THICK SUBCATEGORIES OF THE BOUNDED DERIVED CATEGORY OF A FINITE GROUP

JON F. CARLSON AND SRIKANTH B. IYENGAR

Abstract. A new proof of the classification for tensor ideal thick subcategories of the bounded derived category, and the stable category, of modular representations of a finite group is obtained. The arguments apply more generally to yield a classification of thick subcategories of the bounded derived category of an artinian complete intersection ring. One of the salient features of this work is that it takes no recourse to infinite constructions, unlike the previous proofs of these results.

1. Introduction

In the paper [6], the first author, in collaboration with Benson and Rickard, established a classification for tensor ideal thick subcategories of the stable category of finitely generated kG-modules in the case that G is a finite group and k is a field of characteristic p > 0 dividing the order of G. This result was inspired and influenced by an earlier classification of the thick subcategories of the perfect complexes over a commutative noetherian ring by Hopkins [16]. The statements are remarkably similar: In both cases a subcategory is determined by the support, suitably defined, of the objects in the subcategory.

However, the methods in the two settings were entirely different. At that time it did not seem possible to adapt Hopkins’ methods to modules over group algebras. Instead, the proofs in [6] used idempotent modules and idempotent functors developed by Rickard. Idempotent modules are, in general, infinitely generated, hence the proof had employed constructions from the stable category of all kG-module to obtain a result that spoke only of finitely generated module.

The main purpose of this paper is to show that the results in the two settings are directly related. To this end, we extend Hopkins’ arguments to obtain a classification of the thick subcategories of perfect Differential Graded modules over suitable DG algebras. The result about kG-modules is then deduced from it by a series of reductions, following the paradigm developed in the work of the second author with Avramov, Buchweitz, and Miller [2], and with Benson and Krause [8]. What is more, the arguments require no constructions involving infinitely generated modules, in contrast to the proof in [6].

Date: February 2, 2012.

1991 Mathematics Subject Classification. 20J06 (primary), 20C20, 13D09, 16E45.

Research partially supported by NSF grants DMS-1001102 (JFC) and DMS-0903493 (SBI).

In the reduction mentioned above, it is more natural to work with the bounded derived category of modules over kG; the sought after classification for the stable category is an easy consequence, for the latter is a quotient of the former by the subcategory of perfect complexes. The first step (which comes last in the paper) is to reduce to the case of an elementary abelian p-group; this uses a theorem of the first author [12]. After that, If p = 2 Koszul duality gives an equivalence of triangulated categories between the bounded derived category of k(ℤ/2)^{r}-modules and the derived category of DG modules, with finitely generated cohomology, over a polynomial ring k[y_{1}, …, y_{r}] with generators y_{i} in degree one. Then one can apply (the extension of) Hopkins’ theorem to k[y] to get the desired classification. Even in this case, we take a slightly longer route, that has the merit of being independent of the characteristic of k.

The starting point is the observation that the group algebra of an elementary abelian group is an artinian complete intersection ring. For any such ring R there is an exact functor, constructed in [2], from its bounded derived category to the derived category of DG modules with finitely generated cohomology over a polynomial ring k[x_{1}, …, x_{r}], with generators x_{i} in degree two. Though not an equivalence of categories, the functor is close enough to it that one can relate the thick subcategories in the source and in the target. Hopkins’ Theorem again applies, and yields a classification for thick subcategories of the bounded derived category of R. We note that such a result has been proved for complete intersections of any (Krull) dimension by Stevenson [21], using different techniques involving infinite methods.

Friedlander and Pevtsova [13] have proved a similar classification of thick subcategories for the stable category of modules over a finite group scheme, again using idempotent modules. We have been unable to bring to bear the methods of this paper in that context. The difficulty is that for more general finite group schemes, there is no reduction of such a classification to unipotent abelian subgroup schemes, which are the analogs of the elementary abelian subgroups

Here is an outline of the contents of this paper: Section 3 is devoted to establishing the analogue of Hopkins’ theorem for DG modules over commutative graded rings. This uses a notion of support for DG modules, discussed in Section 2. In Section 4 we classify the thick subcategories of the bounded derived category of DG modules over an exterior algebra, using a BGG correspondence from [2] and the main result of Section 3. The statement on artinian complete intersections is deduced from it, in Section 5. Section 6 is devoted to group algebras.

In this article, we assume that the reader is familiar with basic results on group cohomology, and take [11] as our reference for this topic. We also take the mechanics of triangulated categories as given, and refer the reader to one of the standard texts such as [15] for details. We recall only that a non-empty full subcategory of a triangulated category is thick if it is triangulated (i.e. if two of three objects in a triangle are in the subcategory, then so is the third) and closed under summands. If M is an object in a triangulated category, the smallest thick subcategory containing it is denoted Thick(M).

There are two distinct aspects to any classification of thick subcategories via supports. The first is the realizability problem: Construct an object with support a prescribed closed subset of the ambient variety. This now well-understood to be the more elementary aspect of the classification problem, and there are various solutions that apply in great generality; see, for example, [4]. The second, more challenging, step is to prove that if the support of an object M is contained in the support N, then Thick(M) ⊆ Thick(N). This is where properties specific to the context at hand come in play. In this work, the focus is on this aspect of the classification problem.

Acknowledgments. This collaboration grew out of conversations at the Morningside Center for Mathematics, Beijing, during the workshop “Representation Theory: Cohomology and Support” held in September 2011. It is a pleasure to thank the Morningside Center for the hospitality, and the organizers: Bangming Deng, Henning Krause, Jie Xiao, Hechun Zhang, Bin Zhu, for the invitation to the workshop.

2. Support for DG modules

Let S be a commutative ℤ-graded noetherian ring; we assume S is concentrated in even degrees or 2S = 0. In this article, are gradings are upper; thus S = ⊕_{i∈ℤ}S^{i}. In this section we introduce a notion of support for Differential Graded (henceforth abbreviated to DG) modules over S and describe various ways of computing it.

To begin with, we record a version of Nakayama’s Lemma for graded S-modules. Such results are known in greater generality but the version below is all we need; the proof is similar to the one for ungraded rings, and is sketched for completeness.

Lemma 2.1. Assume S has a unique maximal homogeneous ideal, say 𝔪. If M is a finitely generated graded S-module and 𝔪M = M, then M = 0.

Proof. Pick homogeneous elements {x_{1}, …, x_{n}} that generate M as an S-module. By hypothesis, there exist homogeneous elements {s_{ij}}_{1⩽i,j⩽n} in 𝔪 such that

x_{i} = ∑_{j=1}^{n} s_{ij}x_{j} for each 1 ≤ i ≤ n.

By the determinant trick, it follows that det(I −A)M = 0 where A = (s_{ij}). Observe that det(I − A) = 1 − ∑_{t∈ℤ} s_{t}, where s_{t} ∈ 𝔪_{t}. Thus, for any homogeneous element x in M one gets x = ∑_{t} s_{t}x, so that x = s_{0}x. As S_{0} is a local ring with maximal ideal 𝔪 ∩ S_{0}, the element 1 − s_{0}, which is in S_{0} \ 𝔪, is a unit, and hence x = 0. □

Henceforth we view S as a DG algebra with d^{S} = 0. Given a DG S-module M, its homology H(M) is a graded S-module. We write D(S) for the derived category of DG S-modules, viewed as a triangulated category, and D^{f}(S) for the full subcategory consisting of DG modules M for which the S-module H(M) is finitely generated. The basic constructions and results on DG modules over DG algebra required in this work are all recorded in [2, §3 and §4].

We recall that a DG module S-module F is semi-free if it admits a family (F_{n})_{n∈ℕ} of DG submodules such that F_{n} ⊆ F_{n+1}, and ∪_{n}F_{n} = F, and F_{n+1}/F_{n} is isomorphic to a direct sum of suspensions of S; in particular, forgetting differentials, F is a graded free S-module. When F is semi-free the functors Hom_{S}(F, −) and − ⊗_{S} F defined on the category of DG S-modules preserve quasi-isomorphisms.

Each DG module M over S admits a semi-free resolution: a quasi-isomorphism of DG S-modules F → M with F semi-free. Then assignments

(−⊗_{S}^{L} M) = − ⊗_{S} F and RHom_{S}(M, −) = Hom_{S}(F, −)

then define exact functors on D(S).

We write Spec^{∗}S for the set of homogeneous prime ideals in S. For each 𝔭 in Spec^{∗}S, the homogeneous localization of a graded S-module M at 𝔭 is denoted M_{(𝔭)}. We write k(𝔭) for S_{(𝔭)}/𝔭S_{(𝔭)}, which is a graded field. See Bruns and Herzog [10, §1.5] for the module theory over graded rings.

When M is a DG S-module, and 𝔭 ∈ Spec^{∗}S, the differential on M induces one on M_{(𝔭)}, making it a DG module over S_{(𝔭)}.

Definition 2.2. The support of M ∈ D^{f}(S) is the subset of Spec^{∗}S defined by

Supp_{S}^{∗} M = {𝔭 ∈ Spec^{∗}S | H(M_{(𝔭)}) ≠ 0}

Let Ann_{S} H(M) denote the annihilator of the S-module H(M); it is a homogeneous ideal of S. Since H(M_{(𝔭)}) ≅ H(M)_{(𝔭)}, there are equalities

Supp_{S}^{∗} M = Supp_{S}^{∗} H(M) = {𝔭 ∈ Spec^{∗}S | 𝔭 ⊇ Ann_{S} H(M)} .  (2.3)

The next result provides a description of support that is often more convenient to work with. Its proof is, at the end, an application of Nakayama’s Lemma 2.1.

Theorem 2.4. Let M be a DG S-module in D^{f}(S). For each 𝔭 ∈ Spec^{∗}S one has

H(M_{(𝔭)}) ≠ 0 if and only if H(k(𝔭)⊗_{S}^{L} M) ≠ 0 .

In particular Supp_{S}^{∗} M = {𝔭 ∈ Spec^{∗}S | k(𝔭)⊗_{S}^{L} M ≠ 0 in D(S)}.

Proof. Replacing S by S_{(𝔭)} one may assume S is (graded) local, so that there is a unique maximal homogeneous ideal; call it 𝔪 and let k = S/𝔪. The task is then to verify that, in D(S), if k⊗_{S}^{L} M = 0, then M = 0. Pick a set of elements {s_{1}, …, s_{n}} that generate the ideal 𝔪 and let K be the DG S-module obtained by an iterated mapping cone: K_{0} = S and K_{i} = cone(Σ^{|s_{i}|}K_{i−1} ⟶^{s_{i}} K_{i−1}) for i ≥ 1 and K = K_{n}. Then K is in Thick_{S}(k); see the penultimate paragraph in the proof of [7, Theorem 8.1]. The hypothesis thus implies that K ⊗_{S} M = 0 in D(S). Consider the long exact sequence associated to the exact sequence of DG modules

0 ⟶ K_{i−1} ⊗_{S} M ⟶ K_{i} ⊗_{S} M ⟶ Σ^{|s_{i}|+1}(K_{i−1} ⊗_{S} M) ⟶ 0

Assuming H(K_{i} ⊗ M) = 0 there is an isomorphism

Σ^{|s_{i}|} H(K_{i−1} ⊗_{S} M) ⟶^{s_{i}} H(K_{i−1} ⊗_{S} M)

of finitely generated graded S-modules. Hence H(K_{i−1} ⊗_{S} M) = 0, by Lemma 2.1, for s_{i} is in 𝔪. An iteration then yields H(M) = 0, as desired. □

3. Perfect DG modules over commutative graded algebras

Let S be a commutative ℤ-graded noetherian ring, either concentrated in even degrees or satisfying 2S = 0. As in the previous section, we view S as a DG algebra with differential zero. The main result of this section is a generalization of Hopkins’ theorem that applies to perfect DG modules over S.

Definition 3.1. A DG S-module is said to be perfect if it is in Thick_{S}(S).

A DG module is perfect if and only if it is a retract (that is to say, isomorphic in D^{f}(S) to a direct summand of) a semi-free DG module with a finite free filtration; see [2, Theorem 4.2] for details. We do not need this description in what follows.

When S is concentrated in degree zero, a DG module is nothing other than a complex of S-modules, and a DG module is perfect if and only if it is isomorphic to a bounded complex of finitely generated projective S-modules. Thus the result below generalizes Hopkins’ theorem [16, Theorem 11], see also Neeman [19, Lemma 1.2] and Thomason [23, Theorem 3.14], from rings (which are DG algebras concentrated in degree 0) to DG algebras with zero differential.

Theorem 3.2. If M and N are perfect DG S-modules with Supp_{S}^{∗} M ⊆ Supp_{S}^{∗} N, then M is in Thick_{S}(N).

This result can be deduced from the classification of the localizing subcategories of D(S) proved in [7, §8]. In the interest of avoiding “infinite constructions”, we sketch an alternative proof, mimicking arguments from [16, 19, 23]. The crux is the following ‘Tensor Nilpotence Theorem’ for perfect DG modules that extends [16, Theorem 10] and [19, Theorem 1.1], see also [23, Theorems 3.6,3.8].

Theorem 3.3. Let F and G be perfect DG S-modules. If f: F → X is a morphism of DG S-modules and k(𝔭)⊗_{S}^{L} f = 0 for each 𝔭 ∈ Supp_{S}^{∗} G, then there exists an integer n ≥ 1 such that G⊗_{S}^{L} f^{⊗n} = 0 in D(S).

Here f^{⊗n} denotes the morphism X^{⊗_{S}^{L}n} → Y^{⊗_{S}^{L}n} induced by f.

Proof. The proof proceeds by a series of reductions.

Step 1. Reduction to the case G = S.

This is a formal argument, see [23, Theorem 3.8], exploiting the fact that, since the DG S-module G is perfect, it is a retract of RHom_{S}(G, G)⊗_{S}^{L} G.

Step 2. Reduction to the case F = S.

Indeed, f^{⊗n} = 0 if and only if (f′)^{⊗n} = 0 where f′: S → RHom_{S}(F, X) is the morphism that assigns 1 to f.

For the sequel, we replace X by its semi-free resolution; thus, for example, W ⊗_{S}^{L} f is represented by W ⊗_{S} f for any W ∈ D(S) and f^{⊗n}: S → X^{⊗_{S}^{L} n} is represented by the morphism f^{⊗n}: S → X^{⊗n}. In particular, the following conditions are equivalent.

• f^{⊗n} = 0 in D(S);

• the cycle f(1)^{⊗n} ∈ X^{⊗n} is homologous to 0. Note that morphisms g, h: S → W are homotopic if and only if the cycles g(1)

and h(1) in W_{0} are homologous; in such a situation, we write g(1) ∼ h(1).

Claim 1. Let I, J be homogeneous ideals in S such (S/I) ⊗_{S} f = 0 = (S/J) ⊗_{S} f. If IJ = (0), then f ⊗_{S} f = 0.

Indeed, the hypothesis is equivalent to the existence of cycles a ∈ IX_{0} and b ∈ JX_{0} such that f(1) ∼ a and f(1) ∼ b. It is then readily seen that

(f ⊗ f)(1) = f(1) ⊗ f(1) ∼ a ⊗ b .

Since a ⊗ b ∈ IJ(X_{0} ⊗_{S} X_{0}) = 0, it follows that f ⊗ f = 0.

In what follows, dim S denotes the dimension of Spec^{∗}S.

Claim 2. If d is a non-negative integer such that the result holds for all graded domains S with dim S ≤ d, then it holds for all graded rings S with dim S ≤ d.

Indeed, this is by a straightforward application of Claim 1, given that the minimal primes of S, and so also its nil-radical, are homogeneous; see [10, Lemma 1.5.6].

Next we prove the desired result when dim S is finite, by an induction argument.

Assume dim S = 0. Using the preceding claim we can assume S is a domain, and hence a graded field. In this case the result is obvious.

We now suppose that the result holds for all commutative ℤ-graded noetherian rings of dimension d − 1 or less, and verify that it holds whenever dim S = d ≥ 1; by Claim 2, it suffices to consider the case when S is a domain.

Consider the ideal Ann(f) = {s ∈ S | sf = 0 in D(S)} of S. By hypothesis, k(0) ⊗_{S} f = 0, where k(0) is the graded residue field at the prime ideal (0). Thus, Ann(f) ≠ (0). We may assume f ≠ 0, so there exists a non-zero homogeneous element s ∈ Ann(f) that is not a unit, and hence a non-zero divisor in S.

Consider the morphism S/(s) ⊗_{S} f: S/(s) → S/(s) ⊗_{S} X. Since Spec^{∗}(S/(s)) is naturally identified with the subset of primes 𝔭 ∈ Spec^{∗}S containing s, and for such a 𝔭 the S action on k(𝔭) factors through S/(s), it follows that

k(𝔭) ⊗_{S/(s)} (S/(s) ⊗_{S} f) ≅ k(𝔭) ⊗_{S} f = 0 .

The induction hypothesis thus yields that S/(s) ⊗_{S} f^{⊗n} = 0 for some integer n ≥ 1; that is to say, f^{⊗n}(1) ∼ sx for some x ∈ X. Because sd(x) = d(sx) = 0 and the graded S-module underlying X is free (recall X is semi-free) one gets d(x) = 0, that is to say, x is a cycle. This justifies the second equivalence below:

f^{⊗(n+1)}(1) = (f^{⊗n} ⊗ f)(1) ∼ (sx ⊗ f(1)) = (x ⊗ sf(1)) ∼ (x ⊗ 0) = 0 .

The other equivalence and the equalities are standard. Thus f^{⊗(n+1)} = 0. This completes the proof of the theorem when dim S is finite.

To finish the proof, we consider the chain of ideals

Ann(f) ⊆ Ann(f^{⊗2}) ⊆ Ann(f^{⊗3}) · · · .

Since S is noetherian, there exists an integer n such that Ann(f^{⊗n}) = Ann(f^{⊗i}) for all i ≥ n. We claim that f^{⊗n} = 0, or equivalently, that Ann(f^{⊗n}) = S.

Indeed, if not then there exists a prime ideal 𝔭 ⊇ Ann(f^{⊗n}). The dimension of S_{(𝔭)} is finite and the hypothesis of the theorem apply to the morphism f_{(𝔭)} : S_{(𝔭)} → X_{(𝔭)}, so the already established case yields that for f_{(𝔭)}^{⊗i} = 0 for i ≫ 0 so that

Ann(f^{i})_{(𝔭)} = Ann(f_{(𝔭)}^{⊗i}) = S_{(𝔭)} .

This is a contradiction. □

In the sequel, given a morphism g: X → Y of DG S-modules, we write cone(g) for any DG S-module that fits in an exact triangle X ⟶^{g} Y → cone(g) →, and use the fact that if g = 0, then Y is a retract of cone(g).

Proof of Theorem 3.2. Given Theorem 3.3 the argument is akin to the one for [16, Theorem 7], see also [19, Lemma 1.2] and [23, Lemma 3.14], so we only sketch it.

The natural morphism S → RHom_{S}(N, N) of DG S-modules induces a morphism

g: M → M⊗_{S}^{L} RHom_{S}(N, N) ,

and this gives rise to an exact triangle of perfect DG S-modules:

F ⟶^{f} M ⟶^{g} M⊗_{S}^{L} RHom_{S}(N, N) ⟶

For any 𝔭 ∈ Supp_{S}^{∗} M, one has H(k(𝔭)⊗_{S}^{L} N) ≠ 0, since Supp_{S}^{∗} M ⊆ Supp_{S}^{∗} N; here we use the alternative definition of support from Theorem 2.4. Since k(𝔭) is a graded field, the map k(𝔭)⊗_{S}^{L} g is split-injective, so that k(𝔭)⊗_{S}^{L} f = 0 in D(k(𝔭)). Theorem 3.3 applies to yield that M⊗_{S}^{L} f^{⊗n} = 0 for some positive integer n. Thus, M^{⊗(n+1)} is a retract of M⊗_{S}^{L} cone(f^{⊗n}). A straightforward induction on i shows that cone(f^{⊗i}) is in Thick_{S}(N) for each i ≥ 1, and hence so is M⊗_{S}^{L} cone(f^{⊗i}). In conclusion, M^{⊗(n+1)} is in Thick_{S}(N), and it follows that so is M; this uses the fact that, since M is perfect, it is a retract of the DG S-module RHom_{S}(M, M)⊗_{S}^{L} M, which is isomorphic to RHom_{S}(M, S)⊗_{S}^{L} M^{⊗2}. □

Definition 3.4. A subset V of Spec^{∗}S is specialization closed when the following property holds: If 𝔭 ⊆ 𝔮 are homogenous prime ideals in S and 𝔭 ∈ V, then 𝔮 ∈ V; equivalently, V is a union of closed subsets, in the Zariski topology on Spec^{∗}S.

In what follows, given a subset V of Spec^{∗}S, set

Thick(S)_{V} = {M ∈ Thick(S) | Supp_{S}^{∗} M ⊆ V } ,

viewed as a full subcategory of Thick(S). It is not hard to verify that this is a thick subcategory when V is specialization closed. Theorem 3.2 yields a perfect converse:

Corollary 3.5. If C is a thick subcategory of Thick(S), then there exist a specialization closed subset V ⊆ Spec^{∗}S such that C = Thick(S)_{V}.

Proof. Since the support of any DG S module M in D^{f}(S) is a closed subset of Spec^{∗}S, by (2.3), the subset

V = ⋃_{M∈C} Supp_{S}^{∗} M

is specialization closed. It is immediate from definitions that C ⊆ Thick(S)_{V}.

Let now M be a perfect DG S-module with Supp_{S}^{∗} M ⊆ V. Since Supp_{S}^{∗} M is a closed subset, there exists DG S-modules N_{1}, …, N_{s} in C such that

Supp_{S}^{∗} M ⊆ ⋃_{i} Supp_{S}^{∗}(N_{i})

Note that the subset on the right is the support of the DG S-module N = ⊕_{i}N_{i}, that is also perfect. Thus Theorem 3.2 yields that M is in Thick_{S}(N), and hence in C, since the latter is a thick subcategory. This proves that Thick(S)_{V} ⊆ C. □

The realizability problem in Thick(S) is easily solved: For any closed subset V of Spec^{∗}S, defined by an homogeneous ideal I in S, the perfect DG S-module S//I constructed as in [7, §2.5] has support V. In conjunction with Corollary 3.5, this yields a bijection between thick subcategories of Thick(S) and specialization closed subsets of Spec^{∗}S.

4. DG modules over exterior algebras.

In this section, we sketch the development of a theory of support varieties for DG modules over an exterior algebra, and prove the version of Hopkins’ theorem in that context. A lot of this material is taken from [2, 3].

Suppose that k is a field and that Λ = k⟨z_{1}, …, z_{r}⟩ is an exterior algebra on indeterminates {z_{i}}, all of odd negative degrees; recall that we are using upper gradings, unlike in [2, 3]. We consider Λ as a DG algebra with d^{Λ} = 0, let D(Λ) denote the derived category DG of Λ-modules, and let D^{f}(Λ) denote the full subcategory consisting of DG Λ-modules M such that the Λ-module H(M) is finitely generated. We note that D^{f}(Λ) coincides with Thick_{Λ}(k), the thick subcategory generated by the trivial module k; see [2, Remark 7.5].

Let S be the polynomial algebra k[x_{1}, …, x_{r}] on indeterminates {x_{i}}, where |x_{i}| = −|z_{i}|−1, again viewed as a DG algebra with zero differential; D(S) and D^{f}(S) have the expected meaning. In this case, D^{f}(S) = Thick_{S}(S), that is to say, every DG S-module with finitely generated homology is perfect; see [2, Remark 7.5].

The main result that we need is the following equivalence of categories from [2, Theorem 7.4]. It is a DG analogue of the Bernstein-Gelfand-Gelfand correspondence.

Theorem 4.1. There exists an exact functor h: D(Λ) → D(S) with the property that its restriction to D^{f}(Λ) is an equivalence

h: D^{f}(Λ) ⟶^{≡} D^{f}(S).  □

The construction of the functor h is sketched below. We refer the reader to [2, §7] (note the Corrigendum) for a complete proof of the theorem.

The graded dual Hom_{k}(S, k) is a then a DG S-module. Consider the algebra Λ⊗_{k}S as a DG algebra with zero differential. Let F denote the DG module over it with underlying graded module Λ ⊗_{k} Hom_{k}(S, k) and differential given by multiplication by δ = ∑_{i=1}^{r} z_{i} ⊗ x_{i}. It is a straightforward exercise to show that δ^{2} = 0.

The DG (Λ ⊗_{k} S)-module F is the DG module Hom_{k}(X, k) constructed in [2, §7]. The result below is thus contained in (7.6.4), (7.6.5) and (7.6.3) of [2]. The gist of (1) and (2) is that F is a semi-free resolution of k, viewed as a DG module over Λ.

Proposition 4.2. The following statement hold.

(1) The map ε: F → k induced by augmentations Λ → k and Hom_{k}(S, k) → k, is a morphism of DG Λ-modules and a quasi-isomorphism.

(2) Hom_{Λ}(F, −) preserves quasi-isomorphisms.

(3) The map S → Hom_{Λ}(F, F) induced by the multiplicative action of S on F is a morphism of DG S-modules, and a quasi-isomorphism. □

For any DG Λ-module M, the complex Hom_{Λ}(F, M) is a DG S-module. Hence, thanks to Proposition 4.2(2), one can define an exact functor

h: D(Λ) → D(S) with h(M) = Hom_{Λ}(F, M).

The other parts of Proposition 4.2 imply that this induces an equivalence of triangulated categories between D^{f}(Λ) and D^{f}(S). From this we get a natural notion of support variety for DG Λ-modules.

Definition 4.3. The support of M ∈ D^{f}(Λ) is the subset of Spec^{∗}S defined by

V_{Λ}(M) = Supp_{S}^{∗} h(M) ⊆ Spec^{∗}S .

From (2.3) one then gets the first equality below:

V_{Λ}(M) = Supp_{S}^{∗} H(h(M)) = Supp_{S}^{∗} Ext_{Λ}(k, M) .

The second one is from Proposition 4.2. With this information, we can easily prove the main result of this section.

Theorem 4.4. The following statements hold:

(1) For M, N in D^{f}(Λ) one has V_{Λ}(M) ⊆ V_{Λ}(N) if and only if M is in Thick_{Λ}(N).

(2) For any closed subset V ⊆ Spec^{∗}S there exists M ∈ D^{f}(Λ) with V_{Λ}(M) = V.

Proof. By hypothesis, Supp_{S}^{∗} h(M) ⊆ Supp_{S}^{∗} h(N), so Theorem 3.2 yields that h(M) is in Thick_{S}(h(N)); as h is an equivalence it follows that M is in Thick_{Λ}(N).

For the second statement, we observer that V is the variety of a ideal I ⊂ S. So we let M = h^{−1}(S/I). □

Remark 4.5. The results in this section carry over to the case when the generators {z_{i}} of the exterior algebra Λ are in degree zero; in this case S will be a polynomial algebra on generators of degree −1.

5. Artinian complete intersection rings

Let k be a field, k[z_{1}, …, z_{r}] the polynomial algebra over k in indeterminates z = z_{1}, …, z_{c}, and let f_{1}, …, f_{r} be a regular sequence in (z)^{2}. Set

R = k[z_{1}, …, z_{r}]/(f_{1}, …, f_{r})

Thus R is a complete intersection ring of (Krull) dimension zero. The main result in this section is Theorem 5.6, that is an analogue of Hopkins’ theorem for D^{f}(R), the bounded derived category of finitely generated R-modules.

Since f_{j} is in (z)^{2} for each 1 ≤ j ≤ r, we can write

f_{j} = ∑_{1⩽h⩽i⩽r} c_{hi,j} z_{h}z_{i}  (5.1)

with each c_{hi,j} in k[z_{1}, …, z_{r}]. These elements give rise to structure constants in the Ext-algebra of R, as discovered by Sjödin [20, Theorem 5], and described below.

Remark 5.2. Let R = k{ξ_{1}, …, ξ_{r}, θ_{1}, …, θ_{r}} be the tensor algebra on k, where the {ξ_{i}} and the {θ_{j}} are indeterminates of degree 1 and 2, respectively. Then Ext_{R}(k, k) is this tensor algebra modulo the ideal generated by the relations

ξ_{h}ξ_{i} + ξ_{h}ξ_{i} = − ∑_{j=1}^{r} \overline{c}_{hi,j}θ_{j} for h < i and ξ_{h}^{2} = − ∑_{j=1}^{r} \overline{c}_{hh,j}θ_{j}

θ_{i}ξ_{j} − ξ_{j}θ_{i} = 0 = θ_{i}θ_{j} − θ_{j}θ_{i} for all i, j

where for any c in k[z_{1}, …, z_{r}], we write \overline{c} for its image in the polynomial ring k = k[z_{1}, …, z_{r}]/(z_{1}, …, z_{r}).

Thus, k[θ] = k[θ_{1}, …, θ_{r}] is a polynomial ring in the center of Ext_{R}(k, k), and Ext_{R}(k, k) is a finitely generated (even free) module over it, generated by ξ_{1}, …, ξ_{r}.

Example 5.3. The example to bear in mind is the group algebra of an elementary abelian p-group, E = (ℤ/p)^{r}, over a field of characteristic p. Since

kE ≅ k[z_{1}, …, z_{r}]/(z_{1}^{p}, …, z_{r}^{p})

one can choose c_{hi,j} to be z_{j}^{p−2} when h = i = j, and zero otherwise. Then the description of its Ext-algebra from Remark 5.2 recovers a well-known computation:

Ext_{kE}(k, k) = \begin{cases}k[ξ_{1}, …, ξ_{r}] & if p = 2 \\ k⟨ξ_{1}, …, ξ_{r}⟩ ⊗_{k} k[θ_{1}, …, θ_{r}] & for p odd. \end{cases}

For any complex M of R-modules, the graded k-vector space Ext_{R}(k, M) has a structure of a graded right module over Ext_{R}(k, k), and hence over k[θ]. The next result is due to Gulliksen [14, Theorem 3.1] for general complete intersection rings; we furnish an elementary proof, specific to our context:

Lemma 5.4. There is an equality D^{f}(R) = Thick_{R}(k). In particular, for each M in D^{f}(R), the k[θ]-module Ext_{R}(k, M) is noetherian.

Proof. Let M be in D^{f}(R). Replacing it by a quasi-isomorphic complex, if necessary, we may assume M is a bounded complex of finitely generated R-modules. It then has a finite filtration M ⊇ (z)M ⊇ (z)^{2}M ⊇ ⋯ 0 by sub-complexes, with sub-quotients isomorphic to a complex of k-vector space, of finite rank over k. Such a complex is in Thick_{k}(k) and hence in Thick_{R}(k). It follows that M itself is in Thick_{R}(k). This establishes that D^{f}(R) ⊆ Thick_{R}(k); the reverse inclusion is a tautology.

For the second statement, one has only to note that if N′ → N → N′′ → is an exact triangle of complexes of R-modules, and the k[θ]-modules Ext_{R}(k, N′) and Ext_{R}(k, N′′) are finitely generated, then so is Ext_{R}(k, N). □

This permits one to introduce a notion of support for complexes of R-modules.

Definition 5.5. The support variety of any complex M ∈ D^{f}(R) is the subset

V_{R}(M) = Supp_{k[θ]}^{∗} Ext_{R}(k, M) ⊆ Spec^{∗}k[θ] .

Note that V_{R}(k) = Spec^{∗}k[θ].

One has an exact analogue of Theorem 3.2:

Theorem 5.6. Let M and N be complexes in D^{f}(R). If V_{R}(M) ⊆ V_{R}(N), then M is in Thick_{R}(N).

See Remark 5.12 for antecedents. The result is proved by a reduction to the case of DG modules over exterior algebras, Theorem 4.4, via functors described below.

Construction 5.7. Let K be the Koszul complex on the sequence z_{1}, …, z_{r}, viewed as a DG algebra. It is thus an exterior algebra over R on indeterminates y_{1}, …, y_{r}, with each y_{i} of degree −1, with differential determined by

d(z_{i}) = 0 and d(y_{i}) = z_{i} .

Since K is a finite free complex of R-modules, the assignment M ↦ K ⊗_{R} M induces an exact functor t: D^{f}(R) → D^{f}(K); it is left adjoint to the restriction functor r: D^{f}(K) → D^{f}(R) induced by the morphism of DG algebras R → K.

The proof of the result below only uses the fact that R is an artinian local ring; the complete intersection property is not required. The gist of the statement is that the functors t and r, though not equivalences, come close to that.

Proposition 5.8. One has Thick_{R}(M) = Thick_{R}(rtM) for each M in D^{f}(R).

Proof. It suffices to prove that Thick_{R}(R) = Thick_{R}(K); the stated equality follows by applying −⊗_{R}^{L} M. Since K is a finite free complex, it is in Thick_{R}(R), so it remains to verify that R is in Thick_{R}(K). This is a special case of Theorem 3.2, since K and R are both perfect complexes with the same support—namely, {(z)}— but there is also an elementary argument, exploiting the structure of the Koszul complex; see [17, Lemma 6.0.9]. This is precisely where the hypothesis that R is zero dimensional is used. □

Let Λ be an exterior algebra over k on r generators ξ_{i} in degree 1, and view it as a DG algebra with zero differential.

Lemma 5.9. The assignment ξ_{j} ↦ ∑_{1⩽h⩽i⩽r} c_{hi,j}z_{h}y_{i} induces a morphism Λ → K of DG algebras that is a quasi-isomorphism.

Commentary in lieu of a proof. The said assignment induces a map Φ: Λ → K of graded algebras, by the universal properties of exterior algebras; note that the graded algebra underlying K is strictly graded-commutative, for that too is an exterior algebra. Given (5.1), it is clear that ∑_{h,i} c_{hi,j}z_{h}y_{i} is a cycle in K for each j, so Φ is a morphism of DG algebras. One can complete the proof by invoking the fact that these cycles form a basis for the k-vectorspace H_{1}(K) and that H(K) is the exterior algebra on H_{1}(K); see [22, Theorem 6].

For a proof specific to the case of Example 5.3, see [8, Lemma 7.1]. □

The quasi-isomorphism Λ → K of DG algebras from Lemma 5.9 induces an equivalence i: D^{f}(K) → D^{f}(Λ) of triangulated categories; see [2, 3.6]. This, along with the functors t and r from Construction 5.7 makes for a bridge from R to Λ:

\xymatrix{D^{f}(R) \ar[r]^{t} & D^{f}(K) \ar[l]^{r} \ar[r]_{≡}^{i} & D^{f}(Λ)}  (5.10)

The morphisms R → K ⟵^{≃} Λ of DG algebras induce a homomorphism of kalgebras Ext_{Λ}^{∗}(k, k) → Ext_{R}^{∗}(k, k). This map is one-to-one, and its image is precisely the subalgebra k[θ]. In this way, we identify V_{R}(k) and V_{Λ}(k).

Proposition 5.11. There is an equality V_{R}(M) = V_{Λ}(itM) for each M in D^{f}(R).

Proof. Adjunction gives an isomorphism Ext_{Λ}(k, itM) ≅ Ext_{R}(k, M) and this is compatible with the homomorphism of k-algebras Ext_{Λ}(k, k) → Ext_{R}(k, k). □

Proof of Theorem 5.6. The hypothesis translates to V_{Λ}(itM) ⊆ V_{Λ}(itN), by Proposition 5.11, so Theorem 4.4 yields that itM is in Thick_{Λ}(itN). Since i is an equivalence, this gives that tM is in Thick_{K}(tN), and hence the inclusion below:

Thick_{R}(M) = Thick_{R}(rtM) ⊆ Thick_{R}(rtN) = Thick_{R}(N) .

The equalities are from Proposition 5.8. □

Remark 5.12. Theorem 5.6 carries over to general artinian complete intersection rings, and the proof is identical, except for one step, Lemma 5.9: there may be no morphisms between the DGA algebras K and Λ for the ring R may not contain its residue field as a subring. However, they are still linked by a chain quasiisomorphisms (see, for example, [3, Lemma 6.4]) and that is all that is needed.

A solution to the realizability problem for D^{f}(R) is contained in Bergh [9, Corollary 2.3]; see also [4, Theorem 7.8]: For any closed subset V ⊆ V_{R}(k) there exists a complex M in D^{f}(R) with the property that V_{R}(M) = V. For another proof, closer to the spirit of this paper, see Avramov and Jorgensen [5]. From these and Theorem 5.6, one obtains a bijection between the thick subcategories of D^{f}(R) and specialization closed subsets of V_{R}(k), for any artinian complete intersection ring R; see the proof of Corollary 3.5.

Stevenson [21, Theorem 10.5] has established an analogous classification for any local complete intersection ring; see also [18].

6. Group algebras

The central result in this section is an extension of Hopkins’ Theorem to group algebras. Throughout we assume that G is any finite group and that k is a field of characteristic p > 0 dividing |G|. Keeping in line with notation in previous sections, we write D^{f}(kG) for the bounded derived category of finitely generated kG-modules. This is a tensor triangulated category, where the tensor product M ⊗N of complexes M and N is the complex M ⊗_{k} N with the diagonal G-action.

Given a subgroup E of G and a complex M of kG-modules, we write M↓_{E}^{G} for the complex of kE-modules obtained from M by restriction along the inclusion kE ⊆ kG. Given a complex N of kE-modules, we write N↑_{E}^{G} for complex kG⊗_{kE} N of kG-modules induced from N.

Our main tool is the following result from [12], see also [11, Theorem 8.2.7], that leads to a connection between thick subcategories of the bounded derived category of kG with those of the group algebras of the elementary abelian p-subgroups of G.

Theorem 6.1. There exists a kG-module V and a filtration

{0} = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{t} = k ⊕ V

where for every i = 1, …, t, there is an elementary abelian p-subgroup E_{i} ⊆ G and a finitely generated kE_{i}-module W_{i} such that M_{i}/M_{i−1} ≅ (W_{i})↑_{E_{i}}^{G}. □

In [12, Corollary 2.4] this result was used to prove that the stable category stmod(kG) is generated by modules induced from elementary abelian p-subgroups of G. The same is true for the bounded derived category.

Corollary 6.2. With E_{1}, …, E_{t} the elementary abelian p-subgroups of G from Theorem 6.1, there is an equality

D^{f}(kG) = Thick_{G}(⨁_{i=1}^{t} k↑_{E_{i}}^{G}) .

Proof. Any complex C of kG-modules is a direct summand of C ⊗ (k ⊕ V), which has a filtration by submodules C ⊗ M_{i} with quotients

(C ⊗ M_{i})/(C ⊗ M_{i−1}) ≅ C ⊗ (M_{i}/M_{i−1}) ≅ C ⊗ (W_{i})↑_{E_{i}}^{G} ≅ (C↓_{E_{i}}^{G} ⊗ W_{i})↑_{E_{i}}^{G} ,

the last isomorphism being Frobenius reciprocity. Thus C is in the thick subcategory generated by the (C↓_{E_{i}}^{G} ⊗ W_{i})↑_{E_{i}}^{G}.

It remains to note that when C is in D^{f}(G), the complex C↓_{E_{i}}^{G} ⊗ W_{i} is in D^{f}(E_{i}), and hence in Thick_{E}(k); see Lemma 5.4. □

We recall a construction of cohomological varieties for complexes of kG-modules. The cohomology algebra H^{∗}(G, k) of G, that is to say the k-algebra Ext_{kG}^{∗}(k, k), is

graded commutative and finitely generated, and hence noetherian. As usual, H^{•}(G) denotes ⊕_{i⩾0} H^{2i}(G, k) when p is odd, and all of H^{∗}(G, k) when p = 2. In any case, H^{•}(G) is a commutative noetherian subalgebra of H^{∗}(G, k).

For any complex M of kG-modules, there is a homomorphism of k-algebras

H^{∗}(G, k) → Ext_{G}^{∗}(M, M) ,

so Ext_{G}^{∗}(M, M) is endowed with a structure of a module over H^{•}(G); it is a finitely generated module when M is in D^{f}(kG); see, for example, [11, Theorem 6.5.1]. We note that the result in [11] are stated for kG-modules, but the arguments are easily adapted to apply to any complex in D^{f}(kG). Alternatively, one can deduce the result for complexes from the one for modules using the fact that any M ∈ D^{f}(kG) is in the thick subcategory generated by its total homology module, H(M).

Definition 6.3. The support variety of any complex M ∈ D^{f}(kG) is the subset

V_{G}(M) = Supp_{H^{•}(G)} Ext_{G}^{∗}(M, M) ⊆ Spec^{∗} H^{•}(G) .

Since V_{G}(k) = Spec^{∗} H^{•}(G) the support of M may be viewed as a subset of V_{G}(k).

Remark 6.4. We need support varieties for objects in D^{f}(kG) and so have to work with the affine variety Spec^{∗} H^{•}(G, k). If one is interested only in stmod(kG), then one could consider instead the corresponding projective varieties.

Remark 6.5. Let E be an elementary abelian p-group E of rank r; its group algebra is then isomorphic to the complete intersection R = k[z_{1}, …, z_{r}]/(z_{1}^{p}, …, z_{r}^{p}). For any M in D^{f}(kE), the two notions of support varieties of M in Definitions 6.3 and 5.5 are naturally isomorphic:

V_{E}(M) ≅ V_{R}(M) .

Indeed, to begin with, since E is a p-group, there is an equality of supports:

V_{E}(M) = Supp_{H^{•}(E)} Ext_{E}^{∗}(k, M)

Now let k[θ] be as in Remark 5.2. One has inclusions of k-subalgebras k[θ] ⊆ H^{•}(E), and the k[θ] action on Ext^{∗}(k, M) factors through this inclusion. The induced map

Supp_{H^{•}(E)} Ext_{E}^{∗}(k, M) ⟶^{≅} Supp_{k[θ]} Ext_{E}^{∗}(k, M) = V_{R}(M)

is an isomorphism; see [1, Theorem 7.1].

A thick subcategory C of D^{f}(kG) is tensor ideal provided M ⊗N is in C whenever one of M or N is in C. We write Thick_{G}^{⊗}(N) for the tensor ideal thick subcategory of D^{f}(kG) generated by a complex N.

If D^{f}(kG) = Thick_{G}(k), then we have that Thick_{G}^{⊗}(M) = Thick(M) for any complex M of kG-modules. This happens, for example, if G is a p-group.

With this background we get a version of Hopkins’ Theorem for group algebras.

Theorem 6.6. Let G be a finite group and k a field of characteristic p > 0. If M, N are complexes in D^{f}(kG) with V_{G}(M) ⊆ V_{G}(N), then M is in Thick_{G}^{⊗}(N).

Proof. Let E be an elementary abelian p-subgroup of G. The hypothesis then yields the inclusion below:

V_{E}(M↓_{E}^{G}) = (res_{G,E}^{∗})^{−1} V_{G}(M) ⊆ (res_{G,E}^{∗})^{−1} V_{G}(N) = V_{E}(N↓_{E}^{G})

The equalities are by the Subgroup Restriction Theorem [11, Theorem 9.6.2]. Thus, keeping in mind Remark 6.5, one gets from Theorem 5.6 that M↓_{E}^{G} is in Thick_{E}(N↓_{E}^{G}), and hence, by Frobenius reciprocity, also that

M ⊗ k↑_{E}^{G} ∈ Thick_{G}(N ⊗ k↑_{E}^{G}) ⊆ Thick_{G}^{⊗}(N) .

Let now E_{1}, …, E_{t} the elementary abelian p-subgroups of G from Theorem 6.1. The inclusion above then yields the second inclusion below:

Thick_{G}(M) ⊆ Thick_{G}(⨁_{i=1}^{t}(M ⊗ k↑_{E_{i}}^{G})) ⊆ Thick_{G}^{⊗}(N) .

while the first one is by Corollary 6.2. This completes the proof of the theorem. □

For any specialization closed subset of V of V_{G}(k), we write D^{f}(kG)_{V} for the subcategory consisting of all M in D^{f}(kG) such that V_{G}(M) ⊆ V. Using the properties of support from [11, Proposition 9.7.2], it is not difficult to see that D^{f}(kG)_{V} is a thick tensor ideal subcategory of D^{f}(kG). Arguing as in the proof of Corollary 3.5, and using Theorem 6.6 instead of Theorem 3.2, one get the following result.

Corollary 6.7. If C is a tensor ideal thick subcategory of D^{f}(kG), then there exist a specialization closed subset V of V_{G}(k) such that C = D^{f}(kG)_{V}. □

Now, for any closed subset V ⊆ V_{G}(k), one can construct an object M ∈ D^{f}(kG) with V_{G}(M) = V, either using either the tensor product theorem from [11, Theorem 9.6.4], or the more elementary approach from [4, §5]. Combined with Corollary 6.7 this gives a bijection between tensor ideal thick subcategories of D^{f}(kG) and specialization closed subsets of V_{G}(k).

There is an equivalence D^{f}(kG)/ Thick(kG) ≡ stmod(kG) of triangulated categories. From this and the corollary above, it is easy to deduce the characterization of the tensor ideal thick subcategories of stmod(kG) proved in [6].

References

[1] L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), 71-101.

[2] L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar and C. Miller, Homology of perfect complexes, Advances in Math. 223 (2010), 1731-1781. Corrigendum: Advances in Math. 225 (2010), 3576-3578.

[3] L. L. Avramov and S. B. Iyengar, Cohomology over complete intersections via exterior algebras, Triangulated categories (Leeds, 2006), London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press, Cambridge, 2010, 52-75.

[4] L. L. Avramov and S. B. Iyengar, Constructing modules with prescribed cohomological support, Ill. Jour. Math. 51 (2007), 1-20.

[5] L. L. Avramov and D. A. Jorgensen, Reverse homological algebra over some local rings, in preparation.

[6] D. Benson, J. Carlson and J. Rickard, Thick subcategories of the stable category, Fund. Math. 153 (1997), 59-80.

[7] D. J. Benson, S. B. Iyengar, and H. Krause, Stratifying triangulated categories, J. Topology 4 (2011), 641-666.

[8] D. J. Benson, S. B. Iyengar, and H. Krause, Stratifying modular representations of finite groups, Annals of Math 174 (2011), 1643-1684.

[9] P. Bergh, On support varieties of modules over complete intersections, Proc. Amer. Math. Soc., 135 (2007), 3795-3803.

[10] W. Bruns, J. Herzog, Cohen-Macaulay rings, revised ed., Studies in Advanced Mathematics 39, University Press, Cambridge, 1998.

[11] J. F. Carlson, L. Townsley, L. Valeri-Elizondo, M. Zhang, Cohomology rings of finite groups. With an appendix: Calculations of cohomology rings of groups of order dividing 64, Algebras and Applications, 3. Kluwer Academic Publishers, Dordrecht, 2003.

[12] J. F. Carlson, Cohomology and induction from elementary abelian subgroups, Quart. J. Math. 51 (2000), 169-181.

[13] E. Friedlander and J. Pevtsova, Π-supports for modules for finite group schemes over a field, Duke Math. J., 139 (2007), no. 2, 317-368.

[14] T. H. Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167-183.

[15] D. Happel, Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, Cambridge University Press, Cambridge, 1988.

[16] M. J. Hopkins, Global methods in homotopy theory, Homotopy Theory, Durham 1985, Lecture Notes in Mathematics, vol. 117, Cambridge University Press, 1987.

[17] M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy theory, Mem. AMS, vol. 128, American Math. Society, 1997.

[18] S. B. Iyengar, Thick subcategories and localizing subcategories of derived categories, lectures at workshop “Algebraic triangulated categories and related topics”, RIMS, Kyoto, July

2009. http://www.mi.s.osakafu-u.ac.jp/ kiriko/seminar/09JulRIMS.html

[19] A. Neeman, The chromatic tower for D(R), Topology 31 (1992), 519-532.

[20] G. Sjödin, A set of generators for Ext_{R}(k, k), Math. Scand. 38 (1976), 1-12.

[21] G. Stevenson, Subcategories of singularity categories via tensor actions, preprint 2011;

arXiv:1105.4698

[22] J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14-25.

[23] R. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), 1-27.

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

E-mail address: jfc@math.uga.edu

Department of Mathematics, University of Nebraska, Lincoln, NE 68588, USA

E-mail address: siyengar2@unl.edu