_{M}^{i}(N_{1}) = V_{M}^{i}(N_{3}). In particular, the injective variety

of N equals that of Ω_{Λ}^{−1}(N).

(ii) If N_{2} ∈ ^{⊥}M, then V_{M}^{p}(N_{1}) = V_{M}^{p}(N_{3}). In particular, the projective variety

of N equals that of Ω_{Λ}^{1}(N).

3. Finite generation

The reason why the theories of support varieties for group rings, cocommutative Hopf algebras and complete intersections are all very powerful, is the existence of a central commutative Noetherian ring over which all the cohomology groups are finitely generated (cf. [2], [3], [4], [7], [12], [14]). As shown in [10], a similar theory is obtained for support varieties defined in terms of the Hochschild cohomology ring, when one assumes the existence of such a commutative ring. Motivated by this, we now make the following assumption on the fixed subalgebra H of Ext_{Λ}^{∗}(M, M).

Assumption. The ring H is Noetherian.

A priori, the algebra H is just some unknown graded subalgebra of Ext_{Λ}^{∗}(M, M), and this is of course not satisfactory if we want to do real computations. However, the following result shows that when H is a subalgebra of the center Z(M) of Ext_{Λ}^{∗}(M, M), and we require Ext_{Λ}^{∗}(M, M) to be a finitely generated H-module, then we may take H to be Z(M) itself. Note that Z(M) is a graded algebra. Indeed, suppose η is an element of Z(M), and write η = η_{0} + ⋯ + η_{n}, where η_{i} is an element of Ext_{Λ}^{i}(M, M) for each i. Let θ be any homogeneous element of Ext_{Λ}^{∗}(M, M). Then since ηθ = θη, we see that each η_{i} must commute with θ. Therefore each η_{i} belongs to Z(M), and this shows that Z(M) is a graded algebra.

Proposition 3.1. The following are equivalent.

(i) There exists a commutative Noetherian graded subalgebra R ⊆ Z(M) over which Ext_{Λ}^{∗}(M, M) is a finitely generated module.

(ii) The ring Z(M) is Noetherian, and Ext_{Λ}^{∗}(M, M) is a finitely generated

Z(M)-module. (iii) The ring Ext_{Λ}^{∗}(M, M) is Noetherian and a finitely generated Z(M)-module.

Proof. The implication (ii) ⇒ (i) is obvious. Suppose (i) holds, and let G be an algebra “lying between” R and Ext_{Λ}^{∗}(M, M), i.e. R ⊆ G ⊆ Ext_{Λ}^{∗}(M, M). Then Ext_{Λ}^{∗}(M, M) must be a finitely generated G-module. Moreover, since R is Noetherian and Ext_{Λ}^{∗}(M, M) is a finitely generated R-module, we see that Ext_{Λ}^{∗}(M, M) is a Noetherian ring. This shows the implication (i) ⇒ (iii). Finally, the implication (iii) ⇒ (ii) is [1, Theorem 1]. □

As mentioned in the previous section, the assumption that H_{0} is a local ring is superfluous once we have introduced finiteness conditions. Namely, the following result shows that we may take H to be a polynomial ring over k, so that H_{0} is just k itself. Recall first that if V is a graded k-vector space of finite type (i.e. dim_{k} V_{i} < ∞ for all i), then the rate of growth of V, denoted γ(V), is defined as

γ(V) =^{def} inf{t ∈ ℕ ∪ {0} | ∃a ∈ ℝ such that dim_{k} V_{n} ≤ an^{t−1} for n ≫ 0}.

Proposition 3.2. Let N be a Λ-module, and suppose Ext_{Λ}^{∗}(M, N) (respectively, Ext_{Λ}^{∗}(N, M)) is a finitely generated H-module. Then there exists a polynomial ring k[x_{1}, …, x_{c}] ⊆ H, with c = γ(H), such that H and Ext_{Λ}^{∗}(M, N) (respectively, Ext_{Λ}^{∗}(N, M)) are finitely generated k[x_{1}, …, x_{c}]-modules.

Proof. Follows from the Noether normalization lemma. □

In the following result we characterize precisely when all the cohomology modules are finitely generated over H.

Proposition 3.3. Consider the following conditions.

(i) For all N ∈ mod Λ, the H-module Ext_{Λ}^{∗}(M, N) is finitely generated.

(ii) The H-module Ext_{Λ}^{∗}(M, Λ/ rad Λ) is finitely generated.

(iii) For all N ∈ mod Λ, the H-module Ext_{Λ}^{∗}(N, M) is finitely generated.

(iv) The H-module Ext_{Λ}^{∗}(Λ/ rad Λ, M) is finitely generated. Then the implications (i) ⇔ (ii) and (iii) ⇔ (iv) hold.

Proof. We prove only the implication (ii) ⇒ (i); the implication (iv) ⇒ (iii) is proved analogously. The proof is by induction on the length ℓ(N) of a module N. Since the H-module Ext_{Λ}^{∗}(M, Λ/ rad Λ) is finitely generated, so is Ext_{Λ}^{∗}(M, S) for any simple Λ-module S. Now suppose ℓ(N) > 1, and choose a nonzero proper submodule L of N. The exact sequence

0 → L → N → N/L → 0

induces an exact sequence

Ext_{Λ}^{∗}(M, L) → Ext_{Λ}^{∗}(M, N) → Ext_{Λ}^{∗}(M, N/L)

of H-modules. By assumption, both the end terms are finitely generated Hmodules, hence so is the middle term since H is Noetherian. □

There are situations when finiteness always occurs, regardless of the module M we start with. Namely, when all the cohomology groups of the algebra are finitely generated over a central ring of cohomology operators, as in the following definition.

Definition. The algebra Λ satisfies Fg if there exists a commutative Noetherian graded k-algebra R = ⨁_{i=0}^{∞} R_{i} of finite type (i.e. dim_{k} R_{i} < ∞ for all i) satisfying the following:

(i) For every X ∈ mod Λ there is a graded ring homomorphism

ϕ_{X}: R → Ext_{Λ}^{∗}(X, X).

(ii) For each pair (X, Y) of finitely generated Λ-modules, the scalar actions from R on Ext_{Λ}^{∗}(X, Y) via ϕ_{X} and ϕ_{Y} coincide, and Ext_{Λ}^{∗}(X, Y) is a finitely generated R-module.

As mentioned, this holds if Λ is the group algebra of a finite group, a cocommutative Hopf algebra, a finite dimensional commutative complete intersection, or if the Hochschild cohomology ring of Λ is suitably “nice” (cf. [2], [3], [4], [7], [12], [14], [10], [11], [6]). Now suppose Λ satisfies Fg with respect to a graded ring R as in the definition, and let X be a Λ-module. Then ϕ_{X}(R) is a commutative Noetherian graded subalgebra of the center of Ext_{Λ}^{∗}(X, X). Moreover, for any Y ∈ mod Λ both Ext_{Λ}^{∗}(X, Y) and Ext_{Λ}^{∗}(Y, X) are finitely generated ϕ_{X}(R)-modules.

When Λ satisfies Fg, we may also define support varieties with respect to the ring of cohomology operators. Namely, let R be as in the definition. Given Λ-modules X and Y, we define

V_{R}(X, Y) =^{def} {𝔪 ∈ MaxSpec R | Ann_{R} Ext_{Λ}^{∗}(X, Y) ⊆ 𝔪}.

Is this variety comparable to V_{ϕ_{X}(R)}^{i}(Y) and V_{ϕ_{Y} (R)}^{p}(X), i.e. the varieties defined by the ideals Ann_{ϕ_{X}(R)}^{i}(Y) = Ann_{ϕ_{X}(R)} Ext_{Λ}^{∗}(X, Y) and Ann_{ϕ_{Y} (R)}^{p}(X) = Ann_{ϕ_{Y} (R)} Ext_{Λ}^{∗}(X, Y), respectively? The following result shows that the three varieties V_{ϕ_{X}(R)}^{i}(Y), V_{ϕ_{Y} (R)}^{p}(X) and V_{R}(X, Y) are in fact isomorphic.

Proposition 3.4. Suppose Λ satisfies Fg with respect to a graded ring R as in the definition above, and let X and Y be Λ-modules. Then the varieties V_{ϕ_{X}(R)}^{i}(Y), V_{ϕ_{Y} (R)}^{p}(X) and V_{R}(X, Y) are isomorphic.

Proof. Let 𝔪 be a maximal ideal in R. Since ϕ_{X} (Ann_{R} Ext_{Λ}^{∗}(X, Y)) equals Ann_{ϕ_{X}(R)}^{i} Y, we see that Ann_{R} Ext_{Λ}^{∗}(X, Y) ⊆ 𝔪 if and only if Ann_{ϕ_{X}(R)}^{i} Y ⊆ ϕ_{X}(𝔪). Therefore 𝔪 belongs to V_{R}(X, Y) if and only if ϕ_{X}(𝔪) belongs to V_{ϕ_{X}(R)}^{i}(Y), and this shows that the varieties V_{R}(X, Y) and V_{ϕ_{X}(R)}^{i}(Y) are isomorphic. Similarly the varieties V_{R}(X, Y) and V_{ϕ_{Y} (R)}^{p}(X) are isomorphic. □

As we saw above, when Λ satisfies Fg then for every Λ-module M there exists a commutative Noetherian graded subalgebra H ⊆ Ext_{Λ}^{∗}(M, M) over which Ext_{Λ}^{∗}(Λ/ rad Λ, M) and Ext_{Λ}^{∗}(M, Λ/ rad Λ) are finitely generated. However, the following example shows that this may very well hold for a module even if the algebra does not satisfy Fg.

Example. Suppose Λ is selfinjective, and let M be a nonzero periodic Λ-module, i.e. Ω_{Λ}^{p}(M) ≃ M for some p ≥ 1. Then the first part of the minimal projective resolution of M is a p-fold extension

0 → M → P_{p−1} → ⋯ → P_{0} → M → 0.

Denote this extension by μ, and consider the subalgebra k[μ] of Ext_{Λ}^{∗}(M, M). This subalgebra is a Noetherian ring over which Ext_{Λ}^{∗}(M, M) is finitely generated as a module. In fact, given any Λ-module N, the k[μ]-modules Ext_{Λ}^{∗}(M, N) and Ext_{Λ}^{∗}(N, M) are finitely generated (cf. [15] and [16] for a discussion of these phenomena). As an example, consider the quantum exterior algebra

k⟨x, y⟩/(x^{2}, xy − qyx, y^{2}),

where the element q is a nonzero non-root of unity in k. Let M be a two dimensional vector space with basis {u, v}, say. By defining

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

this vector space becomes a module over the quantum exterior algebra. Moreover, it is not difficult to see that this module is periodic of period one (cf. [5, Example 4.5]). However, by [11] and [6, Theorem 5.5] the algebra does not satisfy Fg, since q is not a root of unity.

We now return to the general theory. A natural question to ask is how big the relative support variety of a module is. For an arbitrary module N, this cannot be answered unless we introduce finiteness conditions, since a priori there is no relationship between H and Ext_{Λ}^{∗}(M, N) or Ext_{Λ}^{∗}(N, M). However, when we introduce finite generation, the situation becomes much more manageable.

Let X be a Λ-module with minimal projective and injective resolutions

⋯ → P_{2} → P_{1} → P_{0} → X → 0,
0 → X → I^{0} → I^{1} → I^{2} → ⋯,

say. Then we define the complexity and plexity of X, denoted cx X and px X, respectively, as

cx X =^{def} inf{t ∈ ℕ ∪ {0} | ∃a ∈ ℝ such that dim_{k} P_{n} ≤ an^{t−1} for n ≫ 0},
px X =^{def} inf{t ∈ ℕ ∪ {0} | ∃a ∈ ℝ such that dim_{k} I^{n} ≤ an^{t−1} for n ≫ 0}.

The complexity and the plexity of a module are not necessarily finite. Also, from the definition we see that cx X = 0 (respectively, px X = 0) if and only if X has finite projective dimension (respectively, finite injective dimension). It is well known that the complexity of X equals γ (Ext_{Λ}^{∗}(X, Λ/ rad Λ)), whereas its plexity equals γ (Ext_{Λ}^{∗}(Λ/ rad Λ, X)). Generalizing this, we define the complexity of the pair (X, Y) of Λ-modules to be γ (Ext_{Λ}^{∗}(X, Y)), and denote it by cx(X, Y). Thus cx X is the complexity of the pair (X, Λ/ rad Λ), whereas px X is the complexity of the pair (Λ/ rad Λ, X). Note that cx(X, Y) ≠ cx(Y, X) in general, that is, the order matters.

It follows from the discussion prior to [4, Proposition 5.3.5] that cx(M, N) ≤ cx M, and similarly cx(M, N) ≤ px N. In particular cx(M, M) is at most cx M and px M. The following example, using the same quantum exterior algebra as in the above example, shows that in general the inequalities might be strict.

Example. Let q be a nonzero non-root of unity in k, and let Γ be the quantum exterior algebra

k⟨x, y⟩/(x^{2}, xy − qyx, y^{2}).

Denote by X the Γ-module Γ(x + y). It was shown in [17] that the complex

⋯ → Γ ⟶^{·(x+q^{2}y)} Γ ⟶^{·(x−qy)} Γ ⟶^{·(x+y)} X → 0

is a minimal projective resolution of X, and that Ext_{Γ}^{n}(X, X) = 0 for n ≫ 0. Thus cx X = 1, whereas cx(X, X) = 0.

The following result shows that equality occurs when finite generation holds. Recall that H is a commutative Noetherian graded subalgebra of Ext_{Λ}^{∗}(M, M).

Proposition 3.5. [4, Proposition 5.3.5] If the H-module Ext_{Λ}^{∗}(M, Λ/ rad Λ) is finitely generated, then

cx M = cx(M, M) = γ(H).

Similarly, if the H-module Ext_{Λ}^{∗}(Λ/ rad Λ, M) is finitely generated , then

px M = cx(M, M) = γ(H).

In particular, if both Ext_{Λ}^{∗}(M, Λ/ rad Λ) and Ext_{Λ}^{∗}(Λ/ rad Λ, M) are finitely generated over H, then cx M = px M.

As for the “size” of the relative support varieties, the following result shows that it is given in terms of the complexity, provided finite generation holds.

Proposition 3.6. If the H-module Ext_{Λ}^{∗}(M, N) is finitely generated, then dim V_{H}^{i}(N) = cx(M, N). Similarly, if the H-module Ext_{Λ}^{∗}(N, M) is finitely generated , then dim V_{H}^{p}(N) = cx(N, M).

Proof. If Ext_{Λ}^{∗}(M, N) is finitely generated over H, then γ (H/ Ann_{H}^{i} N) = γ (Ext_{Λ}^{∗}(M, N)), and so by definition dim V_{H}^{i}(N) = cx(M, N). The other equality is proved similarly. □

4. Wild algebras and complexity

In this section we assume that our field k is algebraically closed. Recall that Λ is of finite representation type if there are only finitely many non-isomorphic indecomposable Λ-modules. Furthermore, recall that Λ is of tame representation type if there exist infinitely many non-isomorphic indecomposable Λ-modules, but they all belong to one-parameter families, and in each dimension there are finitely many such families. Finally, the algebra Λ is of wild representation type if it is not of finite or tame type.

In [9], Crawley-Boevey established a link between the representation type of a selfinjective finite dimensional algebra and the complexities of its modules. Namely, it was shown that for such an algebra, in any dimension only finitely many indecomposable modules are not of complexity one. Using this, Farnsteiner showed in [13] that the complexity of every module of a tame block of a finite group scheme is at most two.

Suppose our algebra Λ is selfinjective and satisfies the “global” finite generation hypothesis Fg defined immediately after Proposition 3.3. That is, suppose there exists a commutative Noetherian graded k-algebra R = ⨁_{i=0}^{∞} R_{i} of finite type satisfying the following:

(i) For every X ∈ mod Λ there is a graded ring homomorphism

ϕ_{X}: R → Ext_{Λ}^{∗}(X, X).

(ii) For each pair (X, Y) of finitely generated Λ-modules, the scalar actions from R on Ext_{Λ}^{∗}(X, Y) via ϕ_{X} and ϕ_{Y} coincide, and Ext_{Λ}^{∗}(X, Y) is a finitely generated R-module.

Then Farnsteiner’s proof still applies; hence Λ is wild if there exists a module of complexity at least three. We end this paper with the following result, which generalizes this.

Theorem 4.1. Suppose Λ is selfinjective, and there exists a Λ-module M satisfying the following:

(i) cx(M, M) ≥ 3,

(ii) there exists a commutative Noetherian graded subalgebra H ⊆ Ext_{Λ}^{∗}(M, M) over which Ext_{Λ}^{∗}(M, M) is a finitely generated module.

Then Λ is of wild representation type.

Proof. Suppose (i) holds. By Proposition 3.2, we may assume that H is a polynomial ring, say H = k[y_{1}, …, y_{n}], where n ≥ 3. Denote the ideal Ann_{H}^{i} N ⊆ H by 𝔞. Since Ext_{Λ}^{∗}(M, M) is a finitely generated H/ 𝔞-module, we may apply the Noether normalization lemma and obtain a new polynomial ring R = k[x_{1}, …, x_{c}] ⊆ H/ 𝔞, with c = cx(M, M), over which Ext_{Λ}^{∗}(M, M) is finitely generated. Moreover, we may assume that the homogeneous elements x_{1}, …, x_{c} are of the same degree, say |x_{i}| = d.

The maximal ideals in R correspond to points (α_{1}, …, α_{c}) ∈ k^{c}. Given an ideal I ⊆ R, we denote its variety by V_{R}(I), thus

V_{R}(I) = {α ∈ k^{c} | f(α) = 0 for all f ∈ I}.

Now for each α ∈ k, denote the element x_{1} + αx_{2} ∈ R by x_{α}. Lifting this element to H gives a homogeneous element η_{α} ∈ Ext_{Λ}^{∗}(M, M) of degree d, from which we obtain a short exact sequence

(†)  0 → M → K_{α} → Ω_{Λ}^{d−1}(M) → 0.

By applying the same proof as in [10, Proposition 4.3], we see that V_{R}^{p}(K_{α}) = V_{R}(x_{1} +αx_{2}). Thus cx(K_{α}, M) = c−1, and K_{α} is not isomorphic to K_{α′} whenever α ≠ α′.

For each α, let K_{α} = K_{α}^{1} ⊕ ⋯ ⊕ K_{α}^{t_{α}} be a decomposition of K_{α} into indecomposable Λ-modules. Moreover, denote the ideal Ann_{R}^{p} K_{α}^{i} ⊆ R, that is, the ideal Ann_{R} Ext_{Λ}^{∗}(K_{α}^{i}, M), by 𝔞_{i}. Then V_{R}^{p}(K_{α}^{i}) is by definition the variety V_{R}(𝔞_{i}), and therefore

V_{R}(x_{1} + αx_{2}) = V_{R}^{p}(K_{α}) = ⋃_{i=1}^{t_{α}} V_{R}^{p}(K_{α}^{i}) = ⋃_{i=1}^{t_{α}} V_{R}(𝔞_{i}) = V_{R}(∏_{i=1}^{t_{α}} 𝔞_{i}),

which in turn implies

∏_{i=1}^{t_{α}} 𝔞_{i} ⊆ \sqrt{∏_{i=1}^{t_{α}} 𝔞_{i}} = \sqrt{(x_{1} + αx_{2})}.

Since the ideal (x_{1} + αx_{2}) is prime, it is equal to its own radical, and contains one of the ideals 𝔞_{1}, …, 𝔞_{t_{α}}, say 𝔞_{1}. However, the variety V_{R}^{p}(K_{α}^{1}) is contained in V_{R}^{p}(K_{α}), and therefore (x_{1} + αx_{2}) = \sqrt{𝔞_{1}}. Consequently \sqrt{𝔞_{1}} = (x_{1} + αx_{2}), and this shows that the varieties V_{R}^{p}(K_{α}^{1}) and V_{R}^{p}(K_{α}) are equal.

The indecomposable Λ-modules {K_{α}^{1}}_{α∈k} are pairwise nonisomorphic, and from the exact sequence (†) we see that dim_{k} K_{α}^{1} ≤ dim_{k} M + dim_{k} Ω_{Λ}^{d−1}(M) for every α ∈ k. Moreover, by construction we know that cx(K_{α}^{1}, M) = cx(K_{α}, M), hence

2 ≤ c − 1 = cx(K_{α}^{1}, M) ≤ cx K_{α}^{1}.

The result now follows from Crawley-Boevey’s result [9, Theorem D]. □

Using Proposition 3.3 and Proposition 3.5, we obtain the following corollaries to Theorem 4.1.

Corollary 4.2. Suppose Λ is selfinjective, and there exists a Λ-module M satisfying the following:

(i) cx M ≥ 3,

(ii) there exists a commutative Noetherian graded subalgebra H ⊆ Ext_{Λ}^{∗}(M, M)

over which Ext_{Λ}^{∗}(M, Λ/ rad Λ) is a finitely generated module. Then Λ is of wild representation type.

Corollary 4.3. Suppose Λ is selfinjective, and there exists a Λ-module M satisfying the following:

(i) px M ≥ 3,

(ii) there exists a commutative Noetherian graded subalgebra H ⊆ Ext_{Λ}^{∗}(M, M)

over which Ext_{Λ}^{∗}(Λ/ rad Λ, M) is a finitely generated module. Then Λ is of wild representation type.

Finally, by applying Proposition 3.1 we obtain the following corollary to Theorem 4.1. It shows that an algebra is wild if it possesses a module whose Ext-algebra is “big enough”, Noetherian and finitely generated over its center.

Corollary 4.4. Suppose Λ is selfinjective, and there exists a Λ-module M satisfying the following:

(i) cx(M, M) ≥ 3, (ii) Ext_{Λ}^{∗}(M, M) is a Noetherian ring and finitely generated as a module over

its center. Then Λ is of wild representation type.

We end this paper with the following example illustrating Theorem 4.1, an example in which the algebra does not satisfy the finite generation hypothesis Fg. Namely, we use the same quantum exterior algebra as in the examples in Section 3

Example. Let q be a nonzero non-root of unity in k, and denote by Γ the quantum exterior algebra

k⟨x, y⟩/(x^{2}, xy − qyx, y^{2}).

Let X be the Γ-module from the example following Proposition 3.4, i.e. X is a two dimensional vector space with basis {u, v}, say, and with scalar multiplication defined by

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

This module is periodic of period one, and so if we denote its projective cover

0 → X → P → X → 0

by μ, the Ext-algebra Ext_{Γ}^{∗}(X, X) is finitely generated as a module over the polynomial subalgebra k[μ]. Now let Λ be the algebra Γ ⊗_{k} Γ ⊗_{k} Γ, and let M be the Λ-module X ⊗_{k} X ⊗_{k} X. Then the Ext-algebra of M is given by

Ext_{Λ}^{∗}(M, M) = Ext_{Γ}^{∗}(X, X)\overline{⊗}_{k} Ext_{Γ}^{∗}(X, X)\overline{⊗}_{k} Ext_{Γ}^{∗}(X, X),

where \overline{⊗} differs from the usual tensor product only in that elements of odd degree anticommute (cf. [8, Chapter XI]). Therefore Ext_{Λ}^{∗}(M, M) is finitely generated as a module over a commutative Noetherian graded subalgebra. Moreover, since cx_{Γ}(X, X) = 1, we see that cx_{Λ}(M, M) = 3. Finally, the algebra Λ does not satisfy Fg. Namely, this algebra is a quantum exterior algebra on six generators, where some of the defining commutators equal q. Then by [11] and [6, Theorem 5.5] Λ does not satisfy Fg, since q is not a root of unity.

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Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

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