HOMOMORPHISMS IN HIGHER

COMPLEXITY QUOTIENT CATEGORIES

Jon F. Carlson^{∗} and Wayne W. Wheeler^{∗}

Abstract. In previous work the authors together with Peter Donovan initiated a study of complexity quotient categories for the modular group algebra of a finite group G. Let M_{c} denote the full subcategory of the stable category consisting of all finitely generated modules having complexity at most c. The previous work studied quotients of the form M_{c}/M_{c−1}; the current paper is devoted to studying quotients M_{c}/M_{d} when c − d ≥ 2. The main tool used here, a chain complex called the modular Koszul complex, gives rise to a spectral sequence that provides a description of homomorphisms in these higher complexity quotient categories. This description is then used to decompose the endomorphism ring of the trivial module by producing a complete set of primitive orthogonal idempotents determined by certain subvarieties of V_{G}(k). The paper also considers connections with Čech cohomology and with the idempotent modules studied by Rickard.

1. Introduction

In [12] the authors together with Peter Donovan began a study of complexity quotient categories for the modular group algebra of a finite group G. That categorical construction provided a new view of the module category that has either directly led to or at least partly inspired the discovery of several interesting methods and results [2, 3, 5, 6, 7, 11, 13, 14, 15, 20]. The main theorem of [12] says that the homomorphisms in certain complexity quotient categories can be obtained by a localization of cohomology. Specifically, let k be an algebraically closed field of characteristic p > 0, and let M_{c} be the full subcategory of the stable category consisting of all finitely generated kG-modules having complexity at most c. Then a morphism from M to N in the quotient category M_{c}/M_{c−1} can be represented by a triple (s, n, f), where n is a nonnegative integer, s : Ω^{n}k ⊗ M → M and f : Ω^{n}k ⊗ M → N are maps in the stable category, s = ζ ⊗ 1_{M} for some ζ ∈ H^{∗}(G, k), and the intersection of the variety of ζ with the variety of M has dimension at most c − 1. The morphism corresponding to the triple (s, n, f) is often represented diagrammatically in the form

(1.1)  M ⟵^{s} Ω^{n}k ⊗ M ⟶^{f} N.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20C20, 20J06.

∗Partially supported by a grant from the NSF.

The condition on the variety of ζ ensures that s is invertible in the quotient category so that this morphism can be thought of as fs^{−1}. Because s ∈ Êxt_{kG}^{n}(M, M) and f ∈ Êxt_{kG}^{n}(M, N), it follows that Hom_{M_{c}/M_{c−1}}(M, N) is really a localization of Êxt_{kG}^{∗}(M, N).

The purpose of this paper is to study a somewhat similar description of homomorphisms in higher complexity quotient categories. In other words, we wish to consider the functor Hom_{M_{c}/M_{d}}(−, −) for c − d ≥ 2, particularly when c is equal to the p-rank of G so that M_{c} is just the stable category stmod-kG of all finitely generated kG-modules. Some recent advances in the theory make this exercise especially relevant. Suppose that X is a full triangulated subcategory of stmod-kG that is closed under taking direct summands and under tensoring with objects of stmod-kG. Then Rickard [20] has shown that the stable category StMod-kG has a corresponding idempotent object F (X). The module F (X) is usually infinitely generated, and it has the property that F (X) ⊗ F (X) ≅ F (X) in StMod-kG. Moreover, the image of the functor − ⊗ F (X) : stmod-kG → StMod-kG is equivalent to the quotient category stmod-kG/X. If X = M_{d}, then it follows that

Hom_{stmod-kG/M_{d}}(k, k) ≅ \underline{Hom}_{kG}(F (X), F (X)).

More generally, suppose that V is a collection of closed homogeneous subvarieties of V_{G}(k) that is closed under finite unions and under specialization. If X is the full subcategory of stmod-kG consisting of all modules whose varieties are in V, then the methods of this paper can be used to determine \underline{Hom}_{kG}(F (X), F (X)). When G is a p-group, every full triangulated subcategory X of stmod-kG that is closed under taking direct summands arises in this way [7].

The first obstacle to obtaining a useful description of morphisms in the quotient categories M_{c}/M_{d} is the problem of finding a collection of modules to play the same role as the modules Ω^{n}k in (1.1). When c − d ≥ 2, the collection of modules {Ω^{n}k | n ≥ 0} is no longer large enough to represent morphisms in the form given by (1.1). Instead, let ρ denote the p-rank of G, set τ = ρ − d, and let Q^{τ} = stmod-kG/M_{d}. If ζ_{1}, …, ζ_{τ} are homogeneous elements of H^{∗}(G, k), then a corresponding module U = U(ζ_{1}, …, ζ_{τ}) and a distinguished homomorphism θ : U → k are defined in Section 2, and the collection of modules {U(ζ_{1}, …, ζ_{τ})} forms an appropriate class for describing the morphisms in Q^{τ}. There is a sense in which the pair (U, θ) represents the ideal I = (ζ_{1}, …, ζ_{τ}) in H^{∗}(G, k). This concept is explored further in [13], where it is shown that under the right circumstances the module U(ζ_{1}, …, ζ_{τ}) is independent of the choice of generators ζ_{1}, …, ζ_{τ} and also that the modules satisfy an analogue of the isomorphism Ω^{m}k ⊗ Ω^{n}k ≅ Ω^{m+n}k in stmod-kG.

In Section 3 we show that Êxt_{kG}^{∗}(U, k) is given as the limit of a spectral sequence whose E_{1} term is a truncated Koszul complex of cohomology rings. Some of the relations involved in changing the parameters ζ_{1}, …, ζ_{τ} are also investigated. The next section is devoted to studying the multiplicative structure on this spectral sequence. These ideas are pulled together in Section 5 to give a description of Hom_{Q^{τ}} (M, N) for finitely generated kG-modules M and N. In the next section we extend the ideas of [12] to obtain a decomposition of End_{Q^{τ}} (k) by producing a complete set of primitive orthogonal idempotents, and in Section 7 we describe the precise connections with the idempotent modules of [20]. Finally, Section 8 discusses a curious relationship between the spectral sequences we have constructed and the Čech cohomology sometimes used in sheaf theory. The section ends by considering examples. Many of the questions that one might ask about morphisms in Q^{τ} seem to be relatively simple when H^{∗}(G, k) is Cohen–Macaulay. In general, however, the answers seem to depend on the higher differentials in the spectral sequences, and many questions remain unresolved.

Some of the work presented in this paper was done while the first author was visiting the Centre for Mathematics and its Applications at the Australian National University. He would like to thank the people at the School of Mathematical Sciences at ANU for the hospitality that made the visit so productive and so much fun. In addition, the authors are grateful to Dave Benson for suggesting the tensor product construction used to define the directed system in Section 7.

2. A new class of modules

Throughout this paper we let G denote a finite group and k an algebraically closed field of characteristic p > 0. For each kG-module M fix an injective hull ι_{M} : M → I(M), and let Ω^{−1}M = Coker ι_{M}. If n > 1, then Ω^{−n}M is defined inductively by setting Ω^{−n}M = Ω^{−1}(Ω^{−n+1}M). For n > 0 the modules Ω^{n}M are defined by a dual procedure, beginning with a projective cover of M.

Now let (P_{∗}, ∂) be a fixed minimal projective resolution of M. Then Ω^{n}M ≅ ∂(P_{n}) for n > 0. If N is any kG-module, then the cohomology Ext_{kG}^{∗}(M, N) is the homology of the cochain complex Hom_{kG}(P_{∗}, N). Thus any cohomology class ζ ∈ Ext_{kG}^{∗}(M, N) is represented by a cocycle ζ′ ∈ Hom_{kG}(P_{n}, N) that induces a map ζ̂ : Ω^{n}M → N, and changing the representative cocycle changes ζ̂ by a homomorphism that factors through a projective module. If M = N = k, then the zero map is the only homomorphism in Hom_{kG}(Ω^{n}k, k) that factors through a projective module; hence ζ and ζ̂ determine each other uniquely. It will usually be convenient not to distinguish between the two, and we will write ζ instead of ζ̂ when no confusion is likely to arise. If ζ ≠ 0, then the module L_{ζ} is defined to be the kernel of ζ : Ω^{n}k → k. Thus there is a short exact sequence

0 ⟶ L_{ζ} ⟶ Ω^{n}k ⟶^{ζ} k ⟶ 0.

The cohomology ring H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) is a finitely generated graded-commutative k-algebra. Its maximal ideal spectrum V_{G}(k) is a homogeneous affine variety of dimension equal to the p-rank of the group G. If M and N are finitely generated kG-modules, then Ext_{kG}^{∗}(M, N) is finitely generated as a module over H^{∗}(G, k). The annihilator J(M) of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k) is a graded ideal, so it determines a homogeneous subvariety V_{G}(M) ⊆ V_{G}(k) consisting of all maximal ideals that contain J(M). The dimension of V_{G}(M) is the complexity cx_{G}(M) of M. If ζ ∈ H^{∗}(G, k) is homogeneous, then the subvariety V_{G}(ζ) ⊆ V_{G}(k) consists of all maximal ideals containing ζ. The following proposition lists some of the fundamental properties of the variety of a module. The reader is referred to [1] or [16] for more details and proofs.

Proposition 2.1.

(1) Let M be a finitely generated kG-module. Then V_{G}(M) = {0} if and only if M is projective. Moreover, V_{G}(M) = V_{G}(M^{∗}) = V_{G}(Ω^{n}M) for all n.

(2) Let M and N be finitely generated kG-modules. Then V_{G}(M ⊕ N) = V_{G}(M) ∪ V_{G}(N) and V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N).

(3) Suppose that 0 → M_{1} → M_{2} → M_{3} → 0 is an exact sequence of finitely generated kG-modules. If {l, m, n} = {1, 2, 3}, then

V_{G}(M_{l}) ⊆ V_{G}(M_{m}) ∪ V_{G}(M_{n}).

(4) If ζ ∈ H^{∗}(G, k) is homogeneous, then V_{G}(L_{ζ}) = V_{G}(ζ).

Let mod-kG denote the category of all finitely generated right kG-modules. If M and N are objects of mod-kG, let PHom_{kG}(M, N) denote the k-subspace of Hom_{kG}(M, N) consisting of all homomorphisms that factor through a projective kG-module. The stable category stmod-kG has the same objects as mod-kG, but the morphisms from M to N are given by

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/ PHom_{kG}(M, N).

When no confusion is likely to arise, we will use the same symbol for a kG-homomorphism and the corresponding morphism in stmod-kG; occasionally, however, it will be necessary to distinguish between them by writing cls(f) for the map in stmod-kG corresponding to the kG-homomorphism f. If Mod-kG denotes the category of all right kG-modules, then there is a corresponding stable category StMod-kG. In this paper we will restrict our attention to finitely generated modules, except in Section 7 when we consider the connections between complexity quotient categories and the idempotent modules of Rickard.

Because kG is a self-injective algebra, the stable category stmod-kG is triangulated by Theorem I.2.6 of [18], and the translation functor is given by Ω^{−1}. The precise connection between short exact sequences in mod-kG and distinguished triangles in stmod-kG is given in the following proposition, which is essentially a restatement of Lemmas I.2.3 and I.2.7 of [18].

Proposition 2.2. Suppose that

0 ⟶ M′ ⟶^{f} M ⟶^{g} M′′ ⟶ 0

is a short exact sequence in mod-kG. Then there is a triangle in stmod-kG of the form

M′ ⟶^{f} M ⟶^{g} M′′ ⟶ Ω^{−1}M′.

Moreover, every distinguished triangle in stmod-kG is isomorphic to one that arises from a short exact sequence in this way.

As an example of the way in which Proposition 2.2 can be used, suppose that ζ is a homogeneous element of H^{∗}(G, k) of degree n, and consider the short exact sequence defining L_{ζ}. This sequence determines a corresponding triangle

L_{ζ} ⟶ Ω^{n}k ⟶^{ζ} k ⟶^{ε} Ω^{−1}L_{ζ}

in stmod-kG. Shifting then gives another triangle of the form

k ⟶^{ε} Ω^{−1}L_{ζ} ⟶ Ω^{n−1}k ⟶^{−Ω^{−1}ζ} Ω^{−1}k,

and Proposition 2.2 implies that there exist a projective module P and a short exact sequence

0 ⟶ k ⟶^{ε′} Ω^{−1}L_{ζ} ⊕ P ⟶ Ω^{n−1}k ⟶ 0

such that ε′ agrees with ε modulo projective homomorphisms. This is the short exact sequence representing ζ as an element of Ext_{kG}^{n}(k, k) ≅ Ext_{kG}^{1}(Ω^{n−1}k, k). The fact that stmod-kG is a triangulated category will play a major part in the remainder of the paper, and the reader is referred to [18] or [21] for more information on triangulated categories.

Suppose that ζ is a homogeneous element of H^{∗}(G, k). Set n = deg ζ, and let I_{ζ} = I(L_{ζ}) be the injective hull of L_{ζ}. Then there is a complex

C(ζ) :  ⋯ ⟶ 0 ⟶ Ω^{n}k ⟶^{ζ′} k ⊕ I_{ζ} ⟶ 0 ⟶ ⋯

in which the module Ω^{n}k is in degree one, the module k⊕I_{ζ} is in degree zero, ζ′ is injective, and ζ′ is congruent to ζ modulo projective homomorphisms. Note that the complex C(ζ) has homology Ω^{−1}L_{ζ} concentrated in degree zero.

More generally, if ζ_{1}, …, ζ_{τ} are homogeneous elements of H^{∗}(G, k), then we define the modular Koszul complex C = C(ζ_{1}, …, ζ_{τ}) by setting

C = C(ζ_{1}, …, ζ_{τ}) = C(ζ_{1}) ⊗ ⋯ ⊗ C(ζ_{τ}).

Let L(ζ_{1}, …, ζ_{τ}) = Ω^{−1}L_{ζ_{1}} ⊗ ⋯ ⊗ Ω^{−1}L_{ζ_{τ}}. Then Proposition 2.1 implies that

V_{G}(L(ζ_{1}, …, ζ_{τ})) = ⋂_{i=1}^{τ} V_{G}(Ω^{−1}L_{ζ_{i}}) = V_{G}(ζ_{1}, …, ζ_{τ}).

Moreover, C has homology L(ζ_{1}, …, ζ_{τ}) concentrated in degree zero, so there is an exact sequence

0 ⟶ C_{τ} ⟶^{δ_{τ}} ⋯ ⟶^{δ_{2}} C_{1} ⟶^{δ_{1}} C_{0} ⟶^{ε} L(ζ_{1}, …, ζ_{τ}) ⟶ 0.

Define U(ζ_{1}, …, ζ_{τ}) = Ker ε, and let θ(ζ_{1}, …, ζ_{τ}) : U(ζ_{1}, …, ζ_{τ}) → C_{0} denote the inclusion map so that there is a short exact sequence

0 ⟶ U(ζ_{1}, …, ζ_{τ}) ⟶^{θ(ζ_{1},...,ζ_{τ})} C_{0} ⟶^{ε} L(ζ_{1}, …, ζ_{τ}) ⟶ 0.

Because C_{0} = C_{0}(ζ_{1})⊗⋯⊗C_{0}(ζ_{τ}) = (k ⊕I_{ζ_{1}})⊗⋯⊗(k ⊕I_{ζ_{τ}}), there is a map α : C_{0} → k given by

α((x_{1}, y_{1}) ⊗ ⋯ ⊗ (x_{τ} , y_{τ})) = x_{1} ⋯ x_{τ}

whenever x_{i} ∈ k and y_{i} ∈ I_{ζ_{i}} for 1 ≤ i ≤ τ. It is easy to see that α is a stable isomorphism, and we will use this map to identify C_{0} with k in stmod-kG throughout the remainder of the paper. With this identification it follows that there is a triangle

U(ζ_{1}, …, ζ_{τ}) ⟶^{θ(ζ_{1},...,ζ_{τ})} k ⟶^{ε} L(ζ_{1}, …, ζ_{τ}) ⟶ Ω^{−1}U(ζ_{1}, …, ζ_{τ})

in stmod-kG, and shifting gives a triangle

k ⟶^{ε} L(ζ_{1}, …, ζ_{τ}) ⟶ Ω^{−1}U(ζ_{1}, …, ζ_{τ}) ⟶^{−Ω^{−1}θ(ζ_{1},...,ζ_{τ})} Ω^{−1}k.

Proposition 2.2 implies that there exist a projective module P and a short exact sequence

E(ζ_{1}, …, ζ_{τ}) : 0 ⟶ k ⟶^{ε′} L(ζ_{1}, …, ζ_{τ}) ⊕ P ⟶ Ω^{−1}U(ζ_{1}, …, ζ_{τ}) ⟶ 0

such that ε′ and ε agree modulo projective homomorphisms. Moreover, if

Ω^{n_{i}}k ⟶^{ζ_{i}} k ⟶^{ε_{i}} Ω^{−1}L_{ζ_{i}} ⟶ Ω^{n_{i}−1}k

is the triangle determined by ζ_{i}, then one can check that ε = ε_{1} ⊗ ⋯ ⊗ ε_{τ}.

We are now ready to prove the main result of this section.

Proposition 2.3. Suppose that ζ_{1}, …, ζ_{τ} ∈ H^{∗}(G, k) are homogeneous, and let I = (ζ_{1}, …, ζ_{τ}) be the ideal of H^{∗}(G, k) generated by ζ_{1}, …, ζ_{τ}.

(1) If M is any finitely generated kG-module, then the exact sequence E(ζ_{1}, …, ζ_{τ})⊗M is split if and only if I ⊆ J(M).

(2) Suppose that

M′ ⟶^{α} M ⟶^{β} M′′ ⟶ Ω^{−1}M′

is a triangle in stmod-kG such that I ⊆ J(M′′). Then there exists a map ψ : U(ζ_{1}, …, ζ_{τ}) ⊗ M → M′ such that the diagram

[figure: \begin{aligned}& U(ζ_{1}, …, ζ_{τ}) ⊗ M \\ M_{′} & M \end{aligned}; ψ; θ(ζ_{1},...,ζ_{τ})⊗1_{M}; α]

commutes in stmod-kG.

Proof. Set E = E(ζ_{1}, …, ζ_{τ}), U = U(ζ_{1}, …, ζ_{τ}), θ = θ(ζ_{1}, …, ζ_{τ}), and L = L(ζ_{1}, …, ζ_{τ}), and let n_{i} = deg ζ_{i} for 1 ≤ i ≤ τ.

To prove (1), first suppose that E ⊗M splits. The class of the extension E is represented by θ under the isomorphism \underline{Hom}_{kG}(U, k) ≅ Ext_{kG}^{1}(Ω^{−1}U, k), so it follows that θ ⊗ 1_{M} : U ⊗ M → k ⊗ M factors through a projective module. But Ω^{n_{i}}k is a summand of C_{1} for 1 ≤ i ≤ τ, and the restriction of δ_{1} to Ω^{n_{i}}k represents ζ_{i}. Thus ζ_{i} ⊗ 1_{M} factors through θ ⊗ 1_{M} and hence factors through a projective module. It follows that ζ_{i} · 1_{M} = 0 in Ext_{kG}^{n_{i}}(M, M), and ζ_{i} ∈ J(M) for 1 ≤ i ≤ τ, as desired.

Conversely, suppose that I ⊆ J(M). We may assume without loss of generality that M has no projective summands. For 1 ≤ i ≤ τ consider the short exact sequence

E_{i} :  0 ⟶ k ⟶^{(ε_{i},μ_{i})} Ω^{−1}L_{ζ_{i}} ⊕ P_{i} ⟶ Ω^{n_{i}−1}k ⟶ 0

representing ζ_{i} as an element of Ext_{kG}^{n_{i}}(k, k) ≅ Ext_{kG}^{1}(Ω^{n_{i}−1}k, k). Because ζ_{i} ∈ J(M) and E_{i} ⊗ M represents ζ_{i} · 1_{M}, it follows that E_{i} ⊗ M must split. Thus there is a kGhomomorphism σ_{i} : Ω^{−1}L_{ζ_{i}} ⊗ M → k ⊗ M such that cls(σ_{i}(ε_{i} ⊗ 1_{M})) = cls(1_{k⊗M}) in stmod-kG. Define σ : Ω^{−1}L_{ζ_{1}} ⊗ ⋯ ⊗ Ω^{−1}L_{ζ_{τ}} ⊗ M → k ⊗ M by setting σ = σ_{1}(1 ⊗ σ_{2}) ⋯ (1 ⊗ ⋯ ⊗ 1 ⊗ σ_{τ}). Then in stmod-kG we have

cls(σ(ε′ ⊗ 1_{M})) = cls(σ_{1}(1 ⊗ σ_{2}) ⋯ (1 ⊗ ⋯ ⊗ 1 ⊗ σ_{τ})(ε_{1} ⊗ ⋯ ⊗ ε_{τ} ⊗ 1_{M}))
= cls(1_{k⊗M}).

Because cls(ε′ ⊗ 1_{M}) splits in the stable category and M has no projective summands, one can check that ε′ ⊗ 1_{M} also splits in mod-kG, and (1) holds.

To prove (2), we first note that by (1) there is a map σ′′ : L ⊗ M′′ → k ⊗ M′′ such that cls(σ′′(ε ⊗ 1_{M′′})) = cls(1_{k⊗M′′}) in stmod-kG, so

cls(σ′′(1_{L} ⊗ β)(ε ⊗ 1_{M})) = cls(σ′′(ε ⊗ 1_{M′′})(1_{k} ⊗ β)) = cls(1_{k} ⊗ β).

Hence there is a commutative diagram

U ⊗ M ⟶^{θ⊗1_{M}} k ⊗ M ⟶^{ε⊗1_{M}} L ⊗ M ⟶ Ω^{−1}U ⊗ M
∥  ↓^{σ′′(1_{L}⊗β)}
M′  ⟶_{α} M ⟶_{β} M′′ ⟶ Ω^{−1}M′

in which the rows are triangles in stmod-kG. Completing this diagram to a morphism of triangles, we get the desired result. □

3. Homomorphisms and the modules U(ζ_{1}, …, ζ_{τ})

In this section we investigate the functor \underline{Hom}_{kG}(U(ζ_{1}, …, ζ_{τ})⊗−, −) when ζ_{1}, …, ζ_{τ} is a collection of homogeneous elements of H^{∗}(G, k). The main tool in this investigation will be called the truncated modular Koszul complex D(ζ_{1}, …, ζ_{τ}) of the collection ζ_{1}, …, ζ_{τ}. If C = C(ζ_{1}, …, ζ_{τ}) is the modular Koszul complex defined in the previous section, then the truncated modular Koszul complex is the complex (D_{∗}, ∂) obtained by setting D_{i} = C_{i+1} for i ≥ 0 and D_{i} = 0 for i < 0; the boundary map is given by ∂_{i} = δ_{i+1} for i > 0 and ∂_{i} = 0 for i ≤ 0. It follows that

H_{n}(D) = \{ \begin{matrix}U(ζ_{1}, …, ζ_{τ}) & if n = 0 \\ 0 & otherwise, \end{matrix}

and there is an exact sequence

0 ⟶ D_{τ−1} ⟶^{∂_{τ−1}} ⋯ ⟶^{∂_{1}} D_{0} ⟶^{ε} U(ζ_{1}, …, ζ_{τ}) ⟶ 0.

It will be very useful to give another description of the complex D based on the fact that Ω^{m}k ⊗ Ω^{n}k ≅ Ω^{m+n}k in stmod-kG for any integers m and n. To avoid technical details, it is convenient to assume that the elements ζ_{1}, …, ζ_{τ} commute with each other. We therefore set

H^{•}(G, k) = \{ \begin{matrix}⨁_{n=0}^{∞} H^{2n}(G, k) & if p is odd \\ H^{∗}(G, k) & if p = 2. \end{matrix}

Then H^{•}(G, k) is a strictly commutative subring of H^{∗}(G, k), and we assume that ζ_{1}, …, ζ_{τ} are homogeneous elements of H^{•}(G, k).

Let T = {1, …, τ}. If S ⊆ T, set deg S = ∑_{i∈S} deg ζ_{i} and W_{S} = Ω^{deg S}k. For 0 ≤ i ≤ τ − 1 set

V_{i} = ⨁_{\begin{matrix}S⊆T \\ |S|=i+1 \end{matrix}} W_{S},

and let μ(i, S) denote the number of distinct elements j ∈ S such that j < i. If i ∈ S, then by abuse of notation we let ζ_{i} : W_{S} → W_{S−{i}} denote a homomorphism representing ζ_{i} in stmod-kG. Then V is a complex of objects in stmod-kG with the boundary map V_{i} → V_{i−1} given on the summand W_{S} by

∑_{j∈S}(−1)^{μ(j,S)}ζ_{j}.

If we regard D as a complex of objects and maps in stmod-kG, then one can check that D ≅ V. The point of this observation is that if M and N are finitely generated kG-modules, then

Êxt_{kG}^{n}(D_{i} ⊗ M, N) ≅ ⨁_{\begin{matrix}S⊆T \\ |S|=i+1 \end{matrix}} Êxt_{kG}^{n}(W_{S} ⊗ M, N) ≅ ⨁_{\begin{matrix}S⊆T \\ |S|=i+1 \end{matrix}} Êxt_{kG}^{n+deg S}(M, N),

and the homomorphism induced on cohomology by ζ_{j} : W_{S} → W_{S−{j}} is just cup product with ζ_{j}. For purposes of computing cohomology, then, it is very convenient to identify the complexes D and V. With this idea in mind we may now prove the main theorem of this section.

Theorem 3.1. Let ζ_{1}, …, ζ_{τ} be homogeneous elements of H^{•}(G, k), and let M and N be finitely generated kG-modules. Then there is a spectral sequence (E, d) converging to the Tate cohomology Êxt_{kG}^{∗}(U(ζ_{1}, …, ζ_{τ}) ⊗ M, N), and the E_{1} term of the spectral sequence is the truncated Koszul complex

E_{1}^{st} = Êxt_{kG}^{t}(D_{s} ⊗ M, N) ≅ ⨁_{\begin{matrix}S⊆T \\ |S|=s+1 \end{matrix}} Êxt_{kG}^{t+deg S}(M, N).

If α ∈ Êxt_{kG}^{t+deg S}(M, N), then

d_{1}α = ∑_{j∈T −S}(−1)^{μ(j,S)}ζ_{j}α,

where ζ_{j}α lies in the summand Êxt_{kG}^{t+deg S+deg ζ_{j}}(M, N) of E_{1}^{s+1,t} corresponding to S ∪{j}.

Proof. Let D_{∗} = D(ζ_{1}, …, ζ_{τ}) be the truncated modular Koszul complex of the elements ζ_{1}, …, ζ_{τ}, and let (P_{∗}, ∂) be a complete projective resolution of k; in other words, P_{∗} is an exact sequence of projective modules such that the image of ∂ : P_{0} → P_{−1} is isomorphic to k. We will consider the two spectral sequences associated to the double complex Hom_{kG}(D_{∗} ⊗ P_{∗} ⊗ M, N). Note that these spectral sequences must converge because the double complex has only a finite number of nonzero columns.

We begin by considering the spectral sequence in which the E_{1} term is obtained by taking the cohomology of E_{0}^{st} = Hom_{kG}(D_{s} ⊗ P_{t} ⊗ M, N) with respect to the horizontal differential (∂ ⊗ 1_{P} ⊗ 1_{M})^{∗}. Because D_{∗} is bounded and P_{t} ⊗ M is projective for all t, the complex D_{∗} ⊗ P_{t} ⊗ M is a direct sum of contractible complexes and complexes with zero boundary. Thus it follows that

E_{1}^{st} = H^{s}(E_{0}^{∗,t}) = Hom_{kG}(H_{s}(D) ⊗ P_{t} ⊗ M, N).

Now the complex H_{s}(D) ⊗ P_{∗} ⊗ M is a complete projective resolution of H_{s}(D) ⊗ M, so

E_{2}^{st} = H^{t}(E_{1}^{s,∗}) = Êxt_{kG}^{t}(H_{s}(D) ⊗ M, N).

Because H_{s}(D) = 0 if s ≠ 0, we see that the E_{2} term is concentrated along the line s = 0. Hence d_{r} = 0 for all r ≥ 2, and the cohomology of the total complex satisfies

H^{t}(Hom_{kG}(D_{∗} ⊗ P_{∗} ⊗ M, N)) = Êxt_{kG}^{t}(H_{0}(D) ⊗ M, N)
= Êxt_{kG}^{t}(U(ζ_{1}, …, ζ_{τ}) ⊗ M, N).

Now consider the spectral sequence E in which the E_{1} term is obtained by taking the cohomology of E_{0}^{st} = Hom_{kG}(D_{s}⊗P_{t}⊗M, N) with respect to the vertical differential (1_{D} ⊗ ∂⊗1_{M})^{∗}. It follows from the previous paragraph that E converges to Êxt_{kG}^{∗}(U(ζ_{1}, …, ζ_{τ})⊗ M, N). Moreover, because D_{s} ⊗ P_{∗} ⊗ M is a complete projective resolution of D_{s} ⊗ M, we have that

E_{1}^{st} = H^{t}(E_{0}^{s,∗}) = Êxt_{kG}^{t}(D_{s} ⊗ M, N).

The differential d_{1} is induced by (∂ ⊗ 1_{P} ⊗ 1_{M})^{∗}, so it is given by the formula set out in the statement of the theorem. This completes the proof. □

We will denote the spectral sequence of Theorem 3.1 by E(ζ_{1}, …, ζ_{τ}; M, N), or simply by E(M, N) if we do not wish to mention the cohomology elements ζ_{1}, …, ζ_{τ} explicitly. The spectral sequence E(ζ_{1}, …, ζ_{τ}; M, N) is clearly functorial in M and N, but in the next result we observe that it enjoys some additional functoriality. The proof is easy and is left to the reader.

Corollary 3.2. Let γ_{1}, …, γ_{σ} and ζ_{1}, …, ζ_{τ} be homogeneous elements of H^{•}(G, k) with corresponding modular Koszul complexes C(γ) = C(γ_{1}, …, γ_{σ}) and C(ζ) = C(ζ_{1}, …, ζ_{τ}). Suppose that M and N are finitely generated kG-modules, and let μ : C(ζ) → C(γ) be a chain map. Then μ induces a map of spectral sequences

μ̂_{r}^{st} : E_{r}^{st}(γ_{1}, …, γ_{σ}; M, N) → E_{r}^{st}(ζ_{1}, …, ζ_{τ}; M, N)

and a compatible kG-homomorphism

μ′ : U(ζ_{1}, …, ζ_{τ}) → U(γ_{1}, …, γ_{σ}).

In particular, the map

(μ′ ⊗ 1_{M})^{∗} : Êxt_{kG}^{n}(U(γ_{1}, …, γ_{σ}) ⊗ M, N) → Êxt_{kG}^{n}(U(ζ_{1}, …, ζ_{τ}) ⊗ M, N)

preserves the filtration determined by the spectral sequence.

Lemma 3.3. Suppose that γ_{1}, …, γ_{σ} are homogeneous elements of H^{•}(G, k) and that ζ ∈ H^{•}(G, k) is a homogeneous element of the ideal generated by γ_{1}, …, γ_{σ}. Then there is a chain map μ : C(ζ) → C(γ_{1}, …, γ_{σ}) such that cls(μ_{0}) = cls(1_{k}) in stmod-kG.

Proof. Let η_{1}, …, η_{σ} be homogeneous elements of H^{•}(G, k) such that ζ = γ_{1}η_{1}+⋯+γ_{σ}η_{σ}. For 1 ≤ i ≤ σ let c_{i} = deg γ_{i} and d_{i} = deg η_{i}, and set m = deg ζ so that m = c_{i} + d_{i}. Let η′_{i} : Ω^{m}k → Ω^{c_{i}}k be a homomorphism representing Ω^{c_{i}}η_{i}. Then the composition γ_{i}η′_{i} : Ω^{m}k → k is the unique kG-homomorphism representing γ_{i}η_{i}, so there is a commutative diagram

Ω^{m}k  ⟶^{ζ} k
( η′_{1} ; ⋮ ; η′_{σ} )↓  ∥
⨁_{i=1}^{σ} Ω^{c_{i}}k ⟶_{(γ_{1},...,γ_{σ})} k.

Set C = C(ζ) and C′ = C(γ_{1}, …, γ_{σ}). By the discussion preceding Theorem 3.1 it follows that there is a diagram

C_{1} ⟶^{∂} C_{0}
μ_{1}↓  ↓^{μ′_{0}}
C′_{1} ⟶^{∂′} C′_{0}

such that cls(μ′_{0}) = cls(1_{k}) in stmod-kG and ∂′μ_{1} − μ′_{0}∂ is a projective homomorphism. Let P be a projective module such that there are kG-homomorphisms f : C_{1} → P and g : P → C′_{0} with gf = ∂′μ_{1} −μ′_{0}∂. The map ∂ is one-to-one by the definition of the complex C = C(ζ), so there is a map f′ : C_{0} → P such that f′∂ = f. Let μ_{0} = μ′_{0} + gf′. Then cls(μ_{0}) = cls(1_{k}) in stmod-kG and μ_{0}∂ = ∂′μ_{1}, so μ : C → C′ is a chain map satisfying the desired conditions. □

Lemma 3.4. Suppose that ζ is a homogeneous element of H^{•}(G, k). Then for any n > 0 there exists a chain map μ : C(ζ^{2}, …, ζ^{2}) = C(ζ^{2})^{⊗n} → C(ζ) such that cls(μ_{0}) = cls(1_{k}) in stmod-kG.

Proof. Let m = deg ζ, and recall that there is a triangle

Ω^{m}k ⟶^{ζ} k ⟶ Ω^{−1}L_{ζ} ⟶ Ω^{m−1}k

in stmod-kG. Proposition 5.9.6 of [1] shows that ζ^{2} annihilates the cohomology of L_{ζ}, so it follows from Proposition 2.3(2) that there is a map ψ : U(ζ^{2}, …, ζ^{2}) → Ω^{m}k such that cls(ζψ) = cls θ(ζ^{2}, …, ζ^{2}) in stmod-kG. Hence there is a diagram

U(ζ^{2}, …, ζ^{2}) ⟶^{θ(ζ^{2},...,ζ^{2})} C_{0}(ζ^{2}, …, ζ^{2})
ψ↓  ↓^{μ′_{0}}
C_{1}(ζ)  ⟶_{∂}  C_{0}(ζ)

that commutes in stmod-kG, and μ′_{0} is congruent to 1_{k} modulo projective homomorphisms. As in the proof of Lemma 3.3, we can now find a kG-homomorphism μ_{0} : C_{0}(ζ^{2}, …, ζ^{2}) → C_{0}(ζ) with cls(μ_{0}) = cls(μ′_{0}) in stmod-kG and ∂ψ = μ_{0}θ(ζ^{2}, …, ζ^{2}) in mod-kG. Let μ_{1} denote the composition

C_{1}(ζ^{2}, …, ζ^{2}) ⟶ U(ζ^{2}, …, ζ^{2}) ⟶^{ψ} C_{1}(ζ).

Then μ : C(ζ^{2}, …, ζ^{2}) → C(ζ) is a chain map with cls(μ_{0}) = cls(1_{k}) in stmod-kG, as desired. □

Remark 3.5. In the situation considered in Lemma 3.4 one might expect that there would be a chain map μ : C(ζ, …, ζ) = C(ζ)^{⊗n} → C(ζ) such that cls(μ_{0}) = cls(1_{k}) in stmod-kG. If p is odd, then ζ annihilates Ext_{kG}^{∗}(L_{ζ}, L_{ζ}) by Theorem 4.1 of [10], and the proof of Lemma 3.4 shows that there is such a chain map. It is interesting to note that this idea can fail, however, when p = 2. For example, suppose that G = (ℤ/2)^{2}. If ζ ∈ H^{1}(G, k) and ζ ∉ H^{1}(G, ℤ/2), then it follows from Remark 11.3 of [9] that ζ does not annihilate the cohomology of L_{ζ}. In this case there is no chain map μ : C(ζ, ζ) → C(ζ) with the required properties.

We end with a result that will be needed for studying complexity quotient categories in the following sections.

Proposition 3.6. Suppose that γ_{1}, …, γ_{σ} are homogeneous elements of H^{•}(G, k). Then for any homogeneous elements ζ_{1}, …, ζ_{τ} ∈ (γ_{1}^{2}, …, γ_{σ}^{2}) there exists a chain map ϕ : C(ζ_{1}, …, ζ_{τ}) → C(γ_{1}, …, γ_{σ}) having the following properties.

(1) ϕ_{0} is congruent to 1_{k} modulo projective homomorphisms.

(2) The induced map ϕ′ : U(ζ_{1}, …, ζ_{τ}) → U(γ_{1}, …, γ_{σ}) satisfies cls(θ(γ_{1}, …, γ_{σ})ϕ′) = cls θ(ζ_{1}, …, ζ_{τ}) in stmod-kG.

Proof. For 1 ≤ i ≤ τ Lemma 3.3 implies that there is a chain map μ^{i} : C(ζ_{i}) → C(γ_{1}^{2}, …, γ_{σ}^{2}) such that cls(μ_{0}^{i}) = cls(1_{k}) in stmod-kG. Then

μ = μ^{1} ⊗ ⋯ ⊗ μ^{τ} : C(ζ_{1}, …, ζ_{τ}) = C(ζ_{1}) ⊗ ⋯ ⊗ C(ζ_{τ}) → C(γ_{1}^{2}, …, γ_{σ}^{2})^{⊗τ}

is a chain map such that cls(μ_{0}) = cls(1_{k}) in stmod-kG. For any chain complexes C and C′ there is a chain map π : C ⊗ C′ → C′ ⊗ C defined by π(x ⊗ y) = (−1)^{(deg x)(deg y)}y ⊗ x, so it follows that there is a corresponding chain map

π′ : C(γ_{1}^{2}, …, γ_{σ}^{2})^{⊗τ} → C(γ_{1}^{2})^{⊗τ} ⊗ ⋯ ⊗ C(γ_{σ}^{2})^{⊗τ} .

Finally, Lemma 3.4 implies that there is a chain map

ν : C(γ_{1}^{2})^{⊗τ} ⊗ ⋯ ⊗ C(γ_{σ}^{2})^{⊗τ} → C(γ_{1}) ⊗ ⋯ ⊗ C(γ_{σ}) = C(γ_{1}, …, γ_{σ}).

Set ϕ = νπ′μ : C(ζ_{1}, …, ζ_{τ}) → C(γ_{1}, …, γ_{σ}). Then it is easy to see that cls(ϕ_{0}) = cls(1_{k}) in stmod-kG, and the induced map ϕ′ : U(ζ_{1}, …, ζ_{τ}) → U(γ_{1}, …, γ_{σ}) satisfies

θ(γ_{1}, …, γ_{σ})ϕ′ = ϕ_{0}θ(ζ_{1}, …, ζ_{τ}).

Hence cls(θ(γ_{1}, …, γ_{σ})ϕ′) = cls θ(ζ_{1}, …, ζ_{τ}) in stmod-kG, as desired. □

4. Multiplicative structure on the spectral sequence

In this section we will show that the spectral sequence described previously carries a multiplicative structure. The properties of this multiplicative structure will be used in the next section to study compositions of morphisms in complexity quotient categories.

Let ζ_{1}, …, ζ_{τ} be nonzero homogeneous elements of H^{•}(G, k), and set n_{i} = deg ζ_{i} for 1 ≤ i ≤ τ. Let T = {1, …, τ}, and let η_{i} = ζ_{i}^{2}. Recall that if I_{j} = I_{ζ_{j}} is the injective hull of L_{ζ_{j}}, then the modular Koszul complex of ζ_{j} has the form

C(ζ_{j}) : ⋯ ⟶ 0 ⟶ Ω^{n_{j}}k ⟶^{ζ′_{j}} k ⊕ I_{j} ⟶ 0 ⟶ ⋯

with Ω^{n_{j}}k in degree one and k⊕I_{j} in degree zero. Suppose that ∅ ≠ S ⊆ T with |S| = i+1 so that 0 ≤ i ≤ τ − 1. If j ∈ T, set

X_{j,S} = \{ \begin{matrix}Ω^{n_{j}}k & if j ∈ S \\ k ⊕ I_{j} & if j ∉ S \end{matrix}

and

X̂_{j,S} = \{ \begin{matrix}Ω^{n_{j}}k ⊗ Ω^{n_{j}}k & if j ∈ S \\ (k ⊕ I_{j}) ⊗ (k ⊕ I_{j}) & if j ∉ S. \end{matrix}

Now let X_{S} = X_{1,S} ⊗ ⋯ ⊗ X_{τ,S} and X̂_{S} = X̂_{1,S} ⊗ ⋯ ⊗ X̂_{τ,S}, and let X_{∅} = X̂_{∅} = 0. Define

D_{i} = ⨁_{\begin{matrix}S⊆T \\ |S|=i+1 \end{matrix}} X_{S}

and

D̂_{i} = ⨁_{\begin{matrix}S⊆T \\ |S|=i+1 \end{matrix}} X̂_{S}.

If |S| ≥ 2 and j ∈ S, let ζ′_{j,S} : X_{S} → X_{S−{j}} be the map given by ζ′_{j} on the factor X_{j,S} and by the identity on all other factors, and let η′_{j,S} : X̂_{S} → X̂_{S−{j}} be the map given by ζ′_{j} ⊗ ζ′_{j} on the factor X̂_{j,S} = X_{j,S} ⊗ X_{j,S} and by the identity on all other factors; if |S| = 1, set ζ′_{j,S} = 0 and η′_{j,S} = 0. Recall that if j ∈ S, then μ(j, S) denotes the number of distinct elements s ∈ S such that s < j. Then D and D̂ are chain complexes with the boundary maps given on X_{S} ⊆ D_{i} and X̂_{S} ⊆ D̂_{i} by

∂ = ∑_{j∈S}(−1)^{μ(j,S)}ζ′_{j,S}

and

ˆ∂ = ∑_{j∈S}(−1)^{μ(j,S)}η′_{j,S}.

In fact, D is just the truncated modular Koszul complex D(ζ_{1}, …, ζ_{τ}), and D̂ is isomorphic to D(η_{1}, …, η_{τ}) as complexes of objects in stmod-kG. Moreover, D̂ is exact except in degree zero, and H_{0}(D̂) is stably isomorphic to U(η_{1}, …, η_{τ}).

Write S = {u_{1}, …, u_{i+1}} with u_{1} < u_{2} < ⋯ < u_{i+1}. If 1 ≤ j ≤ i + 1, let S′_{j} = {u_{1}, …, u_{j}} and S′′_{j} = {u_{j}, …, u_{i+1}}. Then

\begin{aligned}X_{S′_{j}} ⊗ X_{S′′_{j}} & = X_{1,S′_{j}} ⊗ ⋯ ⊗ X_{u_{j}−1,S′_{j}} ⊗ Ω^{n_{u_{j}}} k ⊗ (k ⊕ I_{u_{j}+1}) ⊗ ⋯ ⊗ (k ⊕ I_{τ}) \\ & ⊗ (k ⊕ I_{1}) ⊗ ⋯ ⊗ (k ⊕ I_{u_{j}−1}) ⊗ Ω^{n_{u_{j}}} k ⊗ X_{u_{j}+1,S′′_{j}} ⊗ ⋯ ⊗ X_{τ,S′′_{j}} \end{aligned}

and

\begin{aligned}X̂_{S} & = X_{1,S′_{j}} ⊗ X_{1,S′_{j}} ⊗ ⋯ ⊗ X_{u_{j}−1,S′_{j}} ⊗ X_{u_{j}−1,S′_{j}} ⊗ Ω^{n_{u_{j}}} k ⊗ Ω^{n_{u_{j}}} k \\ & ⊗ X_{u_{j}+1,S′′_{j}} ⊗ X_{u_{j}+1,S′′_{j}} ⊗ ⋯ ⊗ X_{τ,S′′_{j}} ⊗ X_{τ,S′′_{j}}. \end{aligned}

Let σ_{S} : X̂_{S} → X_{S} ⊗X_{S} be the map that permutes the order of the tensor factors according to the equation

σ_{S}(x_{1} ⊗ y_{1} ⊗ ⋯ ⊗ x_{τ} ⊗ y_{τ}) = x_{1} ⊗ ⋯ ⊗ x_{τ} ⊗ y_{1} ⊗ ⋯ ⊗ y_{τ} .

Now let β_{lj} : X_{l,S′_{j}} → k ⊕ I_{l} be the identity map if l ∉ S′_{j} and ζ′_{l} if l ∈ S′_{j}. Similarly, let γ_{lj} : X_{l,S′′_{j}} → k⊕I_{l} be the identity if l ∉ S′′_{j} and ζ′_{l} if l ∈ S′′_{j}. Define α_{ijS} : X̂_{S} → X_{S′_{j}} ⊗X_{S′′_{j}} to be the map given by

α_{ijS} = (1 ⊗ ⋯ ⊗ 1 ⊗ γ_{u_{j}+1,j} ⊗ ⋯ ⊗ γ_{τ,j} ⊗ β_{1j} ⊗ ⋯ ⊗ β_{u_{j}−1,j} ⊗ 1 ⊗ ⋯ ⊗ 1)σ_{S},

and let α_{i} : D̂_{i} → (D ⊗ D)_{i} be the unique map given on the summand X̂_{S} of D̂_{i} by

α_{i} = ∑_{j=1}^{i+1} α_{ijS}.

Proposition 4.1. α is a chain map.

Proof. Fix i with 1 ≤ i ≤ τ − 1, and let S ⊆ T with |S| = i + 1. To prove that ∂α_{i} = α_{i−1} ˆ∂, it suffices to show that this equation holds on the summand X̂_{S}. Write S = {u_{1}, …, u_{i+1}} with u_{1} < u_{2} < ⋯ < u_{i+1}. We begin by observing that if 1 ≤ l < j ≤ i + 1, then (ζ′_{u_{l},S′_{j}} ⊗ 1)α_{ijS} = α_{i−1,j−1,S−{u_{l}}}η′_{u_{l},S}; similarly, if 1 ≤ j < l ≤ i + 1, then (1 ⊗ ζ′_{u_{l},S′′_{j}})α_{ijS} = α_{i−1,j,S−{u_{l}}}η′_{u_{l},S}. Moreover, ζ′_{u_{1},S′_{1}} = 0 and ζ′_{u_{i+1},S′′_{i+1}} = 0, and if j > 1, then (ζ′_{u_{j},S′_{j}} ⊗ 1)α_{ijS} = (1 ⊗ ζ′_{u_{j−1},S′′_{j−1}})α_{i,j−1,S}. Thus on X̂_{S} we have

∂α_{i} = ∂ ∑_{j=1}^{i+1} α_{ijS}

= ∑_{j=1}^{i+1}[(∂ ⊗ 1)α_{ijS} + (−1)^{j−1}(1 ⊗ ∂)α_{ijS}]

= ∑_{j=1}^{i+1} [∑_{l=1}^{j}(−1)^{l−1}(ζ′_{u_{l},S′_{j}} ⊗ 1)α_{ijS} + ∑_{l=j}^{i+1}(−1)^{l−1}(1 ⊗ ζ′_{u_{l},S′′_{j}})α_{ijS}]

= ∑_{j=2}^{i+1} ∑_{l=1}^{j−1}(−1)^{l−1}α_{i−1,j−1,S−{u_{l}}}η′_{u_{l},S} + ∑_{j=1}^{i} ∑_{l=j+1}^{i+1}(−1)^{l−1}α_{i−1,j,S−{u_{l}}}η′_{u_{l},S}

+ ∑_{j=1}^{i+1} [(−1)^{j−1}(ζ′_{u_{j},S′_{j}} ⊗ 1)α_{ijS} + (−1)^{j−1}(1 ⊗ ζ′_{u_{j},S′′_{j}})α_{ijS}]

= ∑_{j=1}^{i} ∑_{l=1}^{i+1}(−1)^{l−1}α_{i−1,j,S−{u_{l}}}η′_{u_{l},S}

+ ∑_{j=2}^{i+1}(−1)^{j−1}(1 ⊗ ζ′_{u_{j−1},S′′_{j−1}})α_{i,j−1,S} + ∑_{j=1}^{i}(−1)^{j−1}(1 ⊗ ζ′_{u_{j},S′′_{j}})α_{ijS}

= α_{i−1} (∑_{l=1}^{i+1}(−1)^{l−1}η′_{u_{l},S})

= α_{i−1} ˆ∂,

and this completes the proof. □

For the remainder of this section we set U = H_{0}(D) = U(ζ_{1}, …, ζ_{τ}), C = C(ζ_{1}, …, ζ_{τ}), θ = θ(ζ_{1}, …, ζ_{τ}), Û = H_{0}(D̂), and Ĉ_{0} = (k ⊕ I_{1}) ⊗ (k ⊕ I_{1}) ⊗ ⋯ ⊗ (k ⊕ I_{τ}) ⊗ (k ⊕ I_{τ}). If j ∈ T, then there is a map X̂_{{j}} → Ĉ_{0} given by ζ′_{j} ⊗ ζ′_{j} on the factor X̂_{j,{j}} and by the identity on all other factors. These maps define a homomorphism ϕ̂ : D̂_{0} → Ĉ_{0} with ϕ̂ˆ∂ = 0, and Im ϕ̂ ≅ Û. If ε̂: D̂_{0} → Û denotes the canonical epimorphism, then it follows that there is a map θ̂ : Û → Ĉ_{0} with θ̂ε̂ = ϕ̂. Let σ : Ĉ_{0} → C_{0} ⊗ C_{0} be the isomorphism given by

σ(x_{1} ⊗ y_{1} ⊗ ⋯ ⊗ x_{τ} ⊗ y_{τ}) = x_{1} ⊗ ⋯ ⊗ x_{τ} ⊗ y_{1} ⊗ ⋯ ⊗ y_{τ} ,

and let ε : D_{0} → U denote the canonical epimorphism.

Proposition 4.2. The chain map α : D̂ → D ⊗ D induces a map u : Û → U ⊗ U such that (θ ⊗ θ)u = σθ̂.

Proof. Because H_{0}(D̂) = Û and H_{0}(D ⊗ D) = U ⊗ U, we know that α induces a map u : Û → U ⊗ U such that uε̂ = (ε ⊗ ε)α_{0}. Let s ∈ T, and set S = {s} so that X̂_{S} is a summand of D̂_{0}. Then the definition of α_{0,1,S} : X̂_{S} → X_{S} ⊗ X_{S} shows that α_{0,1,S} satisfies

α_{0,1,S}(x_{1} ⊗ y_{1} ⊗ ⋯ ⊗ x_{τ} ⊗ y_{τ}) = x_{1} ⊗ ⋯ ⊗ x_{τ} ⊗ y_{1} ⊗ ⋯ ⊗ y_{τ} .

Thus on the summand X̂_{S} we have

(θ ⊗ θ)uε̂ = (θ ⊗ θ)(ε ⊗ ε)α_{0} = (θε ⊗ θε)α_{0,1,S} = σθ̂ε̂.

It follows that (θ ⊗ θ)uε̂ = σθ̂ε̂ on all of D̂_{0}. Because ε̂ is an epimorphism, we see that (θ ⊗ θ)u = σθ̂, as desired. □

Let E_{0} be a double complex with differentials δ_{0}^{st} : E_{0}^{st} → E_{0}^{s,t+1} and δ_{1}^{st} : E_{0}^{st} → E_{0}^{s+1,t}; similarly, let (Ė_{0}, δ̇_{0}, δ̇_{1}) and (Ë_{0}, δ̈_{0}, δ̈_{1}) be double complexes. Recall that a pairing of double complexes is a collection of maps π_{0} : E_{0}^{st} ⊗ Ė_{0}^{s′t′} → Ë_{0}^{s+s′,t+t′} such that

δ̈_{0}^{s+s′,t+t′}π_{0}(x ⊗ y) = π_{0}(δ_{0}^{st}x ⊗ y) + (−1)^{s+t}π_{0}(x ⊗ δ̇_{0}^{s′t′}y)

and

δ̈_{1}^{s+s′,t+t′}π_{0}(x ⊗ y) = π_{0}(δ_{1}^{st}x ⊗ y) + (−1)^{s+t}π_{0}(x ⊗ δ̇_{1}^{s′t′}y).

The following result is a lemma on pairings that will be needed to study the multiplicative structure on the spectral sequences defined in the previous section. If (E, d) is a spectral sequence and g ∈ Ker d_{r}^{st}, then [g] is used to denote the element of E_{r+1}^{st} determined by g.

Lemma 4.3. A pairing π_{0} : E_{0}^{st} ⊗ Ė_{0}^{s′t′} → Ë_{0}^{s+s′,t+t′} of double complexes induces maps π_{r} : E_{r}^{st} ⊗ Ė_{r}^{s′t′} → Ë_{r}^{s+s′,t+t′} satisfying

d̈_{r}^{s+s′,t+t′}π_{r}(g ⊗ f) = π_{r}(d_{r}^{st}g ⊗ f) + (−1)^{s+t}π_{r}(g ⊗ ḋ_{r}^{s′t′}f)

and

π_{r+1}([g] ⊗ [f]) = [π_{r}(g ⊗ f)]

for r ≥ 0. In the limit the maps π_{r} induce maps π_{∞} : E_{∞}^{st} ⊗ Ė_{∞}^{s′t′} → Ë_{∞}^{s+s′,t+t′} in such a way that the isomorphism

E_{∞}^{st} ≅ F^{s}H^{s+t}(Tot E_{0})/F^{s+1}H^{s+t}(Tot E_{0})

preserves products.

Proof. See Theorem 3.9.3 of [1]. □

For the rest of this section let P_{∗} be a complete projective resolution of k. Recall that the completed tensor product P_{∗} ˆ⊗P_{∗} is the chain complex in which

(P ˆ⊗P)_{n} = ∏_{s+t=n} P_{s} ⊗ P_{t}

is a direct product rather than a direct sum; the boundary map is defined on each factor as in the case of the ordinary tensor product. The completed tensor product P_{∗} ˆ⊗P_{∗} is again a complete projective resolution of k. If ε : P_{∗} → k denotes the augmentation map, then we have an augmentation ε ˆ⊗ε : P_{∗} ˆ⊗P_{∗} → k, and there is a diagonal approximation Δ : P_{∗} → P_{∗} ˆ⊗P_{∗} such that (ε ˆ⊗ε)Δ_{0} = ε (see Section VI.5 of [8]).

Suppose that M and N are finitely generated kG-modules, and let E_{0}^{st}(M, N) = Hom_{kG}(D_{s} ⊗ P_{t} ⊗ M, N) so that E_{0}(M, N) is a double complex with differentials δ_{0} : E_{0}^{st}(M, N) → E_{0}^{s,t+1}(M, N) and δ_{1} : E_{0}^{st}(M, N) → E_{0}^{s+1,t}(M, N). Similarly, define a double complex Ê_{0}(M, N) with differentials δ̂_{0} and δ̂_{1} by setting Ê_{0}^{st}(M, N) = Hom_{kG}(D̂_{s} ⊗ P_{t} ⊗ M, N). The spectral sequences associated with these double complexes determine filtrations of Êxt_{kG}^{∗}(U ⊗ M, N) and of Êxt_{kG}^{∗}(Û ⊗ M, N); by abuse of notation we will use F to denote both of these associated filtrations.

If M_{1}, M_{2}, and M_{3} are finitely generated kG-modules, then we wish to define a pairing π_{0} : E_{0}(M_{2}, M_{3}) ⊗ E_{0}(M_{1}, M_{2}) → Ê_{0}(M_{1}, M_{3}). Let σ : D_{∗} ⊗ P_{∗} → P_{∗} ⊗ D_{∗} be the chain map given by

σ(x ⊗ y) = (−1)^{(deg x)(deg y)}y ⊗ x,

and for any integers t and t′ let ϖ_{t,t′} denote the canonical projection of (P ˆ⊗P)_{t+t′} onto the factor P_{t} ⊗ P_{t′}. If g ∈ E_{0}^{st}(M_{2}, M_{3}) = Hom_{kG}(D_{s} ⊗ P_{t} ⊗ M_{2}, M_{3}) and f ∈ E_{0}^{s′t′}(M_{1}, M_{2}) = Hom_{kG}(D_{s′} ⊗ P_{t′} ⊗ M_{1}, M_{2}), let π_{0}(g ⊗ f) ∈ Ê_{0}^{s+s′,t+t′}(M_{1}, M_{3}) = Hom_{kG}(D̂_{s+s′} ⊗ P_{t+t′} ⊗ M_{1}, M_{3}) denote the composition

D̂_{s+s′} ⊗ P_{t+t′} ⊗ M_{1} ⟶^{α⊗ϖΔ⊗1_{M_{1}}} D_{s} ⊗ D_{s′} ⊗ P_{t} ⊗ P_{t′} ⊗ M_{1} ⟶^{1_{D}⊗σ⊗1_{P} ⊗1_{M_{1}}}
D_{s} ⊗ P_{t} ⊗ D_{s′} ⊗ P_{t′} ⊗ M_{1} ⟶^{1_{D}⊗1_{P} ⊗f} D_{s} ⊗ P_{t} ⊗ M_{2} ⟶^{g} M_{3}.

Lemma 4.4. Let M_{1}, M_{2}, and M_{3} be finitely generated kG-modules. Then the map π_{0} : E_{0}(M_{2}, M_{3}) ⊗ E_{0}(M_{1}, M_{2}) → Ê_{0}(M_{1}, M_{3}) is a pairing of double complexes.

Proof. Because Δ is a chain map, we get

δ̂_{0}^{s+s′,t+t′}π_{0}(g ⊗ f)
= (−1)^{s+s′}g(1_{D} ⊗ 1_{P} ⊗ f)(1_{D} ⊗ σ ⊗ 1_{P} ⊗ 1_{M_{1}})(α ⊗ ϖΔ ⊗ 1_{M_{1}})(1D̂ ⊗ ∂ ⊗ 1_{M_{1}})
= (−1)^{s+s′}g(1_{D} ⊗ 1_{P} ⊗ f)(1_{D} ⊗ σ ⊗ 1_{P} ⊗ 1_{M_{1}})(1_{D} ⊗ 1_{D} ⊗ ∂ ⊗ 1_{P} ⊗ 1_{M_{1}})
∘ (α ⊗ ϖΔ ⊗ 1_{M_{1}}) + (−1)^{s+s′+t}g(1_{D} ⊗ 1_{P} ⊗ f)(1_{D} ⊗ σ ⊗ 1_{P} ⊗ 1_{M_{1}})
∘ (1_{D} ⊗ 1_{D} ⊗ 1_{P} ⊗ ∂ ⊗ 1_{M_{1}})(α ⊗ ϖΔ ⊗ 1_{M_{1}})
= (−1)^{s}g(1_{D} ⊗ 1_{P} ⊗ f)(1_{D} ⊗ ∂ ⊗ 1_{D} ⊗ 1_{P} ⊗ 1_{M_{1}})(1_{D} ⊗ σ ⊗ 1_{P} ⊗ 1_{M_{1}})
∘ (α ⊗ ϖΔ ⊗ 1_{M_{1}}) + (−1)^{s+s′+t}g(1_{D} ⊗ 1_{P} ⊗ f)(1_{D} ⊗ 1_{P} ⊗ 1_{D} ⊗ ∂ ⊗ 1_{M_{1}})
∘ (1_{D} ⊗ σ ⊗ 1_{P} ⊗ 1_{M_{1}})(α ⊗ ϖΔ ⊗ 1_{M_{1}})
= (δ_{0}^{st}g)(1_{D} ⊗ 1_{P} ⊗ f)(1_{D} ⊗ σ ⊗ 1_{P} ⊗ 1_{M_{1}})(α ⊗ ϖΔ ⊗ 1_{M_{1}})
+ (−1)^{s+t}g(1_{D} ⊗ 1_{P} ⊗ δ_{0}^{s′t′}f)(1_{D} ⊗ σ ⊗ 1_{P} ⊗ 1_{M_{1}})(α ⊗ ϖΔ ⊗ 1_{M_{1}})
= π_{0}(δ_{0}^{st}g ⊗ f) + (−1)^{s+t}π_{0}(g ⊗ δ_{0}^{s′t′}f).

Using the fact that α is a chain map, a similar computation shows that

δ̂_{1}^{s+s′,t+t′}π_{0}(g ⊗ f) = π_{0}(δ_{1}^{st}g ⊗ f) + (−1)^{s+t}π_{0}(g ⊗ δ_{1}^{s′t′}f),

as desired. □

Recall from Proposition 4.2 that α : D̂ → D ⊗ D induces a map u : Û → U ⊗ U such that (θ ⊗ θ)u = σθ̂.

Theorem 4.5. Let M_{1}, M_{2}, and M_{3} be finitely generated kG-modules, and suppose that 1 ≤ r ≤ ∞. Then π_{0} induces a well-defined map π_{r} : E_{r}^{st}(M_{2}, M_{3}) ⊗ E_{r}^{s′t′}(M_{1}, M_{2}) → Ê_{r}^{s+s′,t+t′}(M_{1}, M_{3}) such that

d̂_{r}^{s+s′,t+t′}π_{r}(g ⊗ f) = π_{r}(d_{r}^{st}g ⊗ f) + (−1)^{s+t}π_{r}(g ⊗ d_{r}^{s′t′}f),

and the isomorphisms

E_{∞}^{st}(M_{i}, M_{i+1}) ≅ F^{s}Êxt_{kG}^{s+t}(U ⊗ M_{i}, M_{i+1})/F^{s+1}Êxt_{kG}^{s+t}(U ⊗ M_{i}, M_{i+1})

and

Ê_{∞}^{st}(M_{i}, M_{i+1}) ≅ F^{s}Êxt_{kG}^{s+t}(Û ⊗ M_{i}, M_{i+1})/F^{s+1}Êxt_{kG}^{s+t}(Û ⊗ M_{i}, M_{i+1})

commute with the product defined by π_{∞}. Moreover, π_{∞} is induced by the product

Êxt_{kG}^{s+t}(U ⊗ M_{2}, M_{3}) ⊗ Êxt_{kG}^{s′+t′}(U ⊗ M_{1}, M_{2}) → Êxt_{kG}^{s+s′+t+t′}(Û ⊗ M_{1}, M_{3})

given by g ⊗ f ↦ g ∘ Ω^{s+t}(1_{U} ⊗ f) ∘ Ω^{s+s′+t+t′}(u ⊗ 1_{M_{1}}).

Proof. The first assertion is an immediate consequence of Lemmas 4.3 and 4.4. To prove the second assertion, suppose that g ∈ E_{0}^{st}(M_{2}, M_{3}) = Hom_{kG}(D_{s} ⊗ P_{t} ⊗ M_{2}, M_{3}) and f ∈ E_{0}^{s′t′}(M_{1}, M_{2}) = Hom_{kG}(D_{s′} ⊗ P_{t′} ⊗ M_{1}, M_{2}), and recall that the product π_{0}(g ⊗ f) ∈ Ê_{0}^{s+s′,t+t′}(M_{1}, M_{3}) = Hom_{kG}(D̂_{s+s′} ⊗ P_{t+t′} ⊗ M_{1}, M_{3}) is defined to be the composition

D̂_{s+s′} ⊗ P_{t+t′} ⊗ M_{1} ⟶^{α⊗ϖΔ⊗1_{M_{1}}} D_{s} ⊗ D_{s′} ⊗ P_{t} ⊗ P_{t′} ⊗ M_{1} ⟶^{1_{D}⊗σ⊗1_{P} ⊗1_{M_{1}}}
D_{s} ⊗ P_{t} ⊗ D_{s′} ⊗ P_{t′} ⊗ M_{1} ⟶^{1_{D}⊗1_{P} ⊗f} D_{s} ⊗ P_{t} ⊗ M_{2} ⟶^{g} M_{3}.

We know that the complexes D_{∗} and D̂_{∗} are exact except in degree zero. Taking homology with respect to the horizontal differentials on the double complexes, we see that the product π_{0} induces a multiplication that is zero unless s = s′ = 0; and if s = s′ = 0, then the induced product is given by

Û ⊗ P_{t+t′} ⊗ M_{1} ⟶^{u⊗ϖΔ⊗1_{M_{1}}} U ⊗ U ⊗ P_{t} ⊗ P_{t′} ⊗ M_{1} ⟶
U ⊗ P_{t} ⊗ U ⊗ P_{t′} ⊗ M_{1} ⟶^{1_{U} ⊗1_{P} ⊗f} U ⊗ P_{t} ⊗ M_{2} ⟶^{g} M_{3}

for all f ∈ Hom_{kG}(U ⊗ P_{t′} ⊗ M_{1}, M_{2}) and g ∈ Hom_{kG}(U ⊗ P_{t} ⊗ M_{2}, M_{3}). If we now take homology with respect to the vertical differentials, then it is easy to see that the product induced by π_{0} is

Ω^{t+t′}(Û ⊗ M_{1}) ⟶^{Ω^{t+t′} (u⊗1_{M_{1}} )} Ω^{t+t′}(U ⊗ U ⊗ M_{1})
⟶^{Ω^{t}(1_{U} ⊗f)} Ω^{t}(U ⊗ M_{2}) ⟶^{g} M_{3}

for all f ∈ Êxt_{kG}^{t′}(U ⊗ M_{1}, M_{2}) and g ∈ Êxt_{kG}^{t}(U ⊗ M_{2}, M_{3}). By Lemma 4.3 the product π_{0} must also induce the map π_{∞} : E_{∞}^{st}(M_{2}, M_{3}) ⊗ E_{∞}^{s′t′}(M_{1}, M_{2}) → Ê_{∞}^{s+s′,t+t′}(M_{1}, M_{3}) on the other spectral sequences coming from the double complexes, and this completes the proof. □

Remark 4.6. We have already observed that D̂ is isomorphic to D(η_{1}, …, η_{τ}) as complexes of objects in stmod-kG. In fact, if M and N are finitely generated kG-modules, then this isomorphism induces an isomorphism of spectral sequences between Ê_{r}(M, N) and E_{r}(η_{1}, …, η_{τ}; M, N) for all r > 0 (but not, of course, for r = 0). Thus if r > 0, then Theorem 4.5 shows that there is a multiplication π_{r} taking E_{r}^{st}(ζ_{1}, …, ζ_{τ}; M_{2}, M_{3}) ⊗ E_{r}^{s′t′}(ζ_{1}, …, ζ_{τ}; M_{1}, M_{2}) into E_{r}^{s+s′,t+t′}(ζ_{1}^{2}, …, ζ_{τ}^{2}; M_{1}, M_{3}) for any finitely generated kGmodules M_{1}, M_{2}, and M_{3}.

5. Homomorphisms in complexity quotient categories

In this section we will describe the precise connections between complexity quotient categories and the spectral sequences studied in Sections 3 and 4. In particular, we will describe how to compose morphisms in complexity quotient categories and will show that this composition corresponds to the multiplicative structure described in the previous section.

Although we have chosen to focus primarily on complexity quotients, it should be noted that the results presented here can easily be adapted to describe morphisms in other quotient categories. In particular, suppose that V is any collection of closed homogeneous subvarieties of V_{G}(k), and let M_{V} denote the full subcategory of stmod-kG consisting of all kG-modules M for which there are varieties V_{1}, …, V_{r} ∈ V with V_{G}(M) ⊆ V_{1} ∪ ⋯ ∪ V_{r}. Then the arguments given in this section can also be used to describe morphisms in the quotient stmod-kG/M_{V}. We will return to this idea in more detail in Section 7.

Recall that if c is a nonnegative integer, then M_{c} denotes the full subcategory of stmodkG consisting of all finitely generated kG-modules of complexity at most c; thus if ρ is the p-rank of G, then M_{ρ} = stmod-kG. Fix an integer τ with 1 ≤ τ ≤ ρ. Then our objective is to study the quotient category Q^{τ} = M_{ρ}/M_{ρ−τ}. To describe the morphisms in Q^{τ}, first suppose that s : U → M is a map in stmod-kG, and consider the triangle

U ⟶^{s} M ⟶ L ⟶ Ω^{−1}U.

If cx_{G}(L) ≤ ρ − τ, then we will say that s is a Q^{τ} -isomorphism.

The objects of the quotient category Q^{τ} = M_{ρ}/M_{ρ−τ} consist of all finitely generated kG-modules; morphisms are given by equivalence classes of pairs of maps in M_{ρ} = stmod-kG. Specifically, a morphism from M to N in Q^{τ} is represented by an object U with a pair of maps s : U → M and f : U → N in stmod-kG such that s is a Q^{τ} - isomorphism. If s′ : U′ → M and f′ : U′ → N are two other such maps in stmod-kG, then (s, U, f) and (s′, U′, f′) represent the same morphism in Q^{τ} if and only if there is an object U′′ with maps in stmod-kG such that the diagram

[figure: \begin{aligned}& U \\ M & U_{′}_{′} & N \\ & U′ \end{aligned}; s; f; s′′; s′; f′]

commutes and s′′ is a Q^{τ} -isomorphism. We will write [s, U, f] for the equivalence class of (s, U, f). It is often useful to think of the morphism [s, U, f] in Q^{τ} represented by (s, U, f) as the composite fs^{−1} : M → N, even though s may not be invertible in stmod-kG. Further background material on localization and quotient categories is available in [21].

If ζ_{1}, …, ζ_{τ} are homogeneous elements of H^{•}(G, k), then for simplicity we will write U(ζ) for U(ζ_{1}, …, ζ_{τ}), θ(ζ) for θ(ζ_{1}, …, ζ_{τ}), Û(ζ) for Û(ζ_{1}, …, ζ_{τ}), and so on. This notation will be used throughout the remaining sections.

Lemma 5.1. Suppose that s : U → M is a Q^{τ} -isomorphism. Then there are homogeneous elements ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) and a Q^{τ} -isomorphism ψ : U(ζ) ⊗ M → U such that sψ = θ(ζ) ⊗ 1_{M} and dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ.

Proof. The morphism s determines a triangle in stmod-kG of the form

U ⟶^{s} M ⟶ L ⟶ Ω^{−1}U.

Because s is a Q^{τ} -isomorphism, dim V_{G}(L) ≤ ρ − τ. Choose homogeneous elements ζ_{1}, …, ζ_{τ} ∈ H^{∗}(G, k) such that dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ and V_{G}(L) ⊆ V_{G}(ζ_{1}, …, ζ_{τ}). Then ζ_{1}, …, ζ_{τ} must lie in H^{•}(G, k) because any homogeneous element of H^{∗}(G, k) that is not in H^{•}(G, k) is nilpotent. Let J(L) denote the annihilator of Ext_{kG}^{∗}(L, L) in H^{∗}(G, k). Because V_{G}(L) ⊆ V_{G}(ζ_{1}, …, ζ_{τ}), it follows that ζ_{i} ∈ \sqrt{J(L)} for all i. Replacing each ζ_{i} by an appropriate power, we may assume that ζ_{i} ∈ J(L) for 1 ≤ i ≤ τ. Then Proposition 2.3(2) implies that there is a map ψ : U(ζ) ⊗ M → U such that sψ = θ(ζ) ⊗ 1_{M}. Because dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ, the map θ(ζ) ⊗ 1_{M} is a Q^{τ} -isomorphism. Moreover, s is a Q^{τ} -isomorphism by assumption, so it follows that ψ is also a Q^{τ} -isomorphism. This completes the proof. □

Proposition 5.2. Let M and N be objects of Q^{τ}. Then any morphism from M to N in Q^{τ} has a representative of the form (θ(ζ) ⊗ 1_{M}, U(ζ) ⊗ M, f) for some homogeneous elements ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) with dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ.

Proof. By the definition of the quotient category Q^{τ} any morphism from M to N is represented by a triple (s, U, g), where s : U → M and g : U → N are maps such that s is a Q^{τ} -isomorphism. By Lemma 5.1 there exist homogeneous elements ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) and a Q^{τ} -isomorphism ψ : U(ζ) ⊗ M → U such that dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ and sψ = θ(ζ) ⊗ 1_{M}. Thus there is a commutative diagram

[figure: \begin{aligned}& U \\ M & U(ζ) ⊗ M & N \\ & U(ζ) ⊗ M \end{aligned}; s; g; θ(ζ)⊗1_{M}; ψ; gψ; θ(ζ)⊗1_{M}; gψ]

in stmod-kG, and it follows that (s, U, g) represents the same morphism in Q^{τ} as (θ(ζ) ⊗ 1_{M}, U(ζ) ⊗ M, gψ). This completes the proof. □

Proposition 5.3. Let M_{1}, M_{2}, and M_{3} be objects of Q^{τ}, and let [s, U, f] : M_{1} → M_{2} and [s′, U′, f′] : M_{2} → M_{3} be maps in Q^{τ}. Then there exist homogeneous elements ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) and maps g_{1} : U(ζ) ⊗ M_{1} → M_{2} and g_{2} : U(ζ) ⊗ M_{2} → M_{3} such that the following conditions are satisfied:

(1) dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ.

(2) [s, U, f] = [θ(ζ) ⊗ 1_{M_{1}}, U(ζ) ⊗ M_{1}, g_{1}] and [s′, U′, f′] = [θ(ζ) ⊗ 1_{M_{2}}, U(ζ) ⊗ M_{2}, g_{2}].

Proof. By Proposition 5.2 there are elements α_{1}, …, α_{τ} and β_{1}, …, β_{τ} in H^{•}(G, k) such that [s, U, f] = [θ(α)⊗1_{M_{1}}, U(α)⊗M_{1}, f_{1}] and [s′, U′, f′] = [θ(β)⊗1_{M_{2}}, U(β)⊗M_{2}, f_{2}] for some maps f_{1} : U(α) ⊗ M_{1} → M_{2} and f_{2} : U(β) ⊗ M_{2} → M_{3}. Moreover, V_{G}(α_{1}, …, α_{τ}) and V_{G}(β_{1}, …, β_{τ}) have dimension ρ − τ. Consider the ideals 𝔞 = (α_{1}^{2}, …, α_{τ}^{2}) and 𝔟 = (β_{1}^{2}, …, β_{τ}^{2}) in H^{∗}(G, k), and note that

V_{G}(𝔞𝔟) = V_{G}(𝔞) ∪ V_{G}(𝔟) = V_{G}(α_{1}, …, α_{τ}) ∪ V_{G}(β_{1}, …, β_{τ})

has dimension ρ − τ. Choose ζ_{1} ∈ 𝔞𝔟 such that V_{G}(ζ_{1}) ≠ V_{G}(k), and for 1 < i ≤ τ choose ζ_{i} ∈ 𝔞𝔟 inductively so that V_{G}(ζ_{i}) contains no component of V_{G}(ζ_{1}, …, ζ_{i−1}) of dimension greater than ρ−τ. Then it follows that ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) and dim V_{G}(ζ_{1}, …, ζ_{i}) = ρ−i for 1 ≤ i ≤ τ. Moreover, Proposition 3.6 implies that there are maps ϕ′_{α} : U(ζ) → U(α) and ϕ′_{β} : U(ζ) → U(β) such that cls(θ(α)ϕ′_{α}) = cls(θ(β)ϕ′_{β}) = cls(θ(ζ)) in stmod-kG. Because dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ, it follows that θ(ζ) : U(ζ) → k is a Q^{τ} -isomorphism and hence so are ϕ′_{α} and ϕ′_{β}. Thus

[s, U, f] = [θ(α) ⊗ 1_{M_{1}}, U(α) ⊗ M_{1}, f_{1}] = [θ(ζ) ⊗ 1_{M_{1}}, U(ζ) ⊗ M_{1}, f_{1}(ϕ′_{α} ⊗ 1_{M_{1}})]

and

[s′, U′, f′] = [θ(β) ⊗ 1_{M_{2}}, U(β) ⊗ M_{2}, f_{2}] = [θ(ζ) ⊗ 1_{M_{2}}, U(ζ) ⊗ M_{2}, f_{2}(ϕ′_{β} ⊗ 1_{M_{2}})],

and this completes the proof. □

In order to describe the composition of morphisms in Q^{τ}, it is enough by Proposition 5.3 to describe the composition [θ(ζ) ⊗ 1_{M_{2}}, U(ζ) ⊗ M_{2}, g_{2}] ∘ [θ(ζ) ⊗ 1_{M_{1}}, U(ζ) ⊗ M_{1}, g_{1}]. By Proposition 4.2 there is a map u(ζ) : Û(ζ) → U(ζ)⊗U(ζ) such that cls((θ(ζ)⊗θ(ζ))u(ζ)) = cls(θ̂(ζ)) in stmod-kG, and it is easy to see that u(ζ) is a Q^{τ} -isomorphism. Because the maps U(ζ)⊗U(ζ)⊗M_{1} → M_{2} given by g_{1}(θ(ζ)⊗1_{U(ζ)} ⊗1_{M_{1}}) and (θ(ζ)⊗1_{M_{2}})(1_{U(ζ)} ⊗g_{1}) are equal, there is a diagram

[figure: \begin{aligned}& & Û(ζ) ⊗ M_{1} \\ & & U(ζ) ⊗ U(ζ) ⊗ M_{1} \\ & U(ζ) ⊗ M_{1} & & U(ζ) ⊗ M_{2} \\ M_{1} & & M_{2} & & M_{3} \end{aligned}; u(ζ)⊗1; θ(ζ)⊗1⊗1; 1_{U(ζ)}⊗g_{1}; θ(ζ)⊗1; g_{1}; θ(ζ)⊗1; g_{2}]

in which the middle diamond commutes in stmod-kG. Thus we get the following result.

Theorem 5.4. Let M_{1}, M_{2}, and M_{3} be objects of Q^{τ}, and let ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) be homogeneous elements such that dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ. If g_{1} : U(ζ) ⊗ M_{1} → M_{2} and g_{2} : U(ζ) ⊗ M_{2} → M_{3} are kG-homomorphisms, then

[θ(ζ) ⊗ 1_{M_{2}}, U(ζ)⊗M_{2}, g_{2}] ∘ [θ(ζ) ⊗ 1_{M_{1}}, U(ζ) ⊗ M_{1}, g_{1}]
= [θ(ζ) ⊗ θ(ζ) ⊗ 1_{M_{1}}, U(ζ) ⊗ U(ζ) ⊗ M_{1}, g_{2}(1_{U(ζ)} ⊗ g_{1})]
= [θ̂(ζ) ⊗ 1_{M_{1}}, Û(ζ) ⊗ M_{1}, g_{2}(1_{U(ζ)} ⊗ g_{1})(u(ζ) ⊗ 1_{M_{1}})].

Note that if the modules M_{1}, M_{2}, and M_{3} and the cohomology elements ζ_{1}, …, ζ_{τ} are given, then all of the data in the formula of Theorem 5.4 are known except for the maps g_{1}, g_{2}, and g_{2}(1_{U(ζ)} ⊗g_{1})(u(ζ)⊗1_{M_{1}}). Thus the composition in Q^{τ} amounts to a collection of maps

\underline{Hom}_{kG}(U(ζ) ⊗ M_{2}, M_{3}) ⊗ \underline{Hom}_{kG}(U(ζ) ⊗ M_{1}, M_{2}) → \underline{Hom}_{kG}(Û(ζ) ⊗ M_{1}, M_{3})

given by g_{2} ⊗ g_{1} ↦ g_{2}(1_{U(ζ)} ⊗ g_{1})(u(ζ) ⊗ 1_{M_{1}}). Then Theorem 4.5 shows that the multiplication

π_{∞} : E_{∞}^{s,−s}(M_{2}, M_{3}) ⊗ E_{∞}^{s′,−s′}(M_{1}, M_{2}) → Ê_{∞}^{s+s′,−s−s′}(M_{1}, M_{3})

is induced by the composition in Q^{τ}.

Let η_{i} = ζ_{i}^{2} for 1 ≤ i ≤ τ. We observed in Section 4 that Û(ζ) is stably isomorphic to U(η). In fact, one can show that there is a stable isomorphism ϕ : U(η) → Û(ζ) such that cls(θ̂(ζ)ϕ) = cls(θ(η)). Hence the composition in Theorem 5.4 can also be expressed in the form

[θ(ζ) ⊗ 1_{M_{2}}, U(ζ) ⊗ M_{2}, g_{2}] ∘ [θ(ζ) ⊗ 1_{M_{1}}, U(ζ) ⊗ M_{1}, g_{1}] = [θ(η) ⊗ 1_{M_{1}}, U(η) ⊗ M_{1}, g_{2}(1_{U(ζ)} ⊗ g_{1})(u(ζ)ϕ ⊗ 1_{M_{1}})].

Let M and N be finitely generated kG-modules, and let ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) be homogeneous elements such that dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ. Then Theorem 3.1 shows that there is a spectral sequence E(M, N) having E_{1}^{st}(M, N) = Êxt_{kG}^{t}(D_{s}(ζ) ⊗ M, N) and converging to Êxt_{kG}^{∗}(U(ζ) ⊗ M, N). This spectral sequence defines a filtration F on Êxt_{kG}^{∗}(U(ζ) ⊗ M, N), and it is possible to use F to define a corresponding filtration (again called F) on Hom_{Q^{τ}} (M, N). The following result is easy to verify, and we omit the proof.

Proposition 5.5. Let M and N be finitely generated kG-modules. For any integer r let F^{r} Hom_{Q^{τ}} (M, N) denote the collection of all morphisms from M to N in Q^{τ} having a representative of the form

M ⟵^{θ(ζ)⊗1_{M}} U(ζ) ⊗ M ⟶^{f} N

for some f ∈ F^{r}\underline{Hom}_{kG}(U(ζ)⊗M, N) and ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) with dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ. Then

(1) F^{r} Hom_{Q^{τ}} (M, N) is a subgroup of Hom_{Q^{τ}} (M, N) under addition.

(2) F^{r} Hom_{Q^{τ}} (M, N) ⊇ F^{r+1} Hom_{Q^{τ}} (M, N) for all r.

(3) F^{0} Hom_{Q^{τ}} (M, N) = Hom_{Q^{τ}} (M, N) and F^{τ} Hom_{Q^{τ}} (M, N) = 0.

Moreover, if M_{1}, M_{2}, and M_{3} are kG-modules, then

F^{r} Hom_{Q^{τ}} (M_{2}, M_{3}) ∘ F^{s} Hom_{Q^{τ}} (M_{1}, M_{2}) ⊆ F^{r+s} Hom_{Q^{τ}} (M_{1}, M_{3}).

In particular, every element of F^{1} End_{Q^{τ}} (M) is nilpotent.

Proposition 5.6. End_{Q^{τ}} (k) is a commutative ring.

Proof. For any two given endomorphisms of k Proposition 5.3 implies that there are homogeneous elements ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) such that the endomorphisms have representatives of the form

k ⟵^{θ(ζ)} U(ζ) ⟶^{f} k

and

k ⟵^{θ(ζ)} U(ζ) ⟶^{g} k.

Then there are commutative diagrams

[figure: \begin{aligned}& & U(ζ) ⊗ U(ζ) \\ & U(ζ) & & U(ζ) \\ k & & k & & k \end{aligned}; θ(ζ)⊗1; 1⊗f; θ(ζ); f; θ(ζ); g]

and

[figure: \begin{aligned}& & U(ζ) ⊗ U(ζ) \\ & U(ζ) & & U(ζ) \\ k & & k & & k, \end{aligned}; 1⊗θ(ζ); g⊗1; θ(ζ); g; θ(ζ); f]

so it follows that

[θ(ζ), U(ζ), g] ∘ [θ(ζ), U(ζ), f] = [θ(ζ) ⊗ θ(ζ), U(ζ) ⊗ U(ζ), g ⊗ f]

and

[θ(ζ), U(ζ), f] ∘ [θ(ζ), U(ζ), g] = [θ(ζ) ⊗ θ(ζ), U(ζ) ⊗ U(ζ), g ⊗ f].

This completes the proof. □

6. The endomorphism ring of the trivial module

A number of different types of theorems appearing in [11, 12, 15] give direct sum decompositions in quotient categories, and it is possible to try to extend these previous results to the quotients Q^{τ} = M_{ρ}/M_{ρ−τ} in many ways. In this section we describe a decomposition of the endomorphism ring of the trivial module analogous to that given in [12]. For these purposes we need a generalization of the idea of a component of a variety.

Definition 6.1. Let V be a homogeneous affine variety, and let n be a nonnegative integer. Suppose that whenever X_{1} and X_{2} are closed homogeneous subvarieties of V such that V = X_{1} ∪X_{2} and dim(X_{1} ∩X_{2}) ≤ n, then either V = X_{1} or V = X_{2}. Then we will say that V is n-irreducible. A closed homogeneous subvariety W ⊆ V will be called an n-component of V if W is a maximal n-irreducible subvariety. We will say that an n-component W of V is large if dim W > n and small if dim W ≤ n.

Note that if n ≥ dim V − 1 and V is n-irreducible, then V is irreducible.

Proposition 6.2. Let V be a homogeneous affine variety, and let n be a nonnegative integer.

(1) If W is a component with dim W ≤ n, then W is an n-component of V.

(2) If V_{1} and V_{2} are distinct n-components of V, then dim(V_{1} ∩ V_{2}) ≤ n.

(3) Every component of V is contained in a unique n-component.

(4) Every n-component is a union of components of V.

Proof. (1) If W is a component of V with dim W ≤ n, then W is certainly n-irreducible. Let V′ be an n-component of V containing W. Let X_{1} = W, and let X_{2} be the union of all components of V′ other than W. Then V′ = X_{1} ∪ X_{2} and dim(X_{1} ∩ X_{2}) ≤ n, so either V′ = X_{1} or V′ = X_{2}. But W ⊆ V′ and W ⊈ X_{2}, so it follows that V′ = X_{1} = W, and W is an n-component of V.

(2) Suppose that V_{1} and V_{2} are n-components of V such that dim(V_{1} ∩ V_{2}) > n. Set V′ = V_{1} ∪ V_{2}, and let X_{1} and X_{2} be closed homogeneous subvarieties of V′ such that V′ = X_{1} ∪X_{2} and dim(X_{1} ∩X_{2}) ≤ n. Because V_{1} is n-irreducible, it follows that either V_{1} ⊆ X_{1} or V_{1} ⊆ X_{2}. By interchanging X_{1} and X_{2} if necessary, we may assume that V_{1} ⊆ X_{1}. Similarly, either V_{2} ⊆ X_{1} or V_{2} ⊆ X_{2}. But dim(V_{1} ∩ V_{2}) > n and dim(X_{1} ∩ X_{2}) ≤ n, so it follows that V_{2} ⊆ X_{1}. Hence V′ ⊆ X_{1}, and V′ is n-irreducible. By the maximality of V_{1} and V_{2}, we conclude that V′ = V_{1} = V_{2}, as desired.

(3) Let W be any component of V. Then W is n-irreducible, so it is contained in an n-component of V. Suppose that V_{1} and V_{2} are n-components of V containing W. If dim W ≤ n, then (1) implies that W = V_{1} = V_{2}; and if dim W > n, then (2) implies that V_{1} = V_{2}.

(4) We will prove that every n-component of V is a union of components by induction on the number of components of V. If V is n-irreducible, then there is nothing to prove, so assume that V is not n-irreducible. Then there are proper closed homogeneous subvarieties X_{1} and X_{2} of V such that V = X_{1} ∪ X_{2} and dim(X_{1} ∩ X_{2}) ≤ n, and we may assume that X_{1} and X_{2} are unions of components of V. If W is any n-component of V, then either W ⊆ X_{1} or W ⊆ X_{2}. Then W is an n-component of either X_{1} or X_{2}, and it follows by induction that W is a union of components of V. □

By Proposition 6.2 it follows that V_{G}(k) has a unique decomposition of the form

(6.3)  V_{G}(k) = V_{1} ∪ ⋯ ∪ V_{r} ∪ W,

where V_{1}, …, V_{r} are all of the large (ρ − τ)-components of V_{G}(k) and W is the union of all small (ρ − τ)-components. For 1 ≤ i ≤ r set d_{i} = dim V_{i} − (ρ − τ).

Lemma 6.4. For 1 ≤ i ≤ r and 1 ≤ j ≤ τ there exist homogeneous elements ζ_{ij} ∈ H^{•}(G, k) satisfying the following conditions.

(1) V_{n} ∪ W ⊆ V_{G}(ζ_{i1}, …, ζ_{iτ}) for all n ≠ i.

(2) dim(V_{i} ∩ V_{G}(ζ_{i1}, …, ζ_{iτ})) ≤ ρ − τ for all i.

(3) If V_{i1}, … V_{in_{i}} are the distinct components of V_{i}, then dim(V_{im} ∩ V_{G}(ζ_{i1}, … ζ_{ij})) = dim V_{im} − j whenever 1 ≤ m ≤ n_{i} and 1 ≤ j ≤ dim V_{im} − (ρ − τ).

(4) ζ_{ij} = 0 if d_{i} < j ≤ τ.

(5) ζ_{ij}ζ_{mn} = 0 for all i, j, m, n with i ≠ m.

(6) deg ζ_{ij} = deg ζ_{mn} for all i, j, m, n.

Proof. Fix i with 1 ≤ i ≤ r, and set d_{im} = dim V_{im} − (ρ − τ) for 1 ≤ m ≤ n_{i}. We first choose elements ζ_{ij} ∈ H^{•}(G, k) for 1 ≤ j ≤ τ by induction on j so that conditions (1), (3), and (4) hold. Let ζ_{i1} ∈ H^{•}(G, k) be any homogeneous element of positive degree such that V_{n} ∪ W ⊆ V_{G}(ζ_{i1}) for all n ≠ i and V_{G}(ζ_{i1}) contains no component of V_{i}. Now assume by induction that 1 < j ≤ d_{i} and ζ_{i1}, …, ζ_{i,j−1} ∈ H^{•}(G, k) have been chosen so that V_{n} ∪ W ⊆ V_{G}(ζ_{i1}, …, ζ_{i,j−1}) whenever n ≠ i and so that dim(V_{im} ∩ V_{G}(ζ_{i1}, …, ζ_{il})) = dim V_{im} − l whenever 1 ≤ m ≤ n_{i} and 1 ≤ l ≤ min{d_{im}, j − 1}. By Proposition 6.2(2) it follows that if n ≠ i and 1 ≤ m ≤ n_{i}, then V_{n} ∪ W does not contain any component of V_{im} ∩ V_{G}(ζ_{i1}, …, ζ_{i,j−1}) of dimension greater than ρ − τ. Thus there is a homogeneous element ζ_{ij} ∈ H^{•}(G, k) such that V_{n} ∪ W ⊆ V_{G}(ζ_{ij}) for all n ≠ i and such that if 1 ≤ m ≤ n_{i}, then V_{G}(ζ_{ij}) does not contain any component of V_{im} ∩ V_{G}(ζ_{i1}, …, ζ_{i,j−1}) of dimension greater than ρ − τ. It follows that V_{n} ∪ W ⊆ V_{G}(ζ_{i1}, …, ζ_{ij}) whenever n ≠ i and that dim(V_{im} ∩ V_{G}(ζ_{i1}, …, ζ_{il})) = dim V_{im} − l whenever 1 ≤ m ≤ n_{i} and 1 ≤ l ≤ min{d_{im}, j}. Thus by induction we have defined ζ_{ij} ∈ H^{•}(G, k) for 1 ≤ j ≤ d_{i}. Finally, if d_{i} < j ≤ τ, set ζ_{ij} = 0 ∈ H^{2}(G, k). Then (1), (3), and (4) are satisfied. Moreover, (3) implies that if 1 ≤ m ≤ n_{i}, then

dim(V_{im} ∩ V_{G}(ζ_{i1}, …, ζ_{iτ})) ≤ dim(V_{im} ∩ V_{G}(ζ_{i1}, …, ζ_{id_{im}})) = dim V_{im} − d_{im} = ρ − τ.

Hence dim(V_{i} ∩ V_{G}(ζ_{i1}, …, ζ_{iτ})) ≤ ρ − τ, and (2) holds.

Finally, fix j and n with 1 ≤ j, n ≤ τ, and suppose that 1 ≤ i, m ≤ r and i ≠ m. Then V_{G}(ζ_{ij}ζ_{mn}) = V_{G}(ζ_{ij}) ∪ V_{G}(ζ_{mn}) = V_{G}(k) by (1), so ζ_{ij}ζ_{mn} is nilpotent. Let s_{ijmn} be a natural number with (ζ_{ij}ζ_{mn})^{s_{ijmn}} = 0, and set N = max{s_{ijmn} | 1 ≤ i, m ≤ r, 1 ≤ j, n ≤ τ, i ≠ m}. Then ζ_{ij}^{N}ζ_{mn}^{N} = 0, so condition (5) is satisfied if we replace ζ_{ij} by ζ_{ij}^{N} for all i

and j. Now set d = lcm {deg ζ_{ij} | 1 ≤ i ≤ r, 1 ≤ j ≤ τ}, and replace ζ_{ij} by ζ_{ij}^{d/ deg ζ_{ij}} for all i and j. Then all the conditions of the lemma are satisfied. □

Lemma 6.5. For 1 ≤ i ≤ r and 1 ≤ j ≤ τ there exist homogeneous elements ζ_{ij} ∈ H^{•}(G, k) having the following properties.

(1) The conditions of Lemma 6.4 are all satisfied.

(2) If α_{j} = ζ_{1j} + ⋯ + ζ_{rj} for 1 ≤ j ≤ τ, then for each i with 1 ≤ i ≤ r there is a map f_{i} : U(α) → k in stmod-kG such that the composition D_{0}(α) → U(α) → k is (ζ_{i1}, …, ζ_{iτ}). Moreover, these maps can be chosen so that f_{1} + ⋯ + f_{r} = θ(α).

Sketch of proof. Let ζ_{ij} ∈ H^{n}(G, k) be any homogeneous elements satisfying the conditions of Lemma 6.4, and set α_{j} = ζ_{1j} + ⋯ + ζ_{rj}. If s ≥ 1, let E(α^{s}) = E(α_{1}^{s}, …, α_{τ}^{s}; k, k) be the spectral sequence given by Theorem 3.1. Fix i with 1 ≤ i ≤ r, and recall that there is a stable isomorphism

D_{0}(α) ≅ ⨁_{j=1}^{τ} Ω^{n}k

so that (ζ_{i1}, …, ζ_{iτ}) ∈ Êxt_{kG}^{0}(D_{0}(α), k) = E_{1}^{0,0}(α). We will show by induction that (ζ_{i1}^{p^{t}}, …, ζ_{iτ}^{p^{t}}) ∈ E_{t+2}^{0,0}(α^{p^{t}}) for all t ≥ 0. Set T = {1, …, τ}. If S = {j, l} ⊆ T with j < l, then Theorem 3.1 shows that the component of d_{1}^{0,0}(ζ_{i1}, …, ζ_{iτ}) lying in the summand Êxt_{kG}^{2n}(k, k) of E_{1}^{1,0}(α) corresponding to S is α_{j}ζ_{il} − α_{l}ζ_{ij} = 0. Thus (ζ_{i1}, …, ζ_{iτ}) ∈ E_{2}^{0,0}(α).

Now assume by induction that t ≥ 1 and that (ζ_{i1}^{p^{t−1}}, …, ζ_{iτ}^{p^{t−1}}) ∈ E_{t+1}^{0,0}(α^{p^{t−1}}). By Remark 4.6 there is a multiplication

π_{t+1} : E_{t+1}^{∗,∗}(α) ⊗ E_{t+1}^{∗,∗}(α) → E_{t+1}^{∗,∗}(α^{2}),

and the differential d_{t+1} is a derivation. If s ≥ 2, then one can generalize this idea to define a product of s factors

π_{t+1}^{(s)} : E_{t+1}^{∗,∗}(α) ⊗ ⋯ ⊗ E_{t+1}^{∗,∗}(α) → E_{t+1}^{∗,∗}(α^{s}).

Taking s = p, we conclude that (ζ_{i1}^{p^{t}}, …, ζ_{iτ}^{p^{t}}) = π_{t+1}^{(p)}((ζ_{i1}^{p^{t−1}}, …, ζ_{iτ}^{p^{t−1}})^{⊗p}) ∈ E_{t+1}^{0,0}(α^{p^{t}}), and one can use the fact that d_{t+1} is a derivation to check that

d_{t+1}(ζ_{i1}^{p^{t}}, …, ζ_{iτ}^{p^{t}}) = d_{t+1}π_{t+1}^{(p)}((ζ_{i1}^{p^{t−1}}, …, ζ_{iτ}^{p^{t−1}})^{⊗p})
= pπ_{t+1}^{(p)}((ζ_{i1}^{p^{t−1}}, …, ζ_{iτ}^{p^{t−1}})^{⊗p−1} ⊗ d_{t+1}(ζ_{i1}^{p^{t−1}}, …, ζ_{iτ}^{p^{t−1}}))
= 0.

Hence (ζ_{i1}^{p^{t}}, …, ζ_{iτ}^{p^{t}}) ∈ E_{t+2}^{0,0}(α^{p^{t}}), as desired.

In particular, it follows that (ζ_{i1}^{p^{τ−2}}, …, ζ_{iτ}^{p^{τ−2}}) ∈ E_{τ}^{0,0}(α^{p^{τ−2}}) = E_{∞}^{0,0}(α^{p^{τ−2}}). Replacing ζ_{ij} by ζ_{ij}^{p^{τ−2}} for all i and j, we may assume that the conditions of Lemma 6.4 are satisfied and that (ζ_{i1}, …, ζ_{iτ}) ∈ E_{∞}^{0,0}(α) for all i. Then there is a map f_{i} ∈ \underline{Hom}_{kG}(U(α), k) = Êxt_{kG}^{0}(U(α), k) corresponding to (ζ_{i1}, …, ζ_{iτ}) under the isomorphism

Êxt_{kG}^{0}(U(α), k)/F^{1}Êxt_{kG}^{0}(U(α), k) ≅ E_{∞}^{0,0}(α).

In other words, the composition

D_{0}(α) ⟶ U(α) ⟶^{f_{i}} k

is just (ζ_{i1}, …, ζ_{iτ}). Because the composition

D_{0}(α) ⟶ U(α) ⟶^{θ(α)} k

is (α_{1}, …, α_{τ}) = ∑_{i=1}^{r}(ζ_{i1}, …, ζ_{iτ}), we can take f_{r} = θ(α) − f_{1} − ⋯ − f_{r−1} so that f_{1} + ⋯ + f_{r} = θ(α), and this gives the desired result. □

Lemma 6.6. For 1 ≤ i ≤ r and 1 ≤ j ≤ τ let ζ_{ij} ∈ H^{•}(G, k) be homogeneous elements satisfying the conditions of Lemma 6.5, and set α_{j} = ζ_{1j} +⋯+ζ_{rj}. Then θ(α) : U(α) → k is a Q^{τ} -isomorphism.

Proof. By the definition of θ(α) there is a triangle

U(α) ⟶^{θ(α)} k ⟶ ⨂_{j=1}^{τ} Ω^{−1}L_{α_{j}} ⟶ Ω^{−1}U(α)

in stmod-kG, so it suffices to show that

ρ − τ ≥ dim V_{G}(⨂_{j=1}^{τ} Ω^{−1}L_{α_{j}}) = dim ⋂_{j=1}^{τ} V_{G}(α_{j}) = dim V_{G}(α_{1}, …, α_{τ}).

Fix i with 1 ≤ i ≤ r, and let 𝔪 ∈ V_{i}. Then for all l ≠ i and all j we have 𝔪 ∈ V_{G}(ζ_{lj}), so 𝔪 ∈ V_{G}(α_{j}) if and only if 𝔪 ∈ V_{G}(ζ_{ij}). Hence V_{i} ∩ V_{G}(α_{1}, …, α_{τ}) = V_{i} ∩ V_{G}(ζ_{i1}, …, ζ_{iτ}) has dimension at most ρ − τ by Lemma 6.4(2). Because this conclusion holds for all i with 1 ≤ i ≤ r, it follows that dim V_{G}(α_{1}, …, α_{τ}) ≤ ρ − τ, as desired. □

For 1 ≤ i ≤ r and 1 ≤ j ≤ τ let ζ_{ij} ∈ H^{•}(G, k) be elements satisfying the conditions of Lemma 6.5, and set α_{j} = ζ_{1j} + ⋯ + ζ_{rj}. By Lemma 6.5(2) there are maps f_{i} : U(α) → k such that f_{1} + ⋯ + f_{r} = θ(α) and the composition

D_{0}(α) ⟶ U(α) ⟶^{f_{i}} k

is (ζ_{i1}, …, ζ_{iτ}) for all i. Moreover, θ(α) is a Q^{τ} -isomorphism by Lemma 6.6, so there is an element e_{i} ∈ End_{Q^{τ}} (k) represented by

k ⟵^{θ(α)} U(α) ⟶^{f_{i}} k.

Proposition 6.7. For 1 ≤ i ≤ r and 1 ≤ j ≤ τ there exist homogeneous elements ζ_{ij} ∈ H^{•}(G, k) and maps f_{i} : U(α) → k such that the conditions of Lemma 6.5 are satisfied and the corresponding elements e_{1}, …, e_{r} ∈ End_{Q^{τ}} (k) are orthogonal idempotents with e_{1} + ⋯ + e_{r} = 1.

Proof. For 1 ≤ i ≤ r and 1 ≤ j ≤ τ let ζ_{ij} ∈ H^{•}(G, k) be any homogeneous elements satisfying the conditions of Lemma 6.5, and let f_{1}, …, f_{r} : U(α) → k be any maps such that f_{1} + ⋯ + f_{r} = θ(α) and the composition

D_{0}(α) ⟶ U(α) ⟶^{f_{i}} k

is (ζ_{i1}, …, ζ_{iτ}) for all i. If e_{i} ∈ End_{Q^{τ}} (k) is represented by

k ⟵^{θ(α)} U(α) ⟶^{f_{i}} k,

then e_{1} + ⋯ + e_{r} is represented by

k ⟵^{θ(α)} U(α) ⟶^{f_{1}+···+f_{r}} k.

But f_{1} + ⋯ + f_{r} = θ(α), so it follows that e_{1} + ⋯ + e_{r} = 1.

Now suppose that 1 ≤ i, j ≤ r and i ≠ j. Then Theorem 5.4 implies that e_{i}e_{j} is represented by

k ⟵^{θ(α)⊗θ(α)} U(α) ⊗ U(α) ⟶^{f_{i}⊗f_{j}} k.

The composition

D_{0}(α) ⊗ D_{0}(α) ⟶ U(α) ⊗ U(α) ⟶^{f_{i}⊗f_{j}} k

is (ζ_{i1}, …, ζ_{iτ}) ⊗ (ζ_{j1}, …, ζ_{jτ}). But by Lemma 6.4(5) we have ζ_{is}ζ_{jt} = 0, so it follows that e_{i}e_{j} ∈ F^{1} End_{Q^{τ}} (k) and hence (e_{i}e_{j})^{τ} = 0 by Proposition 5.5. Let n be a natural number such that p^{n} ≥ τ. Then e_{i}^{p^{n}} is represented by

k ⟵^{θ(α)^{⊗p^{n}}} U(α)^{⊗p^{n}} ⟶^{f_{i}^{⊗p^{n}}} k,

and one can show that this is equivalent to a morphism

k ⟵^{θ(α_{1}^{p^{n}} ,...,α_{τ}^{p^{n}})} U(α_{1}^{p^{n}}, …, α_{τ}^{p^{n}}) ⟶^{f′_{i}} k

in which the composition

D_{0}(α_{1}^{p^{n}}, …, α_{τ}^{p^{n}}) ⟶ U(α_{1}^{p^{n}}, …, α_{τ}^{p^{n}}) ⟶^{f′_{i}} k

is (ζ_{i1}^{p^{n}}, …, ζ_{iτ}^{p^{n}}). Thus if we replace ζ_{ij} by ζ_{ij}^{p^{n}}, α_{j} by α_{j}^{p^{n}}, f_{i} by f′_{i}, and e_{i} by e_{i}^{p^{n}}, then we may assume that e_{i}e_{j} = 0 for all i ≠ j. We then have

e_{i} = e_{i} · 1 = e_{i}(e_{1} + ⋯ + e_{r}) = e_{i}^{2}.

Thus e_{1}, …, e_{r} are orthogonal idempotents,and this completes the proof. □

For the remainder of this section we assume that the elements ζ_{ij} ∈ H^{•}(G, k) and the maps f_{i} : U(α) → k are chosen as in the statement of Proposition 6.7. In particular, the elements e_{1}, …, e_{r} ∈ End_{Q^{τ}} (k) are orthogonal idempotents such that e_{1} + ⋯ + e_{r} = 1, and our objective is to show that e_{1}, …, e_{r} are primitive. We begin with a number of preliminary lemmas. The proof of the first part of the following result is essentially the same as that of Proposition 1.5 of [12]. Because it is short, we have repeated it here.

Proposition 6.8. Let n be an integer such that n > ρ − τ.

(1) If M and N are finitely generated kG-modules such that M ≅ N in Q^{τ}, then V_{G}(M) and V_{G}(N) have the same n-dimensional components.

(2) Suppose that M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ Ω^{−1}M_{1} is a triangle in Q^{τ}. If {i, j, l} = {1, 2, 3}, then any n-dimensional component of V_{G}(M_{l}) is contained in V_{G}(M_{i}) ∪ V_{G}(M_{j}).

Proof. To prove (1), it suffices to consider the case in which there is a triangle in stmod-kG of the form L ⟶ M ⟶ N ⟶ Ω^{−1}L with cx_{G}(L) ≤ ρ−τ. Because V_{G}(N) ⊆ V_{G}(L)∪V_{G}(M) and V_{G}(M) ⊆ V_{G}(L)∪V_{G}(N), it follows that V_{G}(L)∪V_{G}(M) = V_{G}(L)∪V_{G}(N). If n > ρ−τ, then we conclude that V_{G}(M) and V_{G}(N) have the same n-dimensional components, as desired.

To prove (2), we begin by observing that there is a triangle

M′_{1} ⟶ M′_{2} ⟶ M′_{3} ⟶ Ω^{−1}M′_{1}

in stmod-kG such that M′_{i} ≅ M_{i} in Q^{τ} for 1 ≤ i ≤ 3. Then V_{G}(M′_{l}) ⊆ V_{G}(M′_{i}) ∪ V_{G}(M′_{j}), and the result follows from (1). This completes the proof. □

Lemma 6.9. Let β_{1}, …, β_{τ} be homogeneous elements of H^{•}(G, k), and suppose that f : U(β) → k is a map such that the composition D_{0}(β) → U(β) → k is (ζ_{1}, …, ζ_{τ}). If

U(β) ⟶^{f} k ⟶ L ⟶ Ω^{−1}U(β)

is a triangle in stmod-kG, then V_{G}(L) ⊆ V_{G}(β_{1}ζ_{1}, …, β_{τ} ζ_{τ}).

Proof. Set n_{i} = deg β_{i} for 1 ≤ i ≤ τ. Let ε : D_{0}(β) → U(β) denote the canonical epimorphism, and for 1 ≤ j ≤ τ let ι_{j} : Ω^{n_{j}}k → ⊕_{s∈T} Ω^{n_{s}}k ≅ D_{0}(β) denote the inclusion into the j^{th} summand. Consider the triangle of the form

Ω^{n_{j}}k ⟶^{ει_{j}} U(β) ⟶ M_{j} ⟶ Ω^{n_{j}−1}k,

and note that fει_{j} = (ζ_{1}, …, ζ_{τ})ι_{j} = ζ_{j} in stmod-kG. By the Octahedral Axiom there is a commutative diagram of triangles

Ω^{n_{j}}k ⟶^{ει_{j}} U(β) ⟶ M_{j} ⟶ Ω^{n_{j}−1}k
∥  ^{f}↓  ↓  ∥
Ω^{n_{j}}k ⟶^{ζ_{j}}  k  ⟶ Ω^{−1}L_{ζ_{j}} ⟶ Ω^{n_{j}−1}k
↓  ↓
L  L
↓  ↓
Ω^{−1}U(β) ⟶ Ω^{−1}M_{j}.

Then V_{G}(M_{j}) ⊆ V_{G}(Ω^{−1}L_{ζ_{j}}) ∪ V_{G}(L) = V_{G}(ζ_{j}) ∪ V_{G}(L), and V_{G}(L) ⊆ V_{G}(ζ_{j}) ∪ V_{G}(M_{j}).

Applying the argument of the previous paragraph to the map θ(β) : U(β) → k, we see that

V_{G}(M_{j}) ⊆ V_{G}(β_{j}) ∪ V_{G}(L(β_{1}, …, β_{τ})) = V_{G}(β_{j}) ∪ V_{G}(β_{1}, …, β_{τ}) = V_{G}(β_{j}).

Thus it follows in general that V_{G}(L) ⊆ V_{G}(ζ_{j})∪V_{G}(β_{j}) = V_{G}(β_{j}ζ_{j}) for all j, as desired. □

For the next lemma it will be useful to reduce to the situation studied in [15], which we now summarize.

If H is any subgroup of G, then the restriction map res_{G,H} : H^{∗}(G, k) → H^{∗}(H, k) induces a map res_{G,H}^{∗} : V_{H}(k) → V_{G}(k). Let V be any component of V_{G}(k) with dim V > ρ − τ. Then there is a maximal elementary abelian p-subgroup E ⊆ G such that V = res_{G,E}^{∗}(V_{E}(k)). Let N (G, E) denote the full subcategory of stmod-kG consisting of all modules M such that V ⊈ V_{G}(M), and let Q(G, E) denote the quotient stmod-kG/N (G, E). Because dim V > ρ − τ, any diagram

M ⟵^{s} U ⟶^{f} N

representing a morphism in Q^{τ} also represents a morphism in Q(G, E), and there is a canonical functor Q^{τ} → Q(G, E). Let ϕ_{V} : End_{Q^{τ}} (k) → End_{Q(G,E)}(k) denote the ring homomorphism determined by this functor.

Lemma 6.10. Let e′_{1}, e′_{2} ∈ End_{Q^{τ}} (k) be orthogonal idempotents, and set e′ = e′_{1} + e′_{2}. Let

k ⟶^{e′_{i}} k ⟶ L′_{i} ⟶ Ω^{−1}k

and

k ⟶^{e′} k ⟶ L′ ⟶ Ω^{−1}k

be triangles in Q^{τ}, and assume that V is a closed homogeneous subvariety of V_{G}(k) such that one of the following conditions is satisfied:

(1) V is a component of V_{G}(k) such that dim V > ρ − τ and V ⊈ V_{G}(L′).

(2) e′ = 1 and V is a large (ρ − τ)-component of V_{G}(k). Then either V ⊆ V_{G}(L′_{1}) or V ⊆ V_{G}(L′_{2}), but V ⊈ V_{G}(L′_{1}) ∩ V_{G}(L′_{2}).

Proof. First suppose that V is a component of V_{G}(k) such that dim V > ρ − τ and V ⊈ V_{G}(L′). Let E be a maximal elementary abelian p-subgroup of G such that V = res_{G,E}^{∗}(V_{E}(k)). Then L′ ≅ 0 in Q(G, E), so ϕ_{V} (e′) is an idempotent isomorphism. But End_{Q(G,E)}(k) is local by Lemma 4.2 of [15], so it follows that ϕ_{V} (e′) = 1. Then ϕ_{V} (e′_{1}) and ϕ_{V} (e′_{2}) are orthogonal idempotents such that ϕ_{V} (e′_{1})+ϕ_{V} (e′_{2}) = 1, so either ϕ_{V} (e′_{1}) = 0 or ϕ_{V} (e′_{2}) = 0. We may assume without loss of generality that ϕ_{V} (e′_{1}) = 0 and ϕ_{V} (e′_{2}) = 1. Then L′_{1} ≅ 0̸ in Q(G, E) and hence V ⊆ V_{G}(L′_{1}). On the other hand, ϕ_{V} (e′_{2}) = 1 is an isomorphism in Q(G, E), so L′_{2} ≅ 0 in Q(G, E) and hence V ⊈ V_{G}(L′_{2}). Thus we get the desired conclusion when (1) holds.

Now assume that e′ = 1 and V is a large (ρ − τ)-component of V_{G}(k). Then L′ ≅ 0 in Q^{τ}. If V′ is any component of V_{G}(k) with dim V′ > ρ − τ, then V′ ⊈ V_{G}(L′) by Proposition 6.8. Thus V′ satisfies condition (1), and it follows that V′ ⊆ V_{G}(L′_{1})∪V_{G}(L′_{2}). Recall that in (6.3) the subvariety W is the union of all small (ρ − τ)-components of V_{G}(k) so that V_{G}(k) = V_{G}(L′_{1}) ∪ V_{G}(L′_{2}) ∪ W. We claim that dim(V_{G}(L′_{1}) ∩ V_{G}(L′_{2})) ≤ ρ − τ.

Suppose that V_{G}(L′_{1}) ∩ V_{G}(L′_{2}) has a component V′′ with dim V′′ > ρ − τ. Let β_{1}, …, β_{τ} ∈ H^{•}(G, k) be homogeneous elements such that e′_{1} is represented by

k ⟵^{θ(β)} U(β) ⟶^{g_{1}} k,

and let ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) be the elements such that the composition

D_{0}(β) ⟶ U(β) ⟶^{g_{1}} k

is (ζ_{1}, …, ζ_{τ}). Then e′_{2} = 1 − e′_{1} is represented by

k ⟵^{θ(β)} U(β) ⟶^{θ(β)−g_{1}} k,

and the composition

D_{0}(β) ⟶ U(β) ⟶^{θ(β)−g_{1}} k

is (β_{1}−ζ_{1}, …, β_{τ} −ζ_{τ}). Because V′′ ⊆ V_{G}(L′_{1})∩V_{G}(L′_{2}) and dim V′′ > ρ−τ, Proposition 6.8 and Lemma 6.9 imply that

V′′ ⊆ V_{G}(β_{1}ζ_{1}, …, β_{τ} ζ_{τ}) ∩ V_{G}(β_{1}^{2} − β_{1}ζ_{1}, …, β_{τ}^{2} − β_{τ} ζ_{τ})
⊆ V_{G}(β_{1}^{2}, …, β_{τ}^{2}) = V_{G}(β_{1}, …, β_{τ}).

But dim V_{G}(β_{1}, …, β_{τ}) = ρ − τ, so we have a contradiction.

We have now shown that V_{G}(k) = V_{G}(L′_{1}) ∪ V_{G}(L′_{2}) ∪ W and dim(V_{G}(L′_{1}) ∩ V_{G}(L′_{2})) ≤ ρ − τ. Because V is a large (ρ − τ)-component of V_{G}(k), it follows that either V ⊆ V_{G}(L′_{1}) or V ⊆ V_{G}(L′_{2}), but V ⊈ V_{G}(L′_{1}) ∩ V_{G}(L′_{2}). This completes the proof. □

We now return to the study of the specific idempotents e_{1}, …, e_{r} ∈ End_{Q^{τ}} (k) defined previously. For definiteness we concentrate on the idempotent e_{1}.

Lemma 6.11. Let e ∈ End_{Q^{τ}} (k) · e_{1} be an idempotent, and let

k ⟶^{e} k ⟶ L ⟶ Ω^{−1}k

be a triangle in Q^{τ}. Then V_{2} ∪ ⋯ ∪ V_{r} ⊆ V_{G}(L).

Proof. Consider the triangles

T : U(α) ⟶^{f_{1}} k ⟶ L_{1} ⟶ Ω^{−1}U(α)

and

T′ : U(α) ⟶^{θ(α)−f_{1}} k ⟶ L′_{1} ⟶ Ω^{−1}U(α)

in stmod-kG. By Lemma 6.9 we know that

V_{G}(L_{1}) ⊆ V_{G}(α_{1}ζ_{11}, …, α_{τ} ζ_{1τ}) = V_{G}(ζ_{11}^{2}, …, ζ_{1τ}^{2} ) = V_{G}(ζ_{11}, …, ζ_{1τ})

and

V_{G}(L′_{1}) ⊆ V_{G}(α_{1}^{2} − α_{1}ζ_{11}, …, α_{τ}^{2} − α_{τ} ζ_{1τ}).

Now suppose that i > 1, and choose j so that V_{i} ⊈ V_{G}(α_{j}). If V_{i} ⊆ V_{G}(L′_{1}), then V_{i} ⊆ V_{G}(α_{j}^{2} − α_{j}ζ_{1j}) ∩ V_{G}(ζ_{1j}) ⊆ V_{G}(α_{j}^{2}) = V_{G}(α_{j}), and we get a contradiction. Hence V_{i} ⊈ V_{G}(L′_{1}). It is easy to see that the images in Q^{τ} of the triangles T and T′ are isomorphic to

k ⟶^{e_{1}} k ⟶ L_{1} ⟶ Ω^{−1}k

and

k ⟶^{1−e_{1}} k ⟶ L′_{1} ⟶ Ω^{−1}k

respectively, so Lemma 6.10 implies that V_{i} ⊆ V_{G}(L_{1}). Because ee_{1} = e, the Octahedral Axiom gives a commutative diagram of triangles in Q^{τ} of the form

k ⟶^{e_{1}} k ⟶ L_{1} ⟶ Ω^{−1}k
∥  ^{e}↓  ↓  ∥
k ⟶_{e} k ⟶ L ⟶ Ω^{−1}k
↓  ↓
L  L
↓  ↓
Ω^{−1}k ⟶ Ω^{−1}L_{1}.

By Proposition 6.8(2) it follows that V_{i} ⊆ V_{G}(L) for all i > 1, as desired. □

Lemma 6.12. Let e ∈ End_{Q^{τ}} (k) · e_{1} be an idempotent, and let

k ⟶^{e} k ⟶ L ⟶ Ω^{−1}k

be a triangle in Q^{τ}. If V_{1} ⊆ V_{G}(L), then e = 0.

Proof. Let β_{1}, …, β_{τ} ∈ H^{•}(G, k) be homogeneous elements such that e is represented by

k ⟵^{θ(β)} U(β) ⟶^{f} k.

Let 𝔞 = (α_{1}^{2}, …, α_{τ}^{2}) and 𝔟 = (β_{1}^{2}, …, β_{τ}^{2}) be the ideals in H^{•}(G, k) generated by α_{1}^{2}, …, α_{τ}^{2} and by β_{1}^{2}, …, β_{τ}^{2} respectively. Choose γ_{1} ∈ 𝔞𝔟 such that V_{G}(γ_{1}) ≠ V_{G}(k), and for 1 < i ≤ τ choose γ_{i} ∈ 𝔞𝔟 inductively so that V_{G}(γ_{i}) contains no component of V_{G}(γ_{1}, …, γ_{i−1}) of dimension greater than ρ − τ. Then it follows that dim V_{G}(γ_{1}, …, γ_{i}) = ρ − i for 1 ≤ i ≤ τ. In particular, θ(γ) is a Q^{τ} -isomorphism, and by Proposition 3.6 there is a chain map ϕ : C(γ) → C(β) such that the induced map ϕ′ : U(γ) → U(β) satisfies cls(θ(β)ϕ′) = cls(θ(γ)) in stmod-kG. Hence ϕ′ is a Q^{τ} -isomorphism, and e is represented by

k ⟵^{θ(γ)} U(γ) ⟶^{fϕ′} k.

Moreover, W ⊆ V_{G}(α_{1}, …, α_{τ}) ⊆ V_{G}(𝔞𝔟) ⊆ V_{G}(γ_{1}, …, γ_{τ}). Replacing β_{1}, …, β_{τ} by γ_{1}, …, γ_{τ} if necessary, we may assume without loss of generality that W ⊆ V_{G}(β_{1}, …, β_{τ}).

Now let ζ_{1}, …, ζ_{τ} be the homogeneous elements of H^{•}(G, k) such that the composition

D_{0}(β) ⟶ U(β) ⟶^{f} k

is (ζ_{1}, …, ζ_{τ}). We may assume that there is a triangle in stmod-kG of the form

U(β) ⟶^{f} k ⟶ L ⟶ Ω^{−1}U(β),

and Lemmas 6.11 and 6.9 imply that

V_{1} ∪ ⋯ ∪ V_{r} ⊆ V_{G}(L) ⊆ V_{G}(β_{1}ζ_{1}, …, β_{τ} ζ_{τ}).

Hence

V_{G}(k) = V_{1} ∪ ⋯ ∪ V_{r} ∪ W ⊆ V_{G}(β_{1}ζ_{1}, …, β_{τ} ζ_{τ}),

and it follows that β_{1}ζ_{1}, …, β_{τ} ζ_{τ} are all nilpotent. Let N be a natural number such that β_{i}^{N}ζ_{i}^{N} = 0 for 1 ≤ i ≤ τ. Then e = e^{N} is represented by

k ≅ k^{⊗N} ⟵^{θ(β)^{⊗N}} U(β)^{⊗N} ⟶^{g} k,

and the composition D_{0}(β)^{⊗N} → U(β)^{⊗N} → k is (ζ_{1}, …, ζ_{τ})^{⊗N}. It follows that e is also represented by

k ⟵^{θ(β_{1}^{2N} ,...,β_{τ}^{2N})} U(β_{1}^{2N}, …, β_{τ}^{2N}) ⟶^{h} k,

and D_{0}(β_{1}^{2N}, …, β_{τ}^{2N}) → U(β_{1}^{2N}, …, β_{τ}^{2N}) → k is the map (β_{1}^{N}ζ_{1}^{N}, …, β_{τ}^{N}ζ_{τ}^{N}) = (0, …, 0). Hence e ∈ F^{1} End_{Q^{τ}} (k) is nilpotent by Proposition 5.5. Thus e = 0, as desired. □

We are now ready to prove the main result of this section. The notation is that used in Proposition 6.7.

Theorem 6.13. The elements e_{1}, …, e_{r} ∈ End_{Q^{τ}} (k) are a complete set of primitive orthogonal idempotents.

Proof. By Proposition 6.7 we must show that e_{1}, …, e_{r} are primitive, and it clearly suffices to prove that e_{1} is primitive. Suppose, then, that e ∈ End_{Q^{τ}} (k)·e_{1} is a nonzero idempotent, and let

\begin{matrix}k & ⟶^{e} & k & ⟶ & L & ⟶ & Ω^{−1}k, \\ k & ⟶^{1−e} & k & ⟶ & L′ & ⟶ & Ω^{−1}k, \\ k & ⟶^{e_{1}} & k & ⟶ & L_{1} & ⟶ & Ω^{−1}k, \end{matrix}

and

k ⟶^{e_{1}−e} k ⟶ L′_{1} ⟶ Ω^{−1}k

be triangles in Q^{τ}. Because e ≠ 0, Lemma 6.12 implies that V_{1} ⊈ V_{G}(L) and hence V_{1} ⊆ V_{G}(L′) by Lemma 6.10. If V′ is any component of V_{1}, then dim V′ > ρ − τ and V′ ⊆ V_{G}(L′), so Lemma 6.10 shows that V′ ⊈ V_{G}(L). In particular, this argument applies in the case e = e_{1} to show that V′ ⊈ V_{G}(L_{1}). Then another application of Lemma 6.10 shows that V′ ⊆ V_{G}(L′_{1}). Hence V_{1} ⊆ V_{G}(L′_{1}), and Lemma 6.12 implies that e_{1} − e = 0, as desired. □

It should be mentioned that the main results of [3] easily extend to higher complexity quotient categories. Indeed, let StMod-kG denote the stable category of all right kGmodules. Using the definition of complexity given in [5] for infinitely generated modules, we let U_{ρ−τ} be the full subcategory of StMod-kG consisting of modules of complexity at most ρ − τ. Idempotent endomorphisms split in the quotient category StMod-kG/U_{ρ−τ}, so the trivial module has a decomposition

k = π_{1}(k) ⊕ ⋯ ⊕ π_{r}(k)

in which the summand π_{i}(k) is the module corresponding to the idempotent endomorphism e_{i}. The modules π_{i}(k) have the property that π_{i}(k) ⊗ π_{i}(k) ≅ π_{i}(k) in StMod-kG, and π_{i}(k) ⊗ π_{j}(k) ≅ 0 in StMod-kG if j ≠ i. Moreover, if W_{i} denotes the full subcategory of StMod-kG/U_{ρ−τ} consisting of all objects isomorphic to π_{i}(k) ⊗ M for some module M, then

StMod-kG/U_{ρ−τ} = W_{1} × ⋯ × W_{r}.

The reader is referred to [3] for more details.

7. Idempotent modules

In this section we explore some of the connections between quotient categories, the class of modules defined in Section 2, and Rickard’s work on idempotent modules in the stable category [20]. We begin with a brief summary of Rickard’s results.

Recall that StMod-kG denotes the stable category of all right kG-modules, whereas stmod-kG denotes the subcategory of finitely generated modules. A full triangulated subcategory M of stmod-kG is said to be thick if it is closed under taking direct summands. As in [20], we will say that an object N of StMod-kG is M-local if \underline{Hom}_{kG}(M, N) = 0 for all M in M, and we write M^{⊕} for the smallest full triangulated subcategory of StMod-kG that contains M and is closed under arbitrary direct sums. In [20] Rickard shows that for any thick subcategory M of stmod-kG and any kG-module X there is a triangle

T_{M}(X) : E_{M}(X) ⟶ X ⟶ F_{M}(X) ⟶ Ω^{−1}(E_{M}(X))

in StMod-kG such that E_{M}(X) is in M^{⊕} and F_{M}(X) is M-local. Moreover, the triangle T_{M}(X) is unique up to isomorphism.

Now suppose in addition that M ⊗ N is in M whenever M and N are finitely generated and either M or N is in M. Then E_{M}(X) is a filtered colimit of objects in M, and the triangle T_{M}(k) ⊗ X is isomorphic to T_{M}(X). The modules E_{M}(k) and F_{M}(k) are idempotent in the sense that E_{M}(k) ⊗ E_{M}(k) ≅ E_{M}(k) and F_{M}(k) ⊗ F_{M}(k) ≅ F_{M}(k) in StMod-kG, and these modules also have the property that E_{M}(k) ⊗ F_{M}(k) is projective.

Idempotent modules are closely related to the theory of varieties for arbitrary modules developed by Benson, Carlson, and Rickard [5, 6]. If M is any kG-module, then its socalled variety V_{G}(M) is actually a collection of irreducible closed homogeneous subvarieties V of V_{G}(k). Let M_{V} denote the full subcategory of stmod-kG consisting of all finitely generated modules M such that V_{G}(M) ⊆ V, and let M′_{V} denote the full subcategory consisting of all finitely generated modules M such that V_{G}(M) ⊂ V. Then the categories M_{V} and M′_{V} determine idempotent modules E_{V} = E_{M_{V}} (k) and F′_{V} = F_{M′_{V}} (k), and we set κ(V) = E_{V} ⊗ F′_{V}. If M is an arbitrary module, then V_{G}(M) can be defined as the collection of all irreducible closed homogeneous subvarieties V of V_{G}(k) such that M ⊗κ(V) is not projective. Note that V_{G}(κ(V)) = {V } by Lemma 10.4 of [6].

If M is finitely generated, then V_{G}(M) is just the collection of all irreducible closed homogeneous subvarieties of V_{G}(M). For arbitrary modules the sets V_{G}(M) share many of the properties of varieties for finitely generated modules. For example, M is projective if and only if V_{G}(M) = ∅. Moreover, if M and N are any kG-modules, then V_{G}(M ⊕ N) = V_{G}(M) ∪ V_{G}(N) and V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N).

Now let V be a collection of closed homogeneous subvarieties of V_{G}(k) that is closed under finite unions and under specialization; in other words, if V_{1} ⊆ V_{2} are closed subsets of V_{G}(k) with V_{2} ∈ V, then V_{1} ∈ V. Let M_{V} denote the full subcategory of stmod-kG consisting of all kG-modules M with V_{G}(M) ∈ V. Then for any kG-module X the triangle T_{M_{V}} (X) is characterized up to isomorphism by the property that V_{G}(E_{M_{V}} (X)) ⊆ V and V_{G}(F_{M_{V}} (X)) ∩ V = ∅. Moreover, the category M_{V} is closed under tensoring with finitely generated modules; conversely, any thick subcategory of stmod-kG that is closed under this operation must be equal to M_{V} for some collection of varieties V. In fact, if G is a p-group, then every thick subcategory of stmod-kG arises in this way [7].

Our objective in this section is to describe the relationship between the idempotent module F_{M_{V}} (k) and the quotient category stmod-kG/M_{V}. Note that if ρ is the p-rank of the group, τ is an integer with 1 ≤ τ ≤ ρ, and V is the collection of all closed homogeneous subvarieties of V_{G}(k) having dimension at most ρ − τ, then stmod-kG/M_{V} is precisely the complexity quotient category Q^{τ} considered in Sections 5 and 6. Thus the following results will apply in particular to complexity quotient categories, though in fact they are considerably more general.

First consider the special case in which V is the set of all closed homogeneous subvarieties of a fixed variety V ⊆ V_{G}(k). Let ζ_{1}, …, ζ_{τ} be homogeneous elements of H^{•}(G, k) with V_{G}(ζ_{1}, …, ζ_{τ}) = V. Set n_{i} = deg ζ_{i} for 1 ≤ i ≤ τ, and let n be a natural number. Then for each i there is a homomorphism ζ_{i,n} : Ω^{(n+1)n_{i}}k → Ω^{nn_{i}}k representing ζ_{i}. Let I_{i}^{(n)} denote the injective hull of L_{ζ_{i}^{n}} so that the modular Koszul complex C(ζ_{i}^{n}) has the form

C(ζ_{i}^{n}) :  ⋯ ⟶ 0 ⟶ Ω^{nn_{i}}k ⟶ k ⊕ I_{i}^{(n)} ⟶ 0 ⟶ · · · .

Because the map Ω^{(n+1)n_{i}}k → k ⊕ I_{i}^{(n+1)} is injective, it is easy to see that there is a commutative diagram

Ω^{(n+1)n_{i}}k ⟶ k ⊕ I_{i}^{(n+1)}
ζ_{i,n}↓  ↓^{η}
Ω^{nn_{i}}k ⟶ k ⊕ I_{i}^{(n)}

such that η is congruent to 1_{k} modulo projective homomorphisms. Taking the tensor product of the corresponding chain maps C(ζ_{i}^{n+1}) → C(ζ_{i}^{n}), we get a chain map C(ζ_{1}^{n+1}, …, ζ_{τ}^{n+1}) → C(ζ_{1}^{n}, …, ζ_{τ}^{n}), and by Corollary 3.2 there is an induced map θ_{n} : U(ζ_{1}^{n+1}, …, ζ_{τ}^{n+1}) → U(ζ_{1}^{n}, …, ζ_{τ}^{n}). Taking duals now gives a commutative diagram of short exact sequences

0 ⟶ L(ζ_{1}, …, ζ_{τ})^{∗} ⟶ k ⊕ P_{1} ⟶ U(ζ_{1}, …, ζ_{τ})^{∗} ⟶ 0
↓  ↓  ↓^{θ_{1}^{∗}}
0 ⟶ L(ζ_{1}^{2}, …, ζ_{τ}^{2})^{∗} ⟶ k ⊕ P_{2} ⟶ U(ζ_{1}^{2}, …, ζ_{τ}^{2})^{∗} ⟶ 0
↓  ↓  ↓^{θ_{2}^{∗}}
0 ⟶ L(ζ_{1}^{3}, …, ζ_{τ}^{3})^{∗} ⟶ k ⊕ P_{3} ⟶ U(ζ_{1}^{3}, …, ζ_{τ}^{3})^{∗} ⟶ 0
↓  ↓  ↓
⋮  ⋮  ⋮

in which P_{i} is projective for all i ≥ 1 and all maps in the middle column are congruent to 1_{k} modulo projective homomorphisms. Then there is a projective module P (ζ) such that the direct limit of this system has the form

0 ⟶ E(ζ) ⟶ k ⊕ P (ζ) ⟶ F (ζ) ⟶ 0.

Let

T (ζ) :  E(ζ) ⟶^{ϕ} k ⟶^{ψ} F (ζ) ⟶ Ω^{−1}(E(ζ))

denote the corresponding triangle in StMod-kG. A proof similar to that of Theorem 6.1 of [20] now gives the following result.

Proposition 7.1. Suppose that ζ_{1}, …, ζ_{τ} are homogeneous elements of H^{•}(G, k). Let V = V_{G}(ζ_{1}, …, ζ_{τ}), and let M_{V} be the thick subcategory of stmod-kG consisting of all finitely generated kG-modules M such that V_{G}(M) ⊆ V. Then the triangle T (ζ) is isomorphic to T_{M_{V}} (k).

Proof. It suffices to show that E(ζ) is in M_{V}^{⊕} and that F (ζ) is M_{V} -local. First note that if n ≥ 1, then

V_{G}(L(ζ_{1}^{n}, …, ζ_{τ}^{n})^{∗}) = ⋂_{i=1}^{τ} V_{G}(Ω^{−1}L_{ζ_{i}^{n}}) = V_{G}(ζ_{1}^{n}, …, ζ_{τ}^{n}) = V

so that L(ζ_{1}^{n}, …, ζ_{τ}^{n})^{∗} is in M_{V}. Because E(ζ) is a direct limit of objects in M_{V}, it follows that E(ζ) is in M_{V}^{⊕} by Proposition 5.17 of [20].

To prove that F (ζ) is M_{V} -local, let M be an object of M_{V}, and let f : M → F (ζ) be a kG-homomorphism. Because M is finitely generated, there is a positive integer n such that f factors through U(ζ_{1}^{n}, …, ζ_{τ}^{n})^{∗}. For 1 ≤ i ≤ τ we have V_{G}(M^{∗}) = V_{G}(M) ⊆ V ⊆ V_{G}(ζ_{i}), so we may assume without loss of generality that ζ_{i}^{n} annihilates Ext_{kG}^{∗}(M^{∗}, M^{∗}) for all i. After replacing each ζ_{i} by ζ_{i}^{n}, we may therefore assume that f factors through U(ζ_{1}, …, ζ_{τ})^{∗} and that ζ_{1}, …, ζ_{τ} ∈ J(M^{∗}). But because the map ζ_{i,m} : Ω^{(m+1)n_{i}}k → Ω^{mn_{i}}k represents ζ_{i} for all i and m, it follows that the chain map C(ζ_{1}^{m+1}, …, ζ_{τ}^{m+1}) → C(ζ_{1}^{m}, …, ζ_{τ}^{m}) induces the zero map E_{1}(ζ_{1}^{m}, …, ζ_{τ}^{m}; k, M^{∗}) → E_{1}(ζ_{1}^{m+1}, …, ζ_{τ}^{m+1}; k, M^{∗}) on the corresponding spectral sequences. By Corollary 3.2 the map \underline{Hom}_{kG}(U(ζ_{1}^{m}, …, ζ_{τ}^{m}), M^{∗}) → \underline{Hom}_{kG}(U(ζ_{1}^{m+1}, …, ζ_{τ}^{m+1}), M^{∗}) induced by θ_{m} preserves the filtration determined by the spectral sequences. Because this filtration is finite, there is an s ≥ 2 such that θ_{s−1} ⋯ θ_{1} induces the zero map \underline{Hom}_{kG}(U(ζ_{1}, …, ζ_{τ}), M^{∗}) → \underline{Hom}_{kG}(U(ζ_{1}^{s}, …, ζ_{τ}^{s}), M^{∗}). Hence we conclude that θ_{1}^{∗} ⋯ θ_{s−1}^{∗} also induces the zero map \underline{Hom}_{kG}(M, U(ζ_{1}, …, ζ_{τ})^{∗}) → \underline{Hom}_{kG}(M, U(ζ_{1}^{s}, …, ζ_{τ}^{s})^{∗}). Thus f = 0, and F (ζ) is M_{V} - local. This completes the proof. □

Now let V be a collection of closed homogeneous subvarieties of V_{G}(k) that is closed under finite unions and under specialization. For each V ∈ V fix homogeneous elements ζ_{1}, …, ζ_{τ} ∈ H^{•}(G, k) with V_{G}(ζ_{1}, …, ζ_{τ}) = V. Then ζ_{V} = (ζ_{1}, …, ζ_{τ}) determines a short exact sequence

0 ⟶ E(ζ_{V}) ⟶ k ⊕ P (ζ_{V}) ⟶^{f_{V}} F (ζ_{V}) ⟶ 0

as before, so there is an associated map g_{V} : k → F (ζ_{V}) in Mod-kG given by restricting f_{V} to the summand k. If W is a finite subset of V and V ∈ V − W, then there is a map

⨂_{W ∈W} F (ζ_{W}) ⟶ ⨂_{W ∈W∪{V }} F (ζ_{W})

in Mod-kG obtained by tensoring g_{V} with ⨂_{W ∈W} F (ζ_{W}). These objects and homomorphisms form a directed system, and we let F (V) denote the direct limit. Complete the map k → F (V) to a triangle

T (V) :  E(V) ⟶ k ⟶ F (V) ⟶ Ω^{−1}(E(V))

in StMod-kG. Then we obtain the following result.

Proposition 7.2. Let V be a collection of closed homogeneous subvarieties of V_{G}(k) that is closed under finite unions and under specialization. Then the triangle T (V) is isomorphic to T_{M_{V}} (k).

Proof. We will show that V_{G}(E(V)) ⊆ V and that V_{G}(F (V)) ∩ V = ∅. Indeed, suppose that X is any irreducible closed homogeneous subvariety of V_{G}(k). Set κ(X) = E_{X} ⊗ F′_{X} as before, and note that F (V) ⊗ κ(X) is the direct limit of the system of modules ⨂_{W ∈W}(F (ζ_{W}) ⊗ κ(X)).

Suppose that X ∈ V. Then F (ζ_{X}) ⊗ κ(X) is stably isomorphic to F_{X} ⊗ E_{X} ⊗ F′_{X}, and F_{X} ⊗ E_{X} is a projective module by Lemma 8.3(iii) of [6]. Because the modules ⨂_{W ∈W}(F (ζ_{W}) ⊗ κ(X)) with X ∈ W are cofinal in the directed system, it follows that F (V) ⊗ κ(X) is a colimit of projective modules. Hence F (V) ⊗ κ(X) is itself projective, and X ∉ V_{G}(F (V)) so that V_{G}(F (V)) ∩ V = ∅.

Now suppose that X ∉ V, and let V ∈ V. Then there are stable isomorphisms E(ζ_{V}) ⊗ E_{X} ≅ E_{V} ⊗ E_{X} ≅ E_{V ∩X}, and V ∩ X ≠ X because X ∉ V. Hence E(ζ_{V}) ⊗ κ(X) ≅ E_{V ∩X} ⊗ F′_{X} ≅ 0 in StMod-kG by Lemma 8.3(iv) of [6]. Thus the maps

⨂_{W ∈W}(F (ζ_{W}) ⊗ κ(X)) ⟶ ⨂_{W ∈W∪{V }}(F (ζ_{W}) ⊗ κ(X))

are stable isomorphisms for all V ∈ V, so the map

k ⊗ κ(X) ⟶ F (V) ⊗ κ(X)

is also a stable isomorphism. Hence E(V) ⊗ κ(X) is projective, and X ∉ V_{G}(E(V)). Thus V_{G}(E(V)) ⊆ V, and this completes the proof. □

Suppose that V is a collection of closed homogeneous subvarieties of V_{G}(k) that is closed under finite unions and under specialization. Let Υ_{V} be the full subcategory of StModkG consisting of all modules M with V_{G}(M) ⊆ V. Then M_{V} is the full subcategory of Υ_{V} consisting of all finitely generated modules, and the quotient stmod-kG/M_{V} can be identified with a subcategory of StMod-kG/Υ_{V}. In this situation the relationship between idempotent modules and quotient categories is highlighted by the fact that − ⊗ F_{M_{V}} (k) : StMod-kG → StMod-kG is an example of what is sometimes called a localizing functor. In other words, there is a triangle-equivalence F from StMod-kG/Υ_{V} onto the image of − ⊗ F_{M_{V}} (k) giving a commutative diagram

StMod-kG  StMod-kG/Υ_{V}

StMod-kG.

[figure]

−⊗F_{MV} (k)

[figure]

F

[figure]

The essence of this statement is the following result.

Proposition 7.3. Let V be a collection of closed homogeneous subvarieties of V_{G}(k) that is closed under finite unions and under specialization. If M and N are kG-modules, then

Hom_{StMod-kG/Υ_{V}} (M, N) ≅ \underline{Hom}_{kG}(M ⊗ F_{M_{V}} (k), N ⊗ F_{M_{V}} (k)).

Proof. Set Q = StMod-kG/Υ_{V}, E = E_{M_{V}} (k), and F = F_{M_{V}} (k). If U is any kG-module, then V_{G}(U ⊗ E) = V_{G}(U) ∩ V_{G}(E) ⊆ V, so U ⊗ E is in Υ_{V} and hence U ≅ U ⊗ F in Q. Thus it suffices to prove that

Hom_{Q}(M ⊗ F, N ⊗ F) ≅ \underline{Hom}_{kG}(M ⊗ F, N ⊗ F).

Because E ⊗ F is projective, there is a canonical stable isomorphism ϕ : F → F ⊗ F obtained by tensoring the triangle

T_{M_{V}} (k) :  E ⟶ k ⟶ F ⟶ Ω^{−1}E

with F. If [s, U, f] is a morphism from M ⊗ F to N ⊗ F in Q, then there is a triangle

U ⟶^{s} M ⊗ F ⟶ L ⟶ Ω^{−1}U

with L in Υ_{V}. Then V_{G}(L ⊗ F) = V_{G}(L) ∩ V_{G}(F) ⊆ V_{G}(F) ∩ V = ∅ so that L ⊗ F is projective. Thus s ⊗ 1_{F} : U ⊗ F → M ⊗ F ⊗ F is a stable isomorphism. Let Φ : Hom_{Q}(M ⊗ F, N ⊗ F) → \underline{Hom}_{kG}(M ⊗ F, N ⊗ F) be the map given by [s, U, f] ↦ (1_{N} ⊗ ϕ)^{−1}(f ⊗ 1_{F})(s ⊗ 1_{F})^{−1}(1_{M} ⊗ ϕ). Then it is easy to check that Φ is a well-defined isomorphism with an inverse given by g ↦ [1_{M} ⊗ 1_{F}, M ⊗ F, g] for all g : M ⊗ F → N ⊗ F. This completes the proof. □

Remark 7.4. Let V be a collection of closed homogeneous subvarieties of V_{G}(k) that is closed under finite unions and under specialization, and set E = E_{M_{V}} (k) and F = F_{M_{V}} (k). Then one consequence of the results of this section is that Êxt_{kG}^{∗}(F, F) can be described in terms of a direct limit of spectral sequences. For example, suppose that ζ_{1}, …, ζ_{τ} are homogeneous elements of H^{•}(G, k) such that V is the collection of all closed homogeneous subvarieties of V_{G}(ζ_{1}, …, ζ_{τ}). Then Proposition 7.1 implies that Êxt_{kG}^{∗}(k, F) is the limit of the directed system

Êxt_{kG}^{∗}(k, U(ζ_{1}, …, ζ_{τ})^{∗}) → Êxt_{kG}^{∗}(k, U(ζ_{1}^{2}, …, ζ_{τ}^{2})^{∗}) → · · · .

For n ≥ 1 we know that Êxt_{kG}^{∗}(k, U(ζ_{1}^{n}, …, ζ_{τ}^{n})^{∗}) ≅ Êxt_{kG}^{∗}(U(ζ_{1}^{n}, …, ζ_{τ}^{n}), k) is the limit of a spectral sequence, and the maps θ_{n} : U(ζ_{1}^{n+1}, …, ζ_{τ}^{n+1}) → U(ζ_{1}^{n}, …, ζ_{τ}^{n}) correspond to maps between the spectral sequences by Corollary 3.2. The direct limit of these spectral sequences therefore gives another spectral sequence converging to Êxt_{kG}^{∗}(k, F). But \underline{Hom}_{kG}(Ω^{n}E, F) = 0 for any integer n by Propositions 5.4 and 5.5 of [20], so it follows that Êxt_{kG}^{∗}(k, F) ≅ Êxt_{kG}^{∗}(F, F).

8. Examples and connections with Čech cohomology

There is a rather curious relationship between the spectral sequence methods developed here for describing homomorphisms in quotient categories and the Čech cohomology used in sheaf theory. In particular, the colimit of spectral sequences in Remark 7.4 is reminiscent of the colimit appearing in the definition of local cohomology (see [17] or [19]). The similarities become more apparent in Proposition 8.1 below. This result suggests that in the situation described by Proposition 7.1 the module F occurring in the triangle T (ζ_{1}, …, ζ_{τ}) may have an endomorphism ring that is a Čech cohomology ring. It would seem, however, that constructing an isomorphism would require knowing that the spectral sequences collapse at the E_{2} page, at least in some limiting sense. But the higher differentials in the spectral sequences have all of the earmarks of matric Massey products. One can compare them, for example, with the differentials in the hypercohomology spectral sequences of [4]. Thus there are good reasons to think that these higher differentials are not zero.

We begin by recalling the definitions of local and Čech cohomology. If ζ is a homogeneous element of H^{∗}(G, k), then the Koszul complex of ζ is the cochain complex

K(ζ) :  ⋯ ⟶ 0 ⟶ H^{∗}(G, k) ⟶^{ζ} H^{∗}(G, k) ⟶ 0 ⟶ ⋯

concentrated in degrees zero and one. The identity map in degree zero and multiplication by ζ in degree one give a chain map K(ζ^{n}) → K(ζ^{n+1}) for all n ≥ 1, and we define the stabilized Koszul complex K_{∞}(ζ) by setting K_{∞}(ζ) = \underrightarrow{lim} K(ζ^{n}). One can show that K_{∞}(ζ) is isomorphic to the complex

⋯ → 0 → H^{∗}(G, k) → H^{∗}(G, k)[1/ζ] → 0 → · · · .

More generally, set A = H^{∗}(G, k), and suppose that ζ = (ζ_{1}, …, ζ_{τ}) is a sequence of homogeneous elements in A. Then the stabilized Koszul complex of ζ is defined by K_{∞}(ζ) = K_{∞}(ζ_{1}) ⊗_{A} ⋯ ⊗_{A} K_{∞}(ζ_{τ}). If H^{∗}(G, k) is regarded as a complex concentrated in degree zero, then there is a surjective chain map ϕ : K_{∞}(ζ) → H^{∗}(G, k). We define the Čech complex Č(ζ) to be the complex Ker ϕ with a shift in degree so that Č^{n}(ζ) = Ker ϕ^{n+1} for all n. If M is an A-module, then the local cohomology of M at the ideal I = (ζ_{1}, …, ζ_{τ}) is given by

H_{I}^{∗}(M) = H^{∗}(K_{∞}(ζ) ⊗_{A} M),

and the Čech cohomology is

Ȟ_{I}^{∗}(M) = H^{∗}(Č(ζ) ⊗_{A} M).

Both H_{I}^{∗}(M) and Ȟ_{I}^{∗}(M) are independent of the choice of generators of I.

Let M̃ denote the sheaf on Spec A associated to M. The right derived functors of the sections of M̃ with support in V = V_{G}(I) are by definition the cohomology groups H_{V}^{∗} (Spec A, M̃) of Spec A with coefficients in M̃ and supports in V. These derived functors are isomorphic to the local cohomology groups H_{I}^{∗}(M). Moreover, the Čech cohomology has the property that Ȟ_{I}^{n}(M) ≅ H_{I}^{n+1}(M) for n > 0 (see [19]).

Proposition 8.1. Assume that M and N are finitely generated kG-modules and ζ_{1}, …, ζ_{τ} are homogeneous elements of H^{•}(G, k). For any n ≥ 1 let E(ζ^{n}) = E(ζ_{1}^{n}, …, ζ_{τ}^{n}; M, N) be the spectral sequence of Theorem 3.1, and let I be the ideal of H^{∗}(G, k) generated by ζ_{1}, …, ζ_{τ}. Then

\underrightarrow{lim}_{n} E_{2}^{∗,∗}(ζ^{n}) = Ȟ_{I}^{∗}(Êxt_{kG}^{∗}(M, N)).

Proof. Let C^{(n)} = C(ζ_{1}^{n}, …, ζ_{τ}^{n}) be the modular Koszul complex of ζ_{1}^{n}, …, ζ_{τ}^{n}, and let K(ζ^{n}) = K(ζ_{1}^{n}) ⊗_{A} ⋯ ⊗_{A} K(ζ_{τ}^{n}). Then for any integer t the complex Êxt_{kG}^{t}(C^{(n)} ⊗ M, N) is isomorphic to K(ζ^{n}) ⊗_{A} Êxt_{kG}^{t}(M, N). Moreover, except for a shift in degree, the kernel of the surjection

Êxt_{kG}^{t}(C^{(n)} ⊗ M, N) ≅ K(ζ^{n}) ⊗_{A} Êxt_{kG}^{t}(M, N) → A ⊗_{A} Êxt_{kG}^{t}(M, N)

is just the complex E_{1}^{∗,t}(ζ^{n}), and it follows that

\underrightarrow{lim}_{n} E_{1}^{∗,t}(ζ^{n}) ≅ Č(ζ) ⊗_{A} Êxt_{kG}^{t}(M, N).

Because the colimit is an exact functor, we conclude that

\underrightarrow{lim}_{n} E_{2}^{s,∗}(ζ^{n}) ≅ \underrightarrow{lim}_{n} H^{s}(E_{1}^{∗,∗}(ζ^{n})) ≅ Ȟ_{I}^{s}(Êxt_{kG}^{∗}(M, N)),

as desired. □

The spectral sequences of Theorem 3.1 are truncations of hypercohomology spectral sequences. In [4] the hypercohomology sequences were used to develop duality properties of cohomology rings, and the methods were particularly effective when the cohomology rings were Cohen–Macaulay. As the remaining results of this section will show, the Cohen– Macaulay property also has some relevance for our considerations here.

As in previous sections, we let ρ denote the p-rank of G, and we fix an integer τ with 1 ≤ τ ≤ ρ.

Proposition 8.2. Suppose that H^{∗}(G, k) is Cohen–Macaulay, and let ζ_{1}, …, ζ_{τ} be homogeneous elements of H^{•}(G, k) such that dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ. If τ > 1, then E_{r}^{s,t}(ζ_{1}, …, ζ_{τ}; k, k) = 0 whenever r > 1, 0 < s < τ − 1, and t ≥ −s.

Sketch of proof. Because H^{∗}(G, k) is Cohen–Macaulay and dim V_{G}(ζ_{1}, …, ζ_{τ}) = ρ − τ, there exist elements ζ_{τ+1}, …, ζ_{ρ} such that ζ_{1}, …, ζ_{ρ} is a system of parameters for H^{∗}(G, k). Hence ζ_{1}, …, ζ_{τ} is a regular sequence. Let C = C(ζ_{1}, …, ζ_{τ}) be the modular Koszul complex of ζ_{1}, …, ζ_{τ}, and consider the complex

Êxt_{kG}^{∗}(C, k) : ⋯ → 0 → Êxt_{kG}^{∗}(C_{0}, k) → ⋯ → Êxt_{kG}^{∗}(C_{τ} , k) → 0 → · · · .

Because E_{1}^{∗,∗}(ζ_{1}, …, ζ_{τ}; k, k) is obtained by truncating the left-hand term from this complex, the proof of the proposition amounts to showing that the complex Êxt_{kG}^{∗}(C, k) is exact in the appropriate degrees. One can obtain this result by writing Êxt_{kG}^{∗}(C, k) as a direct sum of subcomplexes and using the fact that the differential d_{1} is given in terms of multiplications by the regular elements ζ_{1}, …, ζ_{τ}. A detailed analysis of the argument can be found in [4]. □

Proposition 8.3. Suppose that H^{∗}(G, k) is Cohen–Macaulay and that τ > 1. Let 𝔯 = Rad End_{Q^{τ}} (k). Then 𝔯 = F^{1} End_{Q^{τ}} (k) is an ideal with 𝔯^{2} = 0, and

End_{Q^{τ}} (k)/𝔯 ≅ k.

Proof. Consider an element γ ∈ End_{Q^{τ}} (k) represented by

k ⟵^{θ(α_{1},...,α_{τ})} U(α_{1}, …, α_{τ}) ⟶^{f} k

for some α_{1}, …, α_{τ} ∈ H^{•}(G, k) and f : U(α) → k, and let E = E(α_{1}, …, α_{τ}; k, k) be the spectral sequence determined by α_{1}, …, α_{τ}. By Proposition 8.2 we know that E_{2}^{s,−s} = 0 except when s = 0 or s = τ − 1, and it follows that F^{1} End_{Q^{τ}} (k) = F^{τ−1} End_{Q^{τ}} (k) is an ideal of square zero so that F^{1} End_{Q^{τ}} (k) ⊆ 𝔯. Thus it suffices to show that

End_{Q^{τ}} (k)/F^{1} End_{Q^{τ}} (k) ≅ k.

Let (ζ_{1}, …, ζ_{τ}) be the composition

D_{0}(α) ⟶ U(α) ⟶^{f} k.

Then α_{i}ζ_{j} = α_{j}ζ_{i} for all i ≠ j, and α_{i}, α_{j} is a regular sequence because H^{∗}(G, k) is Cohen–Macaulay. Because α_{j}ζ_{i} ∈ (α_{i}), it follows that ζ_{i} ∈ (α_{i}). But deg α_{i} = deg ζ_{i}, so there is an element c_{i} ∈ k such that ζ_{i} = c_{i}α_{i}. Then c_{i}α_{i}α_{j} = ζ_{i}α_{j} = ζ_{j}α_{i} = c_{j}α_{i}α_{j}, and we conclude that c_{i} = c_{j} for all i and j. Let c = c_{1} = ⋯ = c_{τ}. Then f : U(α) → k is congruent to cθ(α) modulo F^{1}\underline{Hom}_{kG}(U(α), k), and γ ∈ End_{Q^{τ}} (k) is congruent to c modulo F^{1} End_{Q^{τ}} (k). This completes the proof. □

The following result is now an immediate consequence of Propositions 8.3 and 6.7.

Corollary 8.4. Suppose that H^{∗}(G, k) is Cohen–Macaulay. Then it is not possible to decompose the variety as V_{G}(k) = V_{1} ∪ V_{2}, where V_{1} and V_{2} are nonempty unions of components of V_{G}(k) such that dim(V_{1} ∩ V_{2}) ≤ dim V_{G}(k) − 2.

Something like the last result would seem to be true for any finitely generated commutative k-algebra or any commutative complete local k-algebra. The proof is implicit in Lemma 6.4, which depends on no more than the configuration of the components in the variety. Still, the result does not seem to have been noticed in the group cohomology literature.

Corollary 8.4 implies that if G = SL(3, p^{n}) and n > 1, then H^{∗}(G, k) cannot be Cohen– Macaulay. The reason is that a Sylow p-subgroup of SL(3, p^{n}) has two maximal elementary abelian p-subgroups E_{1} and E_{2}. These subgroups have order p^{2n}, intersect in a subgroup of order p^{n}, and are not conjugate. If V_{1} = res_{G,E_{1}}^{∗}(V_{E_{1}}(k)) and V_{2} = res_{G,E_{2}}^{∗}(V_{E_{2}}(k)), then the variety V_{G}(k) decomposes as a union of the 2n-dimensional components V_{1} and V_{2}, and dim(V_{1} ∩ V_{2}) = n.

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Department of Mathematics, University of Georgia, Athens, Georgia 30602 E-mail address: jfc@sloth.math.uga.edu

Department of Mathematics, University of Georgia, Athens, Georgia 30602 E-mail address: www@sloth.math.uga.edu