Decomposing the complexity quotient category

D. J. Benson Department of Mathematics University of Georgia Athens, GA 30602, USA

1 Introduction

In the modular representation theory of finite groups, much recent effort has gone into describing cohomological properties of the category of finitely generated modules. In recent joint work of the author with Jon Carlson and Jeremy Rickard [3], it has become clear that for some purposes the finiteness restriction is undesirable. In particular, in the quotient category of kG-modules by the subcategory of modules of less than maximal complexity, it turns out that finitely generated modules can have infinitely generated summands, and that including these summands in the category repairs the lack of Krull–Schmidt property.

In this paper we investigate the decomposition of the complexity quotient category induced by the decomposition of the trivial module into indecomposable pieces. We stress that in order to decompose the trivial module, it is necessary to include some infinitely generated modules in the category. The indecomposable summands of the trivial module are in one to one correspondence with the conjugacy classes of elementary abelian subgroups of maximal rank. The summand of the category corresponding to a particular elementary abelian subgroup is equivalent to the summand of the corresponding category at the level of the normalizer, so these summands are locally determined.

These decompositions turn out to be related to some observations made in [4, 5] about the lack of commutativity of cohomology rings of modules. In general, although the cohomology ring of a group (i.e., the cohomology ring of the trivial module) is graded commutative, the cohomology ring of a module (under Yoneda composition) can be quite far from being graded commutative. It can have complete matrix rings as quotients, even when the module is simple. We shall see that in the observed cases of this phenomenon, we can always explain this by decomposing the module in the complexity quotient category, possibly into infinitely generated summands. We end with some questions suggested by this observation.

In subsequent work closely related to the work described here, Rickard has produced (infinitely generated) idempotent modules in the stable module category, corresponding to general closed subsets and open subsets of the cohomology variety. These give infinite decompositions of the lower complexity quotient categories.

2 Notation and statement of main results

Throughout the paper, the following notation and conventions will be used. We let G be a finite group, and k be an algebraically closed field of characteristic p dividing |G|. We write Mod(kG) for the category of kG-modules, and mod(kG) for the category of finitely generated kG-modules. We write StMod(kG) and stmod(kG) for the corresponding stable categories, in which homomorphisms which factor through a projective module have been factored out.

Recall from [3] that complexity (Definition 4.3) and varieties (Definition 7.2) are defined for modules in Mod(kG). The main difference from the theory for finitely generated modules is that the variety of a module M is no longer a single subvariety V_{G}(M) of the cohomology variety V_{G}(k), but rather a collection V_{G}(M) of closed homogeneous subvarieties. In the finitely generated case, V_{G}(M) is equal to the collection of closed homogeneous subvarieties of V_{G}(M), so no extra information is present. The complexity of M in general is equal to the maximal dimension of an element of V_{G}(M). The complexity of the trivial module, namely the dimension of V_{G}(k), is equal to the p-rank r = r_{p}(G) of the group G; namely the maximal rank of an elementary abelian p-subgroup E of G (Quillen [11]).

We write M_{c} for the full subcategory of stmod(kG) consisting of modules of complexity at most c, and \underline{U_{c}} for the full subcategory of StMod(kG) consisting of modules of complexity at most c. Each of these subcategories is triangulated, so if c′ < c we may form the quotients M_{c}/M_{c′} and \underline{U_{c}}/\underline{U_{c′}}. If M_{1} and M_{2} are objects in \underline{U_{c}}/\underline{U_{c′}}, we write Hom^{∗}(M_{1}, M_{2}) for the -graded vector space defined by

Hom^{i}(M_{1}, M_{2}) = Hom_{\underline{U_{c}}/\underline{U_{c′}}}(Ω^{i}(M_{1}), M_{2}).

Translation followed by composition gives well defined maps

Hom^{j}(M_{2}, M_{3}) ⊗ Hom^{i}(M_{1}, M_{2}) → Hom^{i+j}(M_{1}, M_{3}).

The obvious functor M_{c}/M_{c′} → \underline{U_{c}}/\underline{U_{c′}} is fully faithful (Proposition 6.2 of [3]). We define M_{c,c′} to be the smallest subcategory of \underline{U_{c}}/\underline{U_{c′}} which contains M_{c}/M_{c′} and is closed under taking direct summands. Then M_{c,c′} is triangulated (Lemma 6.4 of [3]), and the Krull–Schmidt theorem holds in M_{c,c−1} (Theorem 6.7 of [3]). An object M in M_{c,c−1} has a variety V_{G}^{(c)}(M), which is a well defined subvariety of V_{G}(k) all of whose irreducible components have dimension c. See the discussion preceeding Proposition 8.5 of [3] (we have changed the notation from V_{G}^{c}(M) for cosmetic reasons; namely that V_{G}^{r}(M) usually denotes the rank variety of a module).

We are particularly interested in the categories M_{r}/M_{r−1} and M_{r,r−1}, which we regard as full subcategories of \underline{U_{r}}/\underline{U_{r−1}}. We call an isomorphism in \underline{U_{r}}/\underline{U_{r−1}} a “complexity isomorphism”.

Theorem 2.1 The categories M_{r,r−1} and \underline{U_{r}}/\underline{U_{r−1}} decompose as direct sums of categories

M_{r,r−1} = M_{G,E_{1}} ⊕ ⋯ ⊕ M_{G,E_{s}}
\underline{U_{r}}/\underline{U_{r−1}} = \underline{U}_{G,E_{1}} ⊕ ⋯ ⊕ \underline{U}_{G,E_{s}}.

Here, E_{1}, …, E_{s} are representatives of the conjugacy classes in G of elementary abelian p-subgroups of rank r, and M_{G,E_{i}} and \underline{U}_{G,E_{i}} are the full subcategories whose objects are the modules whose varieties are contained in the irreducible component V_{G,E_{i}} = res_{G,E_{i}}^{∗}V_{E_{i}}(k) of the cohomology variety V_{G}(k). Each of the subcategories M_{G,E_{i}} and \underline{U}_{G,E_{i}} is a full triangulated subcategory which is closed under taking tensor products.

Let N_{i} be the normalizer N_{G}(E_{i}) and D_{i} the “diagonalizer”

D_{i} = {g ∈ G | ∃n > 0 s.t. ∀h ∈ E_{i} ghg^{−1} = h^{n}},

and set n_{i} = |N_{i} : D_{i}|.

Theorem 2.2 Induction and restriction

M_{G,E_{i}} \overrightarrow{⟵}M_{N_{i},E_{i}} \underline{U}_{G,E_{i}} \overrightarrow{⟵} \underline{U}_{N_{i},E_{i}}

are equivalences of categories.

Induction and restriction

M_{N_{i},E_{i}} \overrightarrow{⟵}M_{D_{i},E_{i}} \underline{U}_{N_{i},E_{i}} \overrightarrow{⟵} \underline{U}_{D_{i},E_{i}}

have the property that the composites, either way round, are equivalent to the functor taking a module to a direct sum of n_{i} copies of the module.

3 Decomposing the trivial module

We begin by recalling the following theorem:

Theorem 3.1 [6] Let n be an integer divisible by n_{i} = |N_{i} : D_{i}| for 1 ≤ i ≤ s. Then there exists an integer m > 0 such that for all integers t > 0 there exists a kG-module L of complexity less than r = r_{p}(G), a projective kG-module P, and a short exact sequence

0 → L → n.Ω^{mt}(k) ⊕ P → ⨁_{i=1}^{s}(n/n_{i}).k_{D_{i}}↑^{G}→ 0.

Corollary 3.2 In \underline{U_{r}}/\underline{U_{r−1}} we have

n.k ≅ ⨁_{i=1}^{s}(n/n_{i}).k_{D_{i}}↑^{G} .

Proof This follows from the Theorem and the fact that every object in \underline{U_{r}}/\underline{U_{r−1}} is periodic. □

It is proved in Carlson, Donovan and Wheeler [7] that the homomorphisms Hom^{∗}(M, N) in M_{c}/M_{c−1} has the following description. Let S_{M} be the multiplicative subset of H^{∗}(G, k) consisting of the homogeneous elements which do not belong to any prime ideal of dimension c containing the annihilator of Ext_{kG}^{∗}(M, M). Then Hom^{∗}(M, N) is equal to the homogeneous localization Ext_{kG}^{∗}(M, N)S_{M}^{−1}. Using Lemma 6.1 of [3], one can see that the same description holds if M is finitely generated and N is any module.

In the case where M = k and c = r, the ring Hom^{0}(k, k) is a cartesian product of Artinian local rings which are in one to one correspondence with the conjugacy classes of elementary abelian p-subgroups of rank r. The ith direct factor of the endomorphism ring has the property that modulo its radical, it is the function field of the projective variety V̄_{G,E_{i}}.

Now idempotents in \underline{U_{r}}/\underline{U_{r−1}} split (see Section 3 of [3] for a discussion of this), and so the trivial module decomposes as a direct sum of pieces corresponding to the representatives E_{1}, …, E_{s} of the conjugacy classes of elementary abelian p-subgroups of rank r. We write π_{G,E_{i}}(k) for the summand of k corresponding to E_{i}. So in either \underline{U_{r}}/\underline{U_{r−1}} or M_{r,r−1} we have

k ≅ π_{G,E_{1}}(k) ⊕ ⋯ ⊕ π_{G,E_{s}}(k).

The endomorphism ring of each π_{G,E_{i}}(k) is local, so that this is the unique decomposition of k into indecomposable pieces.

Lemma 3.3 If M is an object in M_{c,c−1}, then the variety V_{G}^{(c)}(M) is the union of the c-dimensional components of the variety of the kernel of the map H^{∗}(G, k) → Hom^{∗}(M, M).

Proof Suppose that M is a direct summand of a finitely generated module M_{0}. Let S be the multiplicative subset of H^{∗}(G, k) consisting of the homogeneous elements which do not belong to any prime ideal of dimension c containing the annihilator of Ext_{kG}^{∗}(M_{0}, M_{0}). Then for any finitely generated module N, Hom^{∗}(N, M) ≅ Ext_{kG}^{∗}(N, M)S^{−1} is a direct summand of Hom^{∗}(N, M_{0}) ≅ Ext_{kG}^{∗}(N, M_{0})S^{−1} as a module over H^{∗}(G, k).

The annihilator in H^{∗}(G, k) of Ext_{kG}^{∗}(N, M) has the same c dimensional components as the annihilator of Hom^{∗}(N, M), because of the description of S. The action on Hom^{∗}(N, M) factors through the map H^{∗}(G, k) ≅ Ext_{kG}^{∗}(k, k) → Hom^{∗}(M, M), and Hom^{∗}(M, M) is a direct summand of Hom^{∗}(M_{0}, M). So the smallest annihilator of Hom^{∗}(N, M) for finite dimensional N is the annihilator of Hom^{∗}(M_{0}, M). This has the same annihilator as Hom^{∗}(M, M). □

Lemma 3.4 We have V_{G}^{(r)}(π_{G,E_{i}}(k)) = V_{G,E_{i}}.

Proof This follows from the last lemma, together with the fact that the endomorphism ring Hom^{∗}(π_{G,E_{i}}(k), π_{G,E_{i}}(k)) is equal to the direct factor of H^{∗}(G, k)S^{−1} corresponding to the component of the variety associated to E_{i}. Here, S is the collection of all homogeneous elements of H^{∗}(G, k) which do not belong to any prime ideal of dimension r. □

Lemma 3.5 We have π_{G,E_{i}}(k) ⊗ π_{G,E_{i}}(k) ≅ π_{G,E_{i}}(k), and π_{G,E_{i}}(k) ⊗ π_{G,E_{j}}(k) = 0 for i ≠ j.

Proof According to Proposition 8.2 of [3], the variety of a tensor product of objects in M_{r,r−1} is contained in the intersection of their varieties. It follows that for i ≠ j, V_{G}^{(r)}(π_{G,E_{i}}(k) ⊗ π_{G,E_{j}}(k)) is empty, so that π_{G,E_{i}}(k) ⊗ π_{G,E_{j}}(k) is zero in M_{r,r−1}. It also follows that the variety of π_{G,E_{i}}(k)⊗π_{G,E_{i}}(k) is contained in V_{G,E_{i}}. Now expand out the isomorphism k ⊗ k ≅ k using the direct sum decomposition of k. □

For any object M in \underline{U_{r}}/\underline{U_{r−1}}, we define

π_{G,E_{i}}(M) = π_{G,E_{i}}(k) ⊗ M.

Lemma 3.6 We have π_{G,E_{i}}(π_{G,E_{i}}(M)) = π_{G,E_{i}}(M), π_{G,E_{j}}(π_{G,E_{i}}(M)) = 0 for i ≠ j, and

M ≅ π_{G,E_{1}}(M) ⊕ ⋯ ⊕ π_{G,E_{s}}(M).

Proof This follows immediately from the above lemma. □

We define M_{G,E_{i}} to be the full subcategory of M_{r,r−1} consisting of modules of the form π_{G,E_{i}}(M), for M in M_{r,r−1}.

Theorem 3.7 The category M_{r,r−1} decomposes as a direct sum of triangulated subcategories

M_{r,r−1} = M_{G,E_{1}} ⊕ ⋯ ⊕ M_{G,E_{s}}.

Each M_{G,E_{i}} is closed under taking tensor products, and consists of the objects M in M_{r,r−1} with the property that V_{G}^{(r)}(M) = V_{G,E_{i}}, together with the zero object. Proof This follows immediately from the above properties of the functors π_{G,E_{i}}. □

Similarly, we define \underline{U}_{G,E_{i}} to be the full subcategory of \underline{U_{r}}/\underline{U_{r−1}} consisting of modules of the form π_{G,E_{i}}(M), for M in \underline{U_{r}}/\underline{U_{r−1}}. Theorem 3.8 The category \underline{U_{r}}/\underline{U_{r−1}} decomposes as a direct sum of triangulated subcategories

\underline{U_{r}}/\underline{U_{r−1}} = \underline{U}_{G,E_{1}} ⊕ ⋯ ⊕ \underline{U}_{G,E_{s}}.

Each \underline{U}_{G,E_{i}} is closed under taking tensor products, and consists of the objects M in \underline{U_{r}}/\underline{U_{r−1}} with the property that there is a representative M_{0} in Mod(kG) of the isomorphism class of M in \underline{U_{r}}/\underline{U_{r−1}} with V_{G}(M_{0}) ⊆ V_{G,E_{i}}. □

This completes the proof of Theorem 2.1.

4 Local control

In this section, we demonstrate that the categories M_{r,r−1} and \underline{U_{r}}/\underline{U_{r−1}} are locally determined. Lemma 4.1 We have k_{D_{i}}↑^{G}≅ n_{i}.π_{G,E_{i}}(k) in M_{r,r−1}. Proof By Corollary 3.2 and the decomposition of the trivial module, we have

n.π_{G,E_{1}}(k) ⊕ ⋯ ⊕ n.π_{G,E_{s}}(k) ≅ ⨁_{i=1}^{s}(n/n_{i}).k_{D_{i}}↑^{G} .

By Lemma 3.4 and the Krull–Schmidt theorem for the category M_{r,r−1} (Theorem 6.7 of [3]) this implies that

n.π_{G,E_{i}}(k) ≅ (n/n_{i}).k_{D_{i}}↑^{G},

and the lemma follows by another application of the Krull–Schmidt theorem. □

Lemma 4.2 Suppose that H is a subgroup of G containing E_{i}. Then we have π_{G,E_{i}}(k)↓_{H} ≅ π_{H,E_{i}}(k) in M_{r,r−1}.

Proof This follows from the fact that if we take the idempotent corresponding to π_{G,E_{i}}(k) in the endomorphism ring of k as an object in M_{r,r−1}, and restrict it to H, we get the corresponding idempotent in the endomorphism ring of k there. □

Lemma 4.3 Suppose that H is a subgroup of G containing the diagonalizer D_{i}. Set h = |N_{i} : N_{i} ∩ H|. Then we have π_{H,E_{i}}(k)↑^{G}≅ h.π_{G,E_{i}}(k) in M_{r,r−1}.

Proof Applying Lemma 4.1 to H and to G, and using the fact that the index |N_{H}(E_{i}) : D_{i}| is equal to n_{i}/h, we obtain

(n_{i}/h).π_{H,E_{i}}(k)↑^{G}≅ k_{D_{i}}↑^{H}↑^{G}≅ k_{D_{i}}↑^{G}≅ n_{i}.π_{G,E_{i}}(k).

Now apply the Krull–Schmidt theorem for the category M_{r,r−1} (Theorem 6.7 of [3]). □

Theorem 4.4 Induction and restriction give equivalences of categories

M_{G,E_{i}} \overrightarrow{⟵}M_{N_{i},E_{i}} \underline{U}_{G,E_{i}} \overrightarrow{⟵} \underline{U}_{N_{i},E_{i}}.

Proof The case H = N_{i} of Lemmas 4.2 and 4.3 gives us π_{G,E_{i}}(k)↓_{N_{i}} ≅ π_{N_{i},E_{i}}(k) and π_{N_{i},E_{i}}(k)↑^{G}≅ π_{G,E_{i}}(k). Tensoring with objects in the categories involved proves the theorem. □

Theorem 4.5 The composites of induction and restriction either way round,

M_{N_{i},E_{i}} \overrightarrow{⟵}M_{D_{i},E_{i}} \underline{U}_{N_{i},E_{i}} \overrightarrow{⟵} \underline{U}_{D_{i},E_{i}}

are equivalent to the functor taking a module to a direct sum of n_{i} copies of the module.

Proof Replacing G by N_{i} and H by D_{i} in Lemmas 4.2 and 4.3, we obtain π_{N_{i},E_{i}}(k)↓_{D_{i}} ≅ n_{i}.π_{D_{i},E_{i}}(k) and π_{D_{i},E_{i}}(k)↑^{N_{i}} ≅ n_{i}.π_{N_{i},E_{i}}(k). Tensoring with objects in the categories involved proves the theorem. □

This completes the proof of Theorem 2.2.

Corollary 4.6 If M_{1} and M_{2} are finitely generated kG-modules (or direct summands thereof), then there is a complexity isomorphism between M_{1} and M_{2} if and only if there is a complexity isomorphism between M_{1}↓_{D_{i}} and M_{2}↓_{D_{i}} for each 1 ≤ i ≤ s.

Proof This follows from the above theorem, together with the Krull–Schmidt theorem for M_{r,r−1} (Theorem 6.7 of [3]). □

We do not know whether the corollary holds for all kG-modules.

5 Noncommutative Ext rings

In this section, we describe how the decompositions described in this paper relate to some known examples of noncommutative Ext rings for indecomposable (and even simple) modules. We begin with a theorem which explains some of the observations described in [5]. The notation is the same as in the previous sections.

Theorem 5.1 Let H be a subgroup of G containing D_{i}, and set M = k_{H} ↑^{G} and h = |N_{i} : N_{i} ∩ H|. Then there is an epimorphism of rings from the ring of degree zero elements in the localization Ext_{kG}^{∗}(M, M)S_{M}^{−1} to the ring of h × h matrices over the function field of the projective variety V̄_{G,E_{i}}.

Proof The endomorphism ring Hom^{∗}(M, M) of a module M in M_{r}/M_{r−1} is the homogeneous localization Ext_{kG}^{∗}(M, M)S_{M}^{−1}. Apply this to M = k_{H}↑^{G}. In M_{r,r−1} by Lemma 4.3, M has a summand π_{H,E_{i}}(k)↑^{G}≅ h.π_{G,E_{i}}(k). It follows that Hom^{∗}(M, M) has a summand Mat_{h}(Hom^{∗}(π_{G,E_{i}}(k), π_{G,E_{i}}(k))). The ring Hom^{0}(π_{G,E_{i}}(k), π_{G,E_{i}}(k)) is an Artinian local ring, and modulo its radical it is the function field of V̄_{G,E_{i}}. □

Corollary 5.2 Let H be a subgroup of G containing D_{i}, and set M = k_{H} ↑^{G} and h = |N_{i} : N_{i} ∩ H|. Then there is an epimorphism of rings

Ext_{kG}^{∗}(M, M) → Mat_{h}(k).

Proof This follows from the theorem, using the theory of affine polynomial identity algebras. See for example McConnell and Robson [10]. □

Next, we look at a family of examples from Carlson [4]. These are examples of finitely generated indecomposable modules over an elementary abelian p-group whose cohomology rings map onto matrix rings. Let p ≥ 5, and let G = ⟨x, y⟩ be an elementary abelian p-group of rank two. Let X = x − 1 and Y = y − 1. Let M_{n} (n ≥ 1) be the kG-module of k-dimension (n−1)p+n defined by generators a_{1}, …, a_{n} and relations XY a_{i} = 0 (1 ≤ i ≤ n); X^{p−1}a_{j−1} = Y^{2}a_{j} (2 ≤ j ≤ n − 1); Y^{2}a_{1} = 0; X^{p−1}a_{n−1} = Y a_{n}; and Xa_{n} = 0. Carlson [4] proves that each M_{n} is indecomposable, and that there is a surjective ring homomorphism Ext_{kG}^{∗}(M_{n}, M_{n}) → Mat_{n}(k). We explain this with the following observation.

Proposition 5.3 In the above example, in the quotient category M_{2}/M_{1} the module M_{n} decomposes as a direct sum of n copies of the trivial module.

Proof We define another family of modules M′_{n} as follows. Let M′_{n} (n ≥ 1) be the kG-module of k-dimension (n + 1)p + n defined by generators b_{1}, …, b_{n} and relations XY b_{i} = 0 (1 ≤ i ≤ n − 1); X^{p−1}b_{j−1} = Y^{2}b_{j} (2 ≤ j ≤ n − 1); Y^{2}b_{1} = 0; X^{p−1}b_{n−1} = Y^{3}b_{n}; and XY^{2}b_{n} = 0. We define maps ϕ_{n} : M_{n} → M′_{n} and ψ_{n} : M_{n} → M′_{n−1} ⊕ k which pass down to isomorphisms in M_{2}/M_{1}. Since M_{1} = k, this provides an inductive proof of the proposition.

The map ϕ_{n} : M_{n} → M′_{n} is defined by ϕ_{n}(a_{j}) = b_{j} (1 ≤ j ≤ n−1); ϕ_{n}(a_{n}) = Y^{2}b_{n}. Then ϕ_{n} is injective, and its cokernel is a 2p-dimensional module which is projective on restriction to ⟨x⟩. It follows that the cokernel is in M_{1}, so that ϕ_{n} passes down to an isomorphism in M_{2}/M_{1}.

The map ψ_{n} : M_{n} → M′_{n−1} ⊕ k is defined as follows. Let u be a basis element for the trivial module k. Then ψ_{n}(a_{j}) = b_{j} (1 ≤ j ≤ n − 2); ψ_{n}(a_{n−1}) = Y b_{n}; ψ_{n}(a_{n}) = X^{p−1}b_{n−1} + u. Then ψ_{n} is injective, and its cokernel is a p-dimensional module which is projective on restriction to ⟨x⟩. It follows that the cokernel is in M_{1}, so that ψ_{n} passes down to an isomorphism in M_{2}/M_{1}. □

These observations prompt several speculations. For an elementary abelian pgroup of rank r, could it be true that every direct summand of a finitely generated module in \underline{U_{r}}/\underline{U_{r−1}} is again isomorphic to a finitely generated module; in other words, that M_{r,r−1} = M_{r}/M_{r−1}?

In general, for an indecomposable object in M_{r,r−1}, the endomorphism ring modulo its radical is a division ring; is it always a field? If this is not true in general, is it true for elementary abelian p-groups?

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