SPLENDID EQUIVALENCES: DERIVED CATEGORIES AND PERMUTATION MODULES

JEREMY RICKARD

Abstract. We introduce the notion of a splendid equivalence between blocks of finite groups, which is a strengthening of the notion of an equivalence of derived categories. Such an equivalence is always induced by a two-sided tilting complex whose terms are summands of permutation modules, a feature that is compatible with the belief that these complexes should be related to the l-adic cohomology of Deligne-Lusztig varieties in the case of reductive groups over finite fields. We show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence. In particular, it gives, for principal blocks, a structural explanation of Broué’s concept of an isotypy (a compatible family of perfect isometries). We give several examples, including examples of Morita equivalences that are splendid in an unexpectedly non-trivial way.

1. Introduction and preliminaries

1.1. Introduction. In [2], Broué makes some deep conjectures about equivalences of derived categories for block algebras of finite groups. For example, he conjectures that if a group G has an abelian Sylow p-subgroup P, then the principal blocks of G and N_{G}(P) have equivalent derived categories over a complete discrete valuation ring with residue field of characteristic p.

In that paper he presents a lot of evidence for these conjectures, and subsequent work by many people has provided still more evidence. This is mainly at the level of character theory, where the consequence of a derived equivalence is a character correspondence of a kind which Broué calls a perfect isometry. One striking feature of the known examples of perfect isometries is that they do not occur in isolation: in many examples where blocks of two groups are perfectly isometric, there are also perfect isometries between correponding blocks of local subgroups, and these isometries are compatible in a precise sense. Broué calls such a compatible family of perfect isometries an isotypy.

One feature that that was missing from the derived category picture was a corresponding definition of a compatible family of derived equivalences. We shall, for the case of principal blocks, provide the missing definition: this requires us to tighten the definition of a derived equivalence to what we call, for convoluted reasons that we shall explain later, a splendid equivalence.

Although finding the structural explanation of isotypies was the main motivation for the definition of a splendid equivalence, it turns out that the definition has other splendid consequences. One feature of a splendid equivalence is that it is given by tensoring with a complex of bimodules that are direct summands of permutation modules: in the light of [11], this is compatible with Broué’s belief, expressed first in [2], that in examples involving finite reductive groups this complex should be intimately related to the l-adic cohomology of Deligne-Lusztig varieties.

In Section 2 we give the basic definition and then in Section 3 we give an example involving the principal blocks of A_{5} and A_{4} in characteristic two.

In Section 4 we show how, for groups G and H with the same p-local structure (e.g., a group G with an abelian Sylow p-subgroup P and H = N_{G}(P)), a splendid equivalence between the principal blocks of G and H over a field of characteristic p gives rise in a natural way to splendid equivalences between the principal blocks of centralizers in G and in H of p-subgroups.

The techniques we use to produce these ‘local’ splendid equivalences do not work over a discrete valuation ring, since they involve the ‘Brauer construction’. However, in Section 5 we show that this doesn’t matter; this is because one side-effect of the definition of splendid equivalences is that they can automatically be lifted to a complete d.v.r. from its residue field. Section 6 then describes how a splendid equivalence gives rise to an isotypy.

Another nice feature of the definition is that it even throws new light on Morita equivalence (which is, of course, a simple form of derived equivalence). In Section 7 we show that new insight may be gained into one well-known example of Morita equivalence (arising in p-nilpotent groups) by replacing the bimodule M giving the equivalence by a complex of bimodules isomorphic (in the derived category) to M in order to get a splendid equivalence. This involves us in a study of endo-permutation modules, and we also show how many features of these seem to become clearer when we replace them with complexes.

We hope that Section 7 will convince the reader that replacing a perfectly good Morita equivalence by an equivalence induced by a complex is not necessarily just wilfully perverse. Any readers whom we fail to convince will be unimpressed by Section 8, where we do the same for an equivalence induced by a simple involution of a group algebra: we show that for a finite Coxeter group the self-equivalence of the module category given by tensoring with the sign representation may be realized as a splendid equivalence. In this example, the ‘splendid tilting complex’ arises geometrically, being constructed from the simplicial cochain complex of the Coxeter complex.

1.2. Blocks, symmetric algebras, module categories. We shall be interested in representations of finite groups over fields of prime characteristic and over discrete valuation rings whose residue field has prime characteristic. We shall fix the following notation: O will be a complete discrete valuation ring with maximal ideal M, such that the field of fractions K of O has characteristic zero and the residue field k of O has characteristic p > 0. We shall use R to denote a ring which may be either O or k; this will economize on notation when we are dealing with results that are true for both possible coefficient rings. Also, we shall not be interested in questions of rationality, so we shall always assume that these rings are big enough, in that K is a splitting field for all subgroups of the groups we consider.

Let G be a finite group. Since reduction modulo M gives a bijection between the central idempotents of OG and kG, there is a natural one-to-one correspondence between blocks of OG and of kG, or more generally between algebra summands of OG and of kG. We make the convention that where we denote a block (or sum of blocks) over O by, for example, OA, we shall denote the corresponding block over k by kA, and vice versa.

The modules for group algebras RG in which we shall be most interested in this paper are the p-permutation modules (see, for example, [1]). These are precisely the direct summands of permutation modules; as the name is designed to suggest, they may also be characterized as those modules that are permutation modules on restriction to every p-subgroup of G. Such modules are also known as trivial source modules.

Recall that a block RA of a group algebra RG is a symmetric R-algebra, meaning that RA, considered as a bimodule for itself, is isomorphic to its R-linear dual Hom_{R}(RA, R). This has the consequence that all obvious notions of duality are equivalent, the functors Hom_{R}(−, R) and Hom_{RA}(−, RA) from left RA-modules to right RA-modules being isomorphic. More generally, if M is an RA-RB-bimodule for two blocks (or sums of blocks) RA and RB, then all three RB-RA-bimodules Hom_{R}(M, R), Hom_{RA}(M, RA) and Hom_{RB}(M, RB) are isomorphic.

For any ring Λ, we shall denote by Λ-mod the category of finitely generated left Λmodules. When dealing with functors between full subcategories of module categories or other additive categories, we shall only be interested in additive functors, and so in such situations we shall omit the word ‘additive’.

1.3. Categories of complexes. Let us fix some notation concerning various categories of (co)chain complexes. First of all, it is not very important whether our complexes are chain complexes or cochain complexes, but for the sake of having a definite notational convention, we shall always use cochain complexes (and so the differentials will be degree 1 maps) unless we state otherwise. Since we shall often have to apply contravariant functors to cochain complexes, such as taking duals of complexes of vector spaces, and since this will turn cochain complexes into chain complexes, we shall have to renumber the terms in order to regain a cochain complex. Thus, for example, the dual C^{∨} of a complex C will have degree n term (C^{−n})^{∨}.

We shall usually denote a complex by a single letter such as C, in which case we shall implicitly be using the notation that the degree n term is C^{n} and the degree n differential is d^{n} : C^{n} ⟶ C^{n+1}, or d_{C}^{n} if we need to avoid ambiguity about which complex we mean.

If C is a complex and n is an integer, then C[n] is the complex obtained by ‘shifting C to the left by n places’: more formally, the degree i term of C[n] is C^{i+n} and the degree i differential is (−1)^{n}d_{C}^{i+n}.

We shall always be concerned with bounded complexes, i.e., complexes

… ⟶ X^{n} ⟶ X^{n+1} ⟶ …

such that X^{i} = 0 both for sufficiently large and for sufficiently small i. For any additive category A, then, we shall denote by C^{b}(A) the category of all bounded complexes over A, where the morphisms are just the chain maps. By K^{b}(A), we shall denote the homotopy category of complexes; i.e., the category with the same objects as C^{b}(A) but whose morphisms are chain homotopy classes of chain maps.

If A is an abelian category, then it makes sense to talk of the homology of a complex over A. By a quasi-isomorphism we mean a chain map that induces isomorphisms in homology in every degree. By D^{b}(A) we denote the derived category of A, which is obtained from K^{b}(A) by inverting all quasi-isomorphisms.

We shall regard objects of an additive category A as trivial examples of complexes over A, since to an object X we can associate the complex whose degree zero term is X and all of whose other terms are zero. In this way, A will be regarded as a full subcategory of C^{b}(A), of K^{b}(A), and, if A is an abelian category, of D^{b}(A).

If A is an additive or abelian category, and C and D are (bounded) complexes over A, then the notation Hom_{A}(C, D) has several possible meanings. We shall always use this notation to mean the complex of abelian groups obtained by taking the total complex of the double complex whose degree (i, j) term is Hom_{A}(C^{−i}, D^{j}). If we want to mention the group of homomorphisms between C and D in, for example, the category C^{b}(A), then we shall write Hom_{C^{b}(A)}(C, D). Recall that the degree n homology of Hom_{A}(C, D) is naturally isomorphic to Hom_{K^{b}(A)}(C, D[n]).

We shall have occasion to talk about split complexes, so let us make clear what we mean by this term.

Definition. If A is an additive category, then a complex C over A is split if it is isomorphic in K^{b}(A) to a complex with zero differentials. If A is an abelian category, then this is equivalent to demanding that C is the direct sum of its homology (considered as a complex with zero differential) and a contractible complex.

We end this section with a lemma that will be useful several times in what follows.

Lemma 1.1. (a) Let α : C ⟶ D be a chain map between two bounded complexes over an additive category A. Then α is a chain homotopy equivalence if and only if the chain map

Hom_{A}(X, C) ⟶ Hom_{A}(X, D)

induced by α is a quasi-isomorphism for every object X of A that is a term of either C or D.

(b) Let G be a finite group and let α : C ⟶ D be a chain map between two bounded complexes of p-permutation RG-modules. Then α is a chain homotopy equivalence if and only if the chain map

α^{H} : C^{H} ⟶ D^{H}

between the H-fixed point complexes is a quasi-isomorphism for every subgroup H of G. If H is a collection of subgroups of G such that every term of C and D is a direct sum of direct summands of permutation modules Ind_{H}^{G} R for H ∈ H, then it is only necessary that the maps induced on the H-fixed point complexes for those H that are in H should be quasi-isomorphisms.

Proof. (a) Since a chain map is a chain homotopy equivalence if and only if its mapping cone is contractible and is a quasi-isomorphism if and only if its mapping cone is acyclic, it is enough to show that a bounded complex E of objects of A is contractible if and only if Hom_{A}(X, E) is acyclic for every object X that is a term of E.

If E is contractible, then Hom_{A}(X, E) is contractible, and therefore acyclic, for any object X of A.

Without loss of generality we may and shall suppose that E has no direct summands that are chain homotopy equivalent to E and have fewer non-zero terms. If E is not contractible, then in particular E is non-zero. Let E_{s} be the last non-zero term of E. Since Hom_{A}(E^{s}, E) is acyclic, the map

Hom_{A}(E^{s}, E^{s−1}) ⟶ Hom_{A}(E^{s}, E^{s})

must be surjective. But this means that E^{s−1} ⟶ E^{s} is a split epimorphism, contradicting our assumption about E.

(b) This is just a special case of (a), since for a permutation module X = Ind_{H}^{G} R, the functor Hom_{RG}(X, −) is the same as the H-fixed point functor on RG-modules.

2. Splendid tilting complexes

2.1. Basic definitions. Let RA and RB be two block algebras. The results of [9] show that the derived categories D^{b}(RA-mod) and D^{b}(RB-mod) are equivalent as triangulated categories if and only if there is a bounded complex X of finitelygenerated RA-RB-bimodules and a bounded complex Y of finitely-generated RB-RA-bimodules such that

X ⊗^{L}_{RB} Y ≅ RA

in the derived category of RA-bimodules and

Y ⊗^{L}_{RA} X ≅ RB

in the derived category of RB-bimodules. If such complexes exist, then the functors

Y ⊗^{L}_{RA} − : D^{b}(RA-mod) ⟶ D^{b}(RB-mod)

and

X ⊗^{L}_{RB} − : D^{b}(RB-mod) ⟶ D^{b}(RA-mod)

are equivalences of derived categories. What is more, the complex Y is isomorphic to RHom_{RA}(X, RA) in the derived category of RB-RA-bimodules. The complexes X and Y are then called ‘two-sided tilting complexes’.

Since we are dealing with block algebras, the situation is a little simpler than it is with general rings. Firstly, since the block algebras RA and RB are free Rmodules, projective bimodules for these algebras are projective as left and right modules. Consequently, as in the proof of [9, Corollary 5.5], by taking a projective resolution

P = … ⟶ P^{n} ⟶^{d^{n}} P^{n+1} ⟶ …

of the two-sided tilting complex X and truncating it

… ⟶ 0 ⟶ ker(d^{n}) ⟶ P^{n} ⟶ P^{n+1} ⟶ …

for sufficiently small n we can produce a bounded complex isomorphic to X in the derived category of bimodules with the additional properties that all the terms of this complex are projective as left and as right modules and that all but one of the terms are actually projective as bimodules.

A second property of block algebras that simplifies the situation we have described is that, since RA is a symmetric algebra, the functors Hom_{RA}(−, RA) and Hom_{R}(−, R) are isomorphic, and so we may take Y to be the linear dual X^{∨} = Hom_{R}(X, R) of X.

If we make both of the simplifying assumptions above, then all the terms of both X and Y are projective (and hence flat) as left and as right modules, and so we have

RA ≅ X ⊗^{L}_{RB} Y ≅ X ⊗_{RB} Y ≅ Hom_{RB}(X, X)

and

RB ≅ Y ⊗^{L}_{RA} X ≅ Y ⊗_{RA} X ≅ Hom_{RA}(X, X)

in the appropriate derived categories of bimodules. Also, all but the degree zero terms of X ⊗_{RB} Y and Y ⊗_{RA} X are projective (and hence also injective) bimodules, and so these complexes are split.

To sum up, we have shown that a two-sided tilting complex for block algebras can always be chosen to satisfy the following definition.

Definition. Let Λ and Γ be symmetric R-algebras. A bounded complex X of finitelygenerated Λ-Γ-bimodules is said to be a split-endomorphism two-sided tilting complex if all the terms of X are projective as left and as right modules and

Hom_{Λ}(X, X) ≅ X^{∨} ⊗_{Λ} X ≅ Γ

in the homotopy category of complexes of Γ-bimodules and

Hom_{Γ}(X, X) ≅ X ⊗_{Γ} X^{∨} ≅ Λ

in the homotopy category of complexes of Λ-bimodules.

One important point to notice is that, whereas the property of being a two-sided tilting complex is preserved by changing to a quasi-isomorphic complex, this is not the case for split-endomorphism two-sided tilting complexes. Also, although the argument preceding the definition shows that we always have the option of forcing a two-sided tilting complex to have the split-endomorphism property by taking a complex all but one of whose terms are projective bimodules, there are complexes that have the property for less obvious reasons. We shall see examples later.

We hope in this paper to convince the reader that when looking for derived equivalences between block algebras, the two-sided tilting complex we should be seeking should not only have the split-endomorphism property, but should satisfy even stronger conditions. Let us now give the basic definition.

Definition. Let G and H be finite groups with a common Sylow p-subgroup P, and let RA and RB be block algebras of G and H respectively (or more generally any algebra summands of the group algebras RG and RH). A bounded complex X of finitely-generated RA-RB-bimodules is said to be a splendid tilting complex if:

(i) X is a split-endomorphism tilting complex

(ii) all the terms of X, considered as R[G × H]-modules, are direct summands of relatively ΔP -projective permutation modules (where ΔP is the diagonally embedded subgroup {(π, π) : π ∈ P } of G × H).

Remarks. (a) The name ‘splendid tilting complex’ is an abbreviation of ‘SPLit-ENDomorphism two-sided tilting complex of summands of permutation modules Induced from Diagonal subgroups’.

(b) When we say G and H have a ‘common’ Sylow p-subgroup P, we mean that there is a specified embedding of P into each of G and H. In fact, in all the examples we shall consider in this paper, one of the two groups is a subgroup of the other.

(c) When dealing with blocks whose defect group is smaller than a Sylow psubgroup, it is clear that we must adapt this definition. For example, there are many examples of naturally occurring derived equivalences between blocks of groups that do not have a common Sylow p-subgroup. However, the definition we have given will suffice for all the examples we consider in this paper, so we shall not tempt providence by guessing what the appropriate definition should be for more general blocks.

(d) The condition that the bimodules occurring in a splendid tilting complex should be direct summands of permutation modules is appealing because it suggests that the complexes may have a geometric origin. For example, the simplicial chain complex or cochain complex of a simplicial complex with a group action will automatically be chain homotopy equivalent to a complex of permutation modules: we shall see an example of a splendid tilting complex arising in this way in Section 8. Also, we showed in [11] that the l-adic cohomology with compact support of a quasi-projective variety with a finite group action is naturally realized by a bounded complex of lpermutation modules over Z_{l}. This suggests that the notion of a splendid tilting complex is compatible with Broué’s conjecture [2, Section 6] that derived equivalences in non-describing characteristic for finite reductive groups should be closely related to the l-adic cohomology of Deligne-Lusztig varieties.

It now seems natural to make the following definitions:

Definition. An equivalence of derived categories given by tensoring with a splendid tilting complex is called a splendid equivalence. If there is a splendid equivalence between the derived categories of two block algebras (or more general summands of group algebras), the algebras are said to be splendidly equivalent.

2.2. Simplifying calculations. In order to simplify the calculations we have to carry out when we look at a specific example in the next section, let us point out that a judicious use of abstract nonsense halves the amount of work we have to do. In the following theorem, it is important that we consider single blocks and not sums of blocks, as otherwise it is easy to find examples where the theorem is false.

Theorem 2.1. Let kA and kB be block algebras of groups G and H with a common Sylow p-subgroup P, and let X be a bounded complex of finitely-generated kA-kB-bimodules that are relatively ΔP -projective p-permutation k[G × H]-modules. Let X^{∨} be the dual complex. If X ⊗_{kB} X^{∨} is split with homology concentrated in degree zero and isomorphic to kA as a kA-bimodule, then X is a splendid tilting complex.

Proof. Since block algebras are symmetric, we have

X^{∨} ≅ Hom_{kA}(X, kA) ≅ Hom_{kB}(X, kB),

and so the functors

F = X ⊗_{kB} − : K^{b}(kB-mod) ⟶ K^{b}(kA-mod)

and

F^{∨} = X^{∨} ⊗_{kA} − : K^{b}(kA-mod) ⟶ K^{b}(kB-mod)

are both left and right adjoint to one another.

Consider the natural maps

kB ⟶ X^{∨} ⊗_{kA} X ⟶ kB, (*)

or, equivalently, the unit and counit

id ⟶ F^{∨}F ⟶ id

of the adjunctions (F, F^{∨}) and (F^{∨}, F). Composing with F, or, in terms of bimodules, tensoring with X, the maps

X ⟶ X ⊗_{kB} X^{∨} ⊗_{kA} X ⟶ X

are respectively a split injection and a split surjection, by general properties of adjoint functors [7, Theorem IV.1.1]. Since

X ⊗_{kB} X^{∨} ⊗_{kA} X ≅ kA ⊗_{kA} X ≅ X,

the Krull-Schmidt theorem implies that this split injection and split surjection must both be isomorphisms.

Since kB is a block, and so is indecomposable as a kB-kB-bimodule, if the composition of maps (*) is not an isomorphism it must be nilpotent, in which case it could not give an isomorphism upon tensoring with X. Hence kB must be a bimodule summand of X^{∨} ⊗_{kA} X. Since

X^{∨} ⊗_{kA} X ⊗_{kB} X^{∨} ⊗_{kA} X ≅ X^{∨} ⊗_{kA} kA ⊗_{kA} X ≅ X^{∨} ⊗_{kA} X,

it follows from the Krull-Schmidt theorem that X^{∨} ⊗_{kA} X is actually isomorphic to kB.

3. An example: the principal blocks of A_{4} and A_{5}

Let p = 2, and we shall assume that k is a splitting field for both of the alternating groups A_{4} and A_{5}. Let kB be the principal block algebra of kA_{5}. We shall show that kB and the group algebra kA_{4} are splendidly equivalent, but let us first recall what we need to know about the representation theory of kA_{4} and kA_{5}.

The group algebra kA_{4} has a single block with three simple modules: the trivial module k, and two other one-dimensional modules S_{1} and S_{2}. The indecomposable projective modules have Loewy structures

P_{k} = \begin{aligned}& k \\ S_{1} & & S_{2} \\ & k \end{aligned} , P_{1} = \begin{aligned}& S_{1} \\ S_{2} & & k \\ & S_{1} \end{aligned} , P_{2} = \begin{aligned}& S_{2} \\ k & & S_{1} \\ & S_{2} \end{aligned} .

The principal block of kA_{5} also has three simple modules: the trivial module k and two two-dimensional modules T_{1} and T_{2}; there is also a block of defect zero containing a four-dimensional simple projective module V. The indecomposable projective modules have Loewy structures

Q_{k} = \begin{aligned}& k \\ T_{1} & & T_{2} \\ k & & k \\ T_{2} & & T_{1} \\ & k \end{aligned} , Q_{1} = \begin{matrix}T_{1} \\ k \\ T_{2} \\ k \\ T_{1} \end{matrix} , Q_{2} = \begin{matrix}T_{2} \\ k \\ T_{1} \\ k \\ T_{2} \end{matrix} , Q_{V} = V.

If we identify A_{4} with the normalizer of a Sylow 2-subgroup of A_{5}, then the restrictions of T_{1} and T_{2} to A_{4} are non-split extensions of S_{1} by S_{2} and vice versa; we shall choose our notation so that S_{i} is a quotient of Res_{A_{4}}^{A_{5}}(T_{i}) for i ∈ {1, 2}. The restriction Res_{A_{4}}^{A_{5}}(V) of the projective simple kA_{5}-module is isomorphic to P_{1}.

We shall define a complex X of kB-kA_{4}-bimodules as follows. Only the degree −1 and 0 terms of X are non-zero, and

X = … ⟶ 0 ⟶ X^{−1} ⟶^{d^{−1}} X^{0} ⟶ 0 ⟶ …,

where X^{0} is kB, considered as a kB-kA_{4}-bimodule, X^{−1} is the projective cover (as a bimodule) of the augmentation ideal of kB, and the map d^{−1} is the natural map onto the augmentation ideal. The first thing we shall do is to calculate how X^{−1} decomposes as a direct sum of indecomposable projective bimodules.

Lemma 3.1. Considered as a k[A_{5} × A_{4}]-module,

X^{−1} ≅ (Q_{1} ⊗ P_{1}) ⊕ (Q_{2} ⊗ P_{2}).

Proof. We just need to show that the top of the augmentation ideal of kB is isomorphic to (T_{1} ⊗ S_{1}) ⊕ (T_{2} ⊗ S_{2}).

First, we shall calculate the top of kA_{5}. As a k[A_{5} ×A_{4}]-module, kA_{5} is isomorphic to the permutation module Ind_{ΔA_{4}}^{A_{5}×A_{4}}(k), where ΔA_{4} is the diagonally embedded copy of A_{4}. Thus, by Frobenius reciprocity, the multiplicity of a simple module T ⊗ S in the top of kA_{5} is

dim_{k} Hom_{k[A_{5}×A_{4}]}(kA_{5}, T ⊗ S) = dim_{k} Hom_{kA_{4}}(k, Res_{A_{4}}^{A_{5}}(T) ⊗ S).

This multiplicity is zero except for T ⊗ S isomorphic to k ⊗ k, T_{1} ⊗ S_{1}, T_{2} ⊗ S_{2} or V ⊗ k, for which it is one.

Taking the part which lies in the principal block of kA_{5}, we see that the top of kB is isomorphic to

(k ⊗ k) ⊕ (T_{1} ⊗ S_{1}) ⊕ (T_{2} ⊗ S_{2}),

and so the top of the augmentation ideal of kB must contain (T_{1} ⊗ S_{1}) ⊕ (T_{2} ⊗ S_{2}) as a direct summand. If it were any larger than this, then kA_{5} would have a quotient that was a non-split extension E of k ⊗ k by some other simple k[A_{5} × A_{4}]-module T ⊗ S. Such extensions E only exist for T ⊗ S isomorphic to k ⊗ S_{i} or T_{i} ⊗ k. But, again by Frobenius reciprocity,

Hom_{k[A_{5}×A_{4}]}(kA_{5}, E) ≅ Hom_{kA_{4}}(k, Res_{ΔA_{4}}^{A_{5}×A_{4}}(E)),

which is zero, since for none of the four possibilities for E does Res_{ΔA_{4}}^{A_{5}×A_{4}}(E) contain a trivial submodule.

Proposition 3.2. The complex X is a splendid tilting complex.

Proof. In order to verify the splendour of X it is sufficient, by Theorem 2.1, to show that Hom_{kA_{5}}(X, X) is split with homology isomorphic to kA_{4} concentrated in degree zero. Since only the degree zero term of X is a non-projective bimodule, it is automatic that, if the homology of Hom_{kA_{5}}(X, X) is concentrated in degree zero, then it is split.

As a complex of kA_{5}-modules, X is a projective presentation

Y = … ⟶ 0 ⟶ Y^{−1} ⟶^{d^{−1}} Y^{0} ⟶ 0 ⟶ …

of the trivial module such that, by Lemma 3.1, Y^{−1} does not contain Q_{1} as a summand. Hence there are no non-zero maps k ⟶ Y^{−1}, and so no chain maps Y ⟶ Y [−1]. Every map Y^{−1} ⟶ Y^{0} has image contained in im(d^{−1}), and so factors through d^{−1}. Hence every chain map Y ⟶ Y [1] is homotopic to zero. This proves that the homology of Hom_{kA_{5}}(X, X) is concentrated in degree zero.

Using the Mackey decomposition formula, we see that as kA_{4}-bimodules,

Hom_{kA_{5}}(kA_{5}, kA_{5}) ≅ kA_{5}
≅ Res_{A_{4}×A_{4}}^{A_{5}×A_{5}} Ind_{ΔA_{5}}^{A_{5}×A_{5}}(k)
≅ Ind_{ΔA_{4}}^{A_{4}×A_{4}}(k) ⊕ Z
≅ kA_{4} ⊕ Z,

for some projective bimodule Z. Thus every term of Hom_{kA_{5}}(X, X) is a projective bimodule except for a summand isomorphic to kA_{4} in degree zero. The degree zero homology of Hom_{kA_{5}}(X, X), since this is a split complex, must therefore be isomorphic to the direct sum of kA_{4} and a projective bimodule. To prove that this projective bimodule is in fact zero, it suffices to show that the dimension of the homology is equal to 12, the dimension of kA_{4}.

To calculate this dimension, it is enough to know the structure of X as a complex Y of kA_{5} modules as we did above. We have

Y^{0} ≅ kB ≅ Q_{k} ⊕ Q_{1}^{2} ⊕ Q_{2}^{2}

and, by Lemma 3.1,

Y^{−1} ≅ Q_{1}^{4} ⊕ Q_{2}^{4}.

We can now calculate the dimension of the degree zero homology of Hom_{kA_{5}}(X, X) as the Euler characteristic

− dim_{k} Hom_{kA_{5}}(Y^{0}, Y^{−1}) + dim_{k}(Hom_{kA_{5}}(Y^{0}, Y^{0}) ⊕ Hom_{kA_{5}}(Y^{−1}, Y^{−1}))

− dim_{k} Hom_{kA_{5}}(Y^{−1}, Y^{0}),

which, by the structure of projective kA_{5}-modules, is indeed equal to 12.

Remarks. (i) It will follow from the results of Section 5 that the principal blocks of A_{5} and A_{4} are also splendidly equivalent over O.

(ii) A different method of proof, which is also used for other examples, has been found by Rouquier [12].

4. Local splendid equivalences

In [2], Broué introduced the notion of a ‘perfect isometry’ between the groups of characters of two blocks and showed that a derived equivalence (over O) between blocks induced a perfect isometry. In the same paper he gave a lot of evidence to show that such perfect isometries are very common between blocks with the same ‘local structure’ and that when such an isometry occurs one can also find perfect isometries at the ‘local level’ (between corresponding blocks of centralizers of subgroups of the defect group) that are ‘compatible’ in a precise sense. He was thus led to the definition of an ‘isotypy’ between blocks: a compatible family of perfect isometries. We refer the reader to Broué’s paper for the technical definitions in the case of general blocks, as here we shall only be concerned with principal blocks, where the technicalities of block theory do not appear.

If one believes, as I do, that naturally occurring perfect isometries between blocks are merely the shadows, at the level of characters, of the deeper, structural, phenomena of derived equivalences, then it is very natural to ask what structural phenomenon corresponds to an isotypy: what is the correct definition of a ‘compatible’ family of derived equivalences?

Here we shall answer this question for principal blocks. We shall show that, for principal blocks (with the same local structure), a splendid equivalence automatically gives rise to splendid equivalences at the local level. Initially we shall work over a field k, but the results of Section 5 will imply that everything lifts canonically to O. The local splendid equivalences will be produced using the Brauer construction, so let us first remind the reader of this.

If G is a finite group and Q is a p-subgroup of G, then there is a functor, the ‘Brauer construction with respect to Q’:

—(Q) : kG-mod ⟶ k[N_{G}(P)]-mod

defined by

M(Q) = M^{Q}/( ∑_{R<Q} Tr_{R}^{Q}(Q^{R})),

the quotient of the Q-fixed points of M by the relative traces from all proper subgroups R of Q of the R-fixed points. If M is a k-algebra and the action of G is by algebra automorphisms, then M(Q) is also an algebra, with N_{G}(Q) acting by algebra automorphisms.

If M is a permutation module with a G-stable basis Ω, then M(Q) is naturally isomorphic to the permutation module k[Ω^{Q}] for k[N_{G}(Q)]. In fact, although the definition of M(Q) makes sense for any kG-module M, we shall only be interested in applying the functor to p-permutation modules. One important fact that we shall use is that, for kG-modules M_{1} and M_{2}, there is a natural k[N_{G}(Q)]-module homomorphism

M_{1}(Q) ⊗_{k} M_{2}(Q) ⟶ (M_{1} ⊗_{k} M_{2})(Q)

which, if M_{1} and M_{2} are p-permutation modules, is an isomorphism.

Our results will apply slightly more generally than to principal blocks, but we shall need the following global restriction on the groups involved.

Definition. Finite groups G and H have the same p-local structure if they have a common Sylow p-subgroup P such that whenever Q_{1} and Q_{2} are subgroups of P and θ : Q_{1} ⟶ Q_{2} is an isomorphism, then there is an element g ∈ G such that

θ(q) = q^{g} for all q ∈ Q_{1}

if and only if there is an element h ∈ H such that

θ(q) = q^{h} for all q ∈ Q_{1}.

Notice that if H is a subgroup of G, then H and G have the same local structure precisely when H controls fusion of p-subgroups in G: i.e., when H contains a Sylow p-subgroup P of G and for every subgroup Q < P and element g ∈ G such that Q^{g} < P there is an element c ∈ C_{G}(Q) such that cg ∈ H. This is always true, for example, when P is abelian and H contains N_{G}(P).

We shall now state precisely the theorem we are going to prove, but let us first point out that if Q is a p-subgroup of a finite group G, then, considering the group algebra kG as a k[G × G]-module, kG(ΔQ) is naturally isomorphic, as an algebra, to the group algebra kC_{G}(Q). Thus, more generally, if kA is an algebra summand of kG, then kA(ΔQ) is an algebra summand of kC_{G}(Q): if kA is the principal block of kG, then kA(ΔQ) is, in fact, the principal block of kC_{G}(Q).

Theorem 4.1. Let G and H be finite groups with a common Sylow p-subgroup P and the same p-local structure. If kA and kB are algebra summands of kG and kH respectively, and if X is a splendid tilting complex of kA-kB-bimodules, then for every subgroup Q ≤ P, X(ΔQ) is a splendid tilting complex of kA(ΔQ)-kB(ΔQ)- bimodules.

Remarks. (a) Since the normalizer of ΔQ in G×H contains C_{G}(Q)×C_{H}(Q), X(ΔQ) may be regarded as a complex of kC_{G}(Q)-kC_{H}(Q)-bimodules. It is not hard to see that, regarded like this, it is in fact a complex of kA(ΔQ)-kB(ΔQ)-bimodules.

(b) As particular cases of the theorem, we can take kA and kB to be the whole of the group algebras kG and kH, in which case kA(ΔQ) and kB(ΔQ) are the whole of the group algebras kC_{G}(Q) and kC_{H}(Q), or we can take kA and kB to be the principal blocks of kG and kH, in which case kA(ΔQ) and kB(ΔQ) are the principal blocks of kC_{G}(Q) and kC_{H}(Q).

Before we prove Theorem 4.1, we need a couple of lemmas. I would like to thank Lluis Puig for a suggestion that simplified the proof of the first of these.

Lemma 4.2. Let G be a finite group with a p-subgroup Q. The functors

F : M ⟼ (M^{G})(Q)

and

F′ : M ⟼ (M(ΔQ))^{C_{G}(Q)}

are isomorphic as functors from the category of relatively ΔQ-projective p-permutation k[G × Q]-modules to the category of k-vector spaces.

Proof. Note that

F (M) = M^{G×Q}/( ∑_{R<Q} Tr_{G×R}^{G×Q}(M^{G×R}))

and that

F′(M) = M^{C_{G}(Q)ΔQ}/( ∑_{R<Q} Tr_{C_{G}(Q)ΔR}^{C_{G}(Q)ΔQ})).

For each R < Q, the (G × R)-fixed points of M are contained in the C_{G}(Q)ΔRfixed points, and since a transversal to ΔR in ΔQ is also both a transversal to G × R in G×Q and a transversal to C_{G}(Q)ΔR in C_{G}(Q)ΔQ, these inclusions of fixed points induce a natural map from F (M) to F′(M).

To prove that this natural map is an isomorphism, it is enough to consider the case M_{R} = Ind_{ΔR}^{G×Q}(k) for subgroups R ≤ Q, since every relatively ΔQ-projective p-permutation module is a direct summand of a direct sum of modules of this form.

If R is a proper subgroup of Q, then it easy to calculate that F (M_{R}) and F′(M_{R}) are both zero, and so the map must be an isomorphism. If R = Q, then F (M_{R}) and F′(M_{R}) are both one-dimensional spaces, and it is easy to check that the natural map between them is non-zero, and therefore an isomorphism.

Lemma 4.3. Let G and H be finite groups with a common Sylow p-subgroup P and the same p-local structure, and let X be a relatively ΔP -projective k[G × H]-module. Then, for any subgroup Q ≤ P, X is relatively ΔQ-projective as a k[G × Q]-module.

Proof. It suffices to prove this for X of the form Ind_{ΔP}^{G×H}(Y) for some k[ΔP]-module Y. Then, since there is a set of representatives for the double cosets ΔP \G×H/G×Q consisting of elements of {1} × H, the restriction of X to G × Q is, by Mackey’s decomposition theorem, isomorphic to a direct sum of modules induced from subgroups of the form

(ΔP)^{(1,h)} ∩ G × Q = {(π, π^{h}) : π ∈ P ∩ Q^{h^{−1}}}, (*)

where h ∈ H. Since conjugation by h induces an isomorphism between the subgroups P ∩Q^{h^{−1}} and P^{h}∩Q of P, and since G and H have the same local structure, the same isomorphism is given by conjugation by some element γ ∈ G. Hence, the subgroup (*) of G × Q is conjugate, via (γ, 1), to Δ(P^{h} ∩ Q), a subgroup of ΔQ. Hence X is relatively ΔQ-projective as a k[G × Q]-module.

Proof of Theorem 4.1. We want to prove that the complex

Hom_{kC_{G}(Q)}(X(ΔQ), X(ΔQ))

is split as a complex of kC_{H}(Q)-bimodules, with homology isomorphic to kB(ΔQ) concentrated in degree zero.

As a complex of vector spaces, this may be obtained from Hom_{k}(X, X), regarded as a complex of k[G × H]-modules via the diagonal action, by restricting to G × Q and applying the functor

—(ΔQ)^{C_{G}(Q)} : k[G × Q]-mod ⟶ k-mod. (*)

Since the action of C_{H}(Q)×C_{H}(Q) commutes with the action of G×Q, the k[C_{H}(Q)]- bimodule structure can then be recovered by naturality.

Since every term of X is relatively ΔP -projective, the same is true of every term of Hom_{k}(X, X). By Lemma 4.3, it follows that the restriction of Hom_{k}(X, X) to G × Q is a complex of relatively ΔQ-projective p-permutation modules. By Lemma 4.2, the functor (*) is isomorphic to (-^{G})(Q) on such modules; hence the complex

Hom_{kC_{G}(Q)}(X(ΔQ), X(ΔQ))

is isomorphic (even, by naturality, as a complex of kC_{H}(Q)-bimodules) to

(Hom_{kG}(X, X))(ΔQ),

which, since X is a splendid tilting complex, is split with homology kB(ΔQ) concentrated in degree zero.

Similarly,

Hom_{kC_{H}(Q)}(X(ΔQ), X(ΔQ))

is split with homology kA(ΔQ) concentrated in degree zero, and so X(ΔQ) is a split-endomorphism two-sided tilting complex.

To show that X(ΔQ) is in fact splendid, let us first show that C_{G}(Q) and C_{H}(Q) have a common Sylow p-subgroup. Among all conjugates of Q that are contained in P (since G and H have the same p-local structure, it makes no difference whether we conjugate by elements of G or H) choose one, Q_{0} say, such that C_{P} (Q_{0}) is as large as possible. I claim that C_{P} (Q_{0}) is a Sylow p-subgroup of C_{G}(Q_{0}) and of C_{H}(Q_{0}). To see this, let P′ be a Sylow p-subgroup of Q_{0}.C_{G}(Q_{0}); then P′ is conjugate, by an element g ∈ G, to a subgroup of P. Therefore P contains a Sylow p-subgroup of C_{G}(Q_{0}^{g}) for some conjugate Q_{0}^{g} of Q, and by our choice of Q_{0} it must contain a Sylow p-subgroup of C_{G}(Q_{0}). The same argument applies with C_{H}(Q_{0}) in place of C_{G}(Q_{0}), and so C_{P} (Q_{0}) is a Sylow p-subgroup both of C_{G}(Q_{0}) and of C_{H}(Q_{0}).

These remarks show that we may as well assume that C_{P} (Q) is a Sylow p-subgroup of C_{G}(Q) and C_{H}(Q). Now if Y is the permutation k[G×H]-module with basis given by the cosets of ΔR for some subgroup R ≤ P, then the stabilizer of any basis element is a conjugate ΔR^{(g,h)} of ΔR. Hence the stabilizer in C_{G}(Q) × C_{H}(Q) of any element of the obvious basis of Y (ΔQ) is of the form

ΔS^{(g,h)} = ΔR^{(g,h)} ∩ (C_{G}(Q) × C_{H}(Q)),

where S ≤ R and ΔQ ≤ ΔR^{(g,h)}. Taking a conjugate in C_{G}(Q) × C_{H}(Q), we may as well assume that

ΔS^{(g,h)} ≤ C_{P} (Q) × C_{P} (Q).

Since ΔQ ≤ ΔR^{(g,h)}, Q^{g^{−1}} = Q^{h^{−1}} and conjugation by g agrees with conjugation by h on Q^{g^{−1}}. But also, since G and H have the same p-local structure, there is an element γ of G such that conjugation by γ agrees with conjugation by h on S.Q^{g^{−1}} = S.Q^{h^{−1}}. Hence g^{−1}γ ∈ C_{G}(Q). Conjugating ΔS^{(g,h)} by (g^{−1}γ, 1), we see that Y (ΔQ) has a basis every element of which has a subgroup of Δ(C_{P} (Q)) as its stabilizer. It follows that for any relatively ΔP -projective p-permutation k[G × H]- module W, W (ΔQ) is a relatively Δ(C_{P} (Q))-projective p-permutation module for k[C_{G}(Q) × C_{H}(Q)].

Thus X(ΔQ) is splendid.

5. The lifting of splendid tilting complexes

If X is a splendid tilting complex for two block algebras OA and OB, then it is a triviality to show that X ⊗_{O} k is a splendid tilting complex for the corresponding blocks, kA and kB, over the residue field k. Hence, if blocks are splendidly equivalent over O, they are also splendidly equivalent over k. It is rather less obvious that the converse holds, but that is what we shall show in this section.

The methods that we shall use to prove this are very similar to those that we used in [10], although for the reader’s convenience we shall give a complete proof here. In Theorem 3.3 of that paper we showed that if block algebras kA and kB are derived equivalent, then OA is derived equivalent to some O-algebra Λ such that Λ ⊗_{O} k is isomorphic to kB, bu, as we remarked in [10], we need extra information in order to deduce that this O-algebra is actually isomorphic to OB. If the two-sided tilting complex giving the equivalence of derived categories over k can be chosen to be splendid, then this gives the extra information we need.

Lemma 5.1. Let S be a finite group, and let C be a complex of finitely generated p-permutation kS-modules. If the complex Hom_{kS}(C, C) has homology concentrated in degree zero, then there is a complex C̃ of p-permutation OS-modules, unique up to isomorphism, such that C is isomorphic to C̃ ⊗_{O} k.

Proof. Since the degree n homology of Hom_{kS}(C, C) is isomorphic to the space of chain homotopy classes of chain maps from C to C[n], it follows from the condition on C that every chain map C ⟶ C[n] is homotopic to zero if n ≠ 0.

By an argument of Scott [13], it is known that p-permutation modules and maps between them can be lifted from k to O. Hence, if C has differential d, it is at least possible to lift C to a graded p-permutation OS-module C̃ together with a degree 1 endomorphism d̃_{1}. However, this endomorphism need not satisy d̃_{1}^{2} = 0, and so (C̃, d̃_{1}) need not be a complex. Since d^{2} = 0 and d̃_{1} is a lifting of d, we do at least have d̃_{1}^{2} = 0 (mod M), where M is the maximal ideal of O. We shall show inductively that we can produce a sequence d̃_{i} of degree 1 endomorphisms of C̃ such that d̃_{i+1} = d̃_{i} (mod M^{i}) and d̃_{i}^{2} = 0 (mod M^{i}). Hence, since O is complete, we can take the limit d̃ of the d̃_{i} so that (C̃, d̃) is a complex lifting (C, d).

Suppose, then, that we have produced d̃_{1},…, d̃_{i} with the properties described. Let π be a generator of the ideal M, so

d̃_{i}^{2} = π^{i}α

for some degree 2 graded endomorphism of C̃. Hence

d̃_{i}(π^{i}α) = (π^{i}α)d̃_{i},

and so d̃_{i}α = αd̃_{i}. Reducing modulo M, we see that the reduction of α modulo M commutes with d, and so is a chain map C ⟶ C[2]. We have seen that the conditions we imposed on C imply that all such chain maps are homotopic to zero, and so if we take a chain homotopy and lift it to a degree 1 graded endomorphism h of C̃, we have

α = d̃_{i}h + hd̃_{i} (mod M).

If we now define

d̃_{i+1} = d̃_{i} − π^{i}h,

then

d̃_{i+1}^{2} = d̃_{i}^{2} − π^{i}(d̃_{i}h + hd̃_{i}) + π^{2i}h^{2},

which is zero modulo M^{i+1}.

To prove uniqueness of C̃, suppose that C̃′ is another lifting of C to a complex of p-permutation modules. An isomorphism

C̃ ⊗_{O} k ⟶ C̃′ ⊗_{O} k

can certainly be lifted to an isomorphism of graded modules

β_{1} : C̃ ⟶ C̃′,

and β_{1}d = dβ_{1} (mod M). Suppose that there is a sequence of isomorphisms β_{1},…, β_{k} of graded modules such that β_{i} and β_{i+1} agree modulo M^{i} and β_{i}d = dβ_{i} (mod M^{i}). Then

β_{k}d − dβ_{k} = π^{k}ϕ

for some degree 1 graded homomorphism ϕ. Since

π^{k}(dϕ − ϕd) = 0,

ϕ is a map of complexes C̃ ⟶ C̃′[1]. Therefore, since every map of complexes C ⟶ C[1] is homotopic to zero, there is a map of graded modules h : C̃ ⟶ C̃′ such that

ϕ = hd − dh (mod M).

If we now define β_{k+1} to be β_{k} − π^{k}h, then

β_{k+1}d − dβ_{k+1} = 0 (mod M^{k+1}).

Taking the limit of the β_{k}, we get an isomorphism of complexes β : C̃ ⟶ C̃′.

We shall now apply this to the case of a splendid tilting complex.

Theorem 5.2. Let kA and kB be algebra summands of finite group algebras kG and kH respectively, and let X be a splendid tilting complex for kA and kB. Then there is a splendid tilting complex X̃ for OA and OB with X̃ ⊗_{O} k ≅ X. Moreover, X̃ is unique up to isomorphism.

Proof. Let S = G×H, and regard X as a complex of kS-modules. Then the complex Hom_{kS}(X, X) is obtained from the complex Hom_{kA}(X, X) by taking H-fixed points. But since this latter complex is split as a complex of kB-bimodules with homology concentrated in degree zero, the fixed point complex also has homology concentrated in degree zero.

It now follows from Lemma 5.1 that we can lift X to a complex X̃ of p-permutation OA-OB-bimodules.

The natural map of complexes

ϵ : kB ⟶ Hom_{kA}(X, X)

is a chain homotopy equivalence, so the map

ϵ^{K} : (kB)^{K} ⟶ Hom_{kA}(X, X)^{K}

induced on the fixed point complexes for any subgroup K of H × H is a quasiisomorphism.

Since taking fixed points commutes with reduction mod M for p-permutation modules over O, ϵ^{K} is the same as the map obtained by reduction mod M from the natural map

ϵ̃^{K} : (OB)^{K} ⟶ Hom_{OA}(X̃, X̃)^{K}.

By the Universal Coefficient Theorem, any map between complexes of finitely generated free O-modules that becomes a quasi-isomorphism upon reduction mod M is itself a quasi-isomorphism.

Therefore, by Lemma 1.1, the natural chain map

OB ⟶ Hom_{OA}(X̃, X̃)

is a chain homotopy equivalence.

Similarly, Hom_{OB}(X̃, X̃) is split with homology concentrated in degree zero and isomorphic to OA, and so X̃ is a splendid tilting complex. The claim about uniqueness follows from the corresponding statement in Lemma 5.1.

6. Splendid equivalences and isotypies

As we have already remarked, the local splendid equivalences of Section 4 are the analogue at the level of derived categories of the local perfect isometries involved in Broué’s notion of an isotypy. For the sake of completeness we include in this short section a more precise description of the connection. However, we should point out that these results appear, with a sketch of the proof, in Broué’s survey [3], and so there is little new in this section.

Since we shall be dealing with characters, we shall have to consider derived equivalences over O rather than k, so let us start by stating the O-analogue of Theorem 4.1, whose proof follows immediately from Theorems 4.1 and 5.2.

Theorem 6.1. Let G and H be finite groups with a common Sylow p-subgroup P and the same p-local structure. Let OA and OB be algebra summands of OG and OH respectively, and for each subgroup Q ≤ P let OA_{Q} and OB_{Q} be the algebra summands of OC_{G}(Q) and OC_{H}(Q) which lift kA(ΔQ) and kB(ΔQ) respectively. If X is a splendid tilting complex of OA-OB-bimodules, then for every subgroup Q ≤ P there is a splendid tilting complex X_{Q} of OA_{Q}-OB_{Q}-bimodules, unique up to isomorphism, such that X_{Q} ⊗_{O} k ≅ (X ⊗_{O} k)(ΔQ).

Let us now fix some notation.

Given a finite group G, we shall denote by CF (G, K) the space of class functions on G taking values in K, the field of fractions of O, and we shall denote by CF_{princ}(G, K) the subspace spanned by the characters belonging to the principal block. Given two finite groups G and H, giving a linear map

I : CF_{princ}(G, K) ⟶ CF_{princ}(H, K)

is equivalent to giving a class function μ_{I} ∈ CF_{princ}(G × H, K), where

μ_{I} = ∑_{ζ∈Irr_{princ}(G)} ζ ⊗ I(ζ),

Irr_{princ}(G) being the set of irreducible characters of G belonging to the principal block.

We can now recall the definition [2, 3] of an isotypy, at least for principal blocks of groups with the same local structure; for the more general definition, we refer the reader to Broué’s paper [2].

Definition. Let G and H be finite groups with a common Sylow p-subgroup P and the same p-local structure. An isotypy between the principal blocks of OG and OH is a collection of perfect isometries

I(Q) : CF_{princ}(C_{G}(Q), K) ⟶ CF_{princ}(C_{H}(Q), K),

one for each subgroup Q ≤ P, with the following compatibility condition. Whenever Q ≤ P, x, y ∈ C_{P} (Q), x′ ∈ C_{G}(x) and y′ ∈ C_{H}(y), where x′ and y′ are both p′- elements,

μ_{I(Q)}(xx′, yy′) = \begin{cases}0 & if x and y are not conjugate in G, \\ μ_{I(Q⟨x⟩)}(x′, y′) & if x = y. \end{cases}

The following well-known lemma describes the effect of the Brauer construction on characters of p-permutation modules.

Lemma 6.2. Let Γ be a finite group, let γ and γ′ be commuting elements of Γ with γ a p-element and γ′ a p′-element, and let X be a p-permutation OΓ-module. Let X_{γ} be the p-permutation OC_{Γ}(γ)-module such that X_{γ} ⊗_{O} k is isomorphic to the module obtained by applying the Brauer construction with respect to the cyclic subgroup ⟨γ⟩ to X ⊗_{O} k, and let χ and χ_{γ} be the characters of X and X_{γ} respectively. Then

χ(γγ′) = χ_{γ}(γ′).

Proof. The statement of the lemma only involves the structure of X as a module for the cyclic group

⟨γγ′⟩ = ⟨γ⟩ × ⟨γ′⟩,

and since every p-permutation module for this group is a direct sum of modules of the form Y ⊗ V, where Y is an indecomposable permutation O⟨γ⟩-module and V is an O⟨γ′⟩-module, we shall assume, with no loss of generality, that Γ is generated by γγ′ and that X = Y ⊗ V. But in this case,

X_{γ} ≅ \begin{cases}V & if Y is the trivial module O, \\ 0 & otherwise. \end{cases}

The statement on characters follows since the character of Y vanishes on γ unless Y is the trivial module.

The following theorem, giving the connection between splendid equivalences and isotypies for principal blocks, follows easily from Theorem 4.1 and Lemma 6.2, using the fact [2] that a derived equivalence over O induces a perfect isometry on the level of characters. The theorem is not stated in the most general form possible: we leave to the reader the formulation for those non-principal blocks covered by Theorem 6.1.

Theorem 6.3. Let G and H be finite groups with a common Sylow p-subgroup P and the same local structure. If the principal blocks of OG and OH are splendidly equivalent, then there is an isotypy between these blocks.

7. Endo-permutation Modules and p-nilpotent Groups

7.1. Endo-permutation modules and complexes of permutation modules. Let P be a finite p-group, and let us recall the definition [4] of an endo-permutation RP -module.

Definition. An endo-permutation RP -module is a finitely-generated RP -module M such that End_{R}(M), regarded as a RP -module via the diagonal action of P, is a permutation module.

We are going to study the relationship between endo-permutation modules and modules that can be realized as the homology of a particular sort of complex of permutation modules, according to the following definition, which is not dissimilar to the definition of a splendid tilting complex. In fact, we shall give the definition for arbitrary finite groups, where it is more natural to allow p-permutation modules in the complex.

Definition. Let G be a finite group. An endo-split permutation resolution of a finitely-generated RG-module M is a bounded complex X_{M} of p-permutation RGmodules together with an isomorphism between M and the degree zero homology of X_{M}, such that

(i) the homology of M is concentrated in degree zero, and

(ii) the complex Hom_{R}(X_{M}, X_{M}) is split as a complex of RG-modules (with G acting diagonally).

Note that if P is a p-group and the RP -module M has an endo-split permutation resolution, then Hom_{R}(M, M) is a permutation module for RP, being the homology of the split complex Hom_{R}(X_{M}, X_{M}) of permutation modules. In other words, M must be an endo-permutation module. A natural question to ask is whether the converse is true; i.e., does every endo-permutation module for every p-group have an endo-split permutation resolution?

Over k we do not know of any examples of endo-permutation modules for which this is not so, but the only positive results we have are based on known classifications of certain types of such modules; in particular, we shall show how Dade’s classification [4, 5] of ‘capped’ endo-permutation modules for abelian p-groups leads to a positive answer in that case.

Let us make some remarks about endo-permutation modules over O. There is good news and bad news. The bad news is that over O there are very simple examples of such modules which do not have an endo-split permutation resolution. For example, if p > 2 and P is cyclic of order p, a non-trivial rank one OP module does not have an endo-split permutation resolution. This is clear since the character of such a module is not a linear combination of characters of permutation modules.

The good news is that the following proposition is true.

Proposition 7.1. Let G be a finite group, and let M be a kG-module that has an endo-split permutation resolution X_{M}. Then M can be lifted to an OG-module with an endo-split permutation resolution .

Proof. Exactly as in Section 5, the complex X_{M} can be lifted (uniquely, up to isomorphism) to give a complex X̃_{M} of p-permutation OG modules. By the universal coefficient theorem, the homology of X̃_{M} is concentrated in degree zero and is an O-free OG-module M̃ with M̃ ⊗ k ≅ M.

Since Hom_{O}(X̃_{M}, X̃_{M}) is a complex of p-permutation OG-modules which is a lifting of Hom_{k}(X_{M}, X_{M}), a split complex of p-permutation kG-modules with homology concentrated in degree zero, it follows that Hom_{O}(X̃_{M}, X̃_{M}) is itself split.

In the introduction to [5], Dade raises the question of liftability of endo-permutation modules. Proposition 7.1 answers this question for modules which have an endo-split permutation resolution, and moreover gives a canonical lifting once the complex X_{M} is given, since the lifting X̃_{M} is unique up to isomorphism.

7.2. Endo-permutation modules for abelian p-groups. Using Dade’s classification [5] of ‘capped’ endo-permutation modules for abelian p-groups P (i.e., endopermutation modules containing a summand with vertex P), we shall prove the following theorem.

Theorem 7.2. Let P be a finite abelian p-group, and let M be an endo-permutation kP -module with an indecomposable summand having vertex P. Then M has an endosplit permutation resolution.

Before proving this theorem, we shall prove four easy lemmas that we shall need in the proof. These lemmas are true for arbitrary finite groups G.

Lemma 7.3. If Q is a quotient group of G and the kQ-module M has an endosplit permutation resolution X_{M}, then the inflation of X_{M} to G is an endo-split permutation resolution of M considered as a kG-module.

Proof. Since Hom_{k}(X_{M}, X_{M}) is split as a complex of kQ-modules, it is certainly split as a complex of kG-modules.

Lemma 7.4. If M and N are kG-modules with endo-split permutation resolutions X_{M} and X_{N}, then

X_{M⊗N} = X_{M} ⊗ X_{N}

is an endo-split permutation resolution of M ⊗ N.

Proof. Certainly X_{M⊗N} has homology isomorphic to M ⊗ N concentrated in degree zero and

Hom_{k}(X_{M⊗N}, X_{M⊗N}) ≅ Hom_{k}(X_{M}, X_{M}) ⊗ Hom_{k}(X_{N}, X_{N}),

which is split as a complex of kG-modules.

Lemma 7.5. If M and N are kG-modules such that M ⊕ N has an endo-split permutation resolution, then M and N have endo-split permutation resolutions.

Proof. Let X_{M⊕N} be an endo-split permutation resolution of M ⊕ N. Since the complex Hom_{k}(X_{M⊕N}, X_{M⊕N}) is split, taking homology commutes with taking Gfixed points, and so the natural map from the endomorphism ring of X_{M⊕N} in the category K^{b}(kG-mod) to the endomorphism ring of M ⊕ N in the category of kGmodules is an isomorphism (since the former ring is just the degree zero homology of Hom_{kG}(X_{M⊕N}, X_{M⊕N})). Therefore the idempotent endomorphism of M ⊕ N that projects onto the summand M lifts to an endomorphism α of the complex X_{M⊕N} which is idempotent up to chain homotopy. Since X_{M⊕N} is a complex of finite dimensional vector spaces, it must decompose as the direct sum of

X_{M} = ⋂_{i=1}^{∞} im(α^{i})

and

X_{N} = ⋃_{i=1}^{∞} ker(α^{i}).

The degree zero homology modules of X_{M} and X_{N} are isomorphic to M and N respectively, and so X_{M} and X_{N} are endo-split permutation resolutions of M and N.

Remark. Since it is not in general true, even for abelian p-groups, that a direct sum of endo-permutation modules is itself an endo-permutation module, the converse to Lemma 7.5 is not true in general.

Lemma 7.6. If M is a kG-module with an endo-split permutation resolution X_{M}, then any kG-module that is a direct sum of modules of the form

Ind_{H}^{G} Res_{H}^{G}(M),

for subgroups H of G, also has an endo-split permutation resolution.

Proof. Suppose

N = ⨁_{i} Ind_{H_{i}}^{G} Res_{H_{i}}^{G}(M).

Let us define

X_{N} = ⨁_{i} Ind_{H_{i}}^{G} Res_{H_{i}}^{G}(X_{M}),

so that X_{N} has homology isomorphic to N concentrated in degree zero. Then Hom_{k}(X_{N}, X_{N}), considered as a complex of kG-modules via the diagonal action of G, is isomorphic to a direct sum of complexes of the form

Res_{ΔG}^{G×G} Ind_{H_{i}×H_{j}}^{G×G} Res_{H_{i}×H_{j}}^{G×G} Hom_{k}(X_{M}, X_{M}),

which, by Mackey’s decomposition theorem, is isomorphic to a direct sum of complexes of the form

Ind_{Δ(H_{i}∩H_{j}^{π})}^{ΔG} Hom_{k}(X_{M}, X_{M})

for elements π ∈ G, which are split since Hom_{k}(X_{M}, X_{M}) is split as a complex of modules for k[ΔG] and so is certainly split as a complex of modules for k[Δ(H_{i} ∩ H_{j}^{π})].

Proof of Theorem 7.2. By Dade’s result [4, Theorem 6.10] determining the possible structures of endo-permutation kP -modules with an indecomposable summand N with vertex P, every such module is a direct summand of a direct sum of modules of the form

Ind_{Q}^{P} Res_{Q}^{P}(N)

for subgroups Q of P. Therefore, using Lemmas 7.6 and 7.5, we reduce the proof to the case where M is indecomposable.

By Dade’s classification [5, Theorem 12.5] of indecomposable endo-permutation modules with vertex P for an abelian p-group P, every such module is a direct summand of a tensor product of inflations of modules of the form Ω^{n}(k) for quotient groups of P. Therefore, successively using Lemmas 7.3, 7.4 and 7.5, we reduce the proof to the case M = Ω^{n}(k).

So let M = Ω^{n}(k). If n is positive let X_{M} be the complex

… ⟶ 0 ⟶ P^{1−n} ⟶ … ⟶ P^{0} ⟶ k ⟶ 0 ⟶ …

obtained by truncating a minimal projective resolution of k shifted in degree so that P^{1−n} is the degree zero term. If n is negative let X_{M} be the complex

… ⟶ 0 ⟶ k ⟶ I^{0} ⟶ … ⟶ I^{1−n} ⟶ 0 ⟶ …

obtained by truncating a minimal injective resolution of k shifted in degree so that I^{1−n} is the degree zero term. If n = 0 let X_{M} be the complex with the single nonzero term k in degree zero. In all of these cases X_{M} has homology isomorphic to M concentrated in degree zero, and Hom_{k}(X_{M}, X_{M}) is split, since its homology is concentrated in degree zero and every term is projective apart from the degree zero term. Thus X_{M} is an endo-split permutation resolution of M.

7.3. Slashed modules. Let us make some remarks about the relationship between endo-split resolutions and Dade’s construction of the ‘slashed modules’ of endopermutation modules [4, Section 4]. If M is an endo-permutation kP -module, then one can apply the Brauer construction to End_{k}(M) with respect to a subgroup Q ≤ P. As Dade remarked, the resulting algebra End_{k}(M)(Q) is a simple algebra, and since the ring homomorphism kP ⟶ End_{k}(M) induces a map

kC_{P} (Q) = (kP)(Q) ⟶ End_{k}(M)(P),

the simple End_{k}(M)(P) module has the structure of a kC_{P} (Q)-module; this is what Dade calls the slashed module M\Q. Of course, End_{k}(M)(P) has an action of N_{P} (Q) on it, not just C_{P} (Q), and Puig [8] and Dade independently showed that this lifts to an action of N_{P} (Q) on the module M\Q.

In the case of an endo-permutation kP -module with an endo-split resolution X_{M}, the construction of the slashed module can be carried out by a much more direct use of the Brauer construction. Applying the Brauer construction to X_{M}, we get a complex X_{M}(Q) such that

Hom_{k}(X_{M}(Q), X_{M}(Q)) ≅( Hom_{k}(M, M))(Q),

which has homology concentrated in degree zero. It follows that X_{M} has homology concentrated in a single degree (not necessarily in degree zero), and so X_{M}(Q)[i] is an endo-split resolution of some kN_{P} (Q)-module for some integer i: this module is precisely Dade’s slashed module. Notice that this construction gives Puig’s and Dade’s action of N_{P} (Q) immediately, albeit only for modules with an endo-split resolution.

7.4. p-nilpotent groups. We are now going to show how one very well-known example of Morita equivalence in block theory is an example of a splendid equivalence, even though the bimodule that induces the equivalence of module categories is not, in general, a p-permutation module.

Let G = SP be a finite p-nilpotent group, i.e., the semi-direct product of a normal p′-subgroup S by a p-group P. Let us assume k is a splitting field for G. Then as an algebra, kG is the skew group algebra of P acting by conjugation on kS, which is a product of matrix rings. Thus it is easy to see that if χ is a character of S that is stable under the action of P, then there is a corresponding block kB_{χ} of G which, as an algebra, is the skew group algebra of P acting on a single matrix ring. Since P is a p-group, it is easy to verify that this block algebra is isomorphic to the tensor product of this matrix ring with kP, and so is Morita equivalent to kP.

In more group theoretic terms, we can take a kS-module M_{χ} realizing the character χ. This can be lifted to a kG-module (which we shall also call M_{χ}), again using the fact that P is a p-group. The Morita equivalence from kP to kB_{χ} is given by taking a kP -module, which may be regarded as a kG-module with trivial S-action, and taking the tensor product over k with M_{χ}. In other words, the bimodule that induces the Morita equivalence is

N_{χ} = Ind_{Δ_{P} G}^{G×P}(M_{χ}),

where Δ_{P} G is the copy

{(σπ, π) : σ ∈ S, π ∈ P }

of G embedded ‘diagonally’ in G × P.

In general N_{χ} is not a p-permutation module for k[G × P], and so at first sight it appears that this simple example of a Morita equivalence, and hence of a derived equivalence, is not splendid. However, we know that in the case of a derived equivalence we have to choose a tilting complex carefully within its isomorphism class in the derived category in order to ensure that it is a splendid tilting complex, and there is no obvious reason why, in the case of a Morita equivalence (when the tilting complex is isomorphic to a single bimodule), this careful choice should itself be a single bimodule. In fact we shall show that, at least when P is abelian and so Theorem 7.2 applies, we can find a splendid tilting complex with homology isomorphic to N_{χ} concentrated in degree zero.

We shall start by proving the following lemma about the module M_{χ}.

Lemma 7.7. With the notation above, suppose that P is abelian. Then M_{χ}, considered as a kG-module, has an endo-split permutation resolution.

Proof. Since End_{k}(M_{χ}) is one of the matrix algebras occurring as a summand of kS, which is a permutation kP -module, the restriction of M_{χ} to P is an endo-permutation kP -module. Hence, by Theorem 7.2, M_{χ} has an endo-split permutation resolution X_{M} as a kP -module. We can induce to G to get a complex

Y_{M} = Ind_{P}^{G}(X_{M})

of permutation kG-modules whose homology is isomorphic to Ind_{P}^{G}(M_{χ}) and concentrated in degree zero.

Consider Hom_{k}(Y_{M}, Y_{M}). As a complex of kG-modules, this is isomorphic to

Res_{ΔG}^{G×G} Ind_{P ×P}^{G×G}(Hom_{k}(X_{M}, X_{M})),

which, by Mackey’s decomposition theorem, is isomorphic to the direct sum of the complexes

Ind_{P ∩P^{σ}}^{G} Hom_{k}(X_{M}, X_{M}^{σ}) (*)

as σ runs over all the elements of S. But an easy calculation shows that P ∩ P^{σ} = C_{P} (σ), the subgroup of P that centralizes σ, and so X_{M} is isomorphic to X_{M}^{σ} as a complex of k[P ∩ P^{σ}]-modules. Since Hom_{k}(X_{M}, X_{M}) is split as a complex of kP - modules, it is certainly split as a complex of kC_{P} (σ)-modules, and so the complexes (*) are split as complexes of kG-modules. Hence Y_{M} is an endo-split permutation resolution of Ind_{P}^{G}(M_{χ}).

Since M_{χ} is a direct summand of Ind_{P}^{G}(M_{χ}), it follows from Lemma 7.5 that, considered as a kG-module, M_{χ} has an endo-split permutation resolution.

We can now use this endo-split resolution to produce a splendid tilting complex. We keep the same notation.

Theorem 7.8. If P is abelian, there is a splendid tilting complex of kB_{χ}-kP -bimodules.

Proof. By Lemma 7.7, there is an endo-split permutation resolution X′_{M} of the kGmodule M_{χ}. we define the complex X′_{N} of kB_{χ}-kP -bimodules by

X′_{N} = Ind_{Δ_{P} G}^{G×P}(X′_{M}),

and we claim that this is a splendid tilting complex.

Clearly the homology of X′_{N} is isomorphic to N_{χ} concentrated in degree zero, so X′_{N} is at least a two-sided tilting complex. Also

Hom_{k}(X′_{N}, X′_{N}) ≅ Hom_{k}(Ind_{Δ_{P} G}^{G×P}(X′_{M}), Ind_{Δ_{P} G}^{G×P}(X′_{M}))
≅ Ind_{Δ_{P} G×Δ_{P} G}^{G×P ×G×P}(Hom_{k}(X′_{M}, X′_{M})),

whose restriction to the subgroup

ΔG × P × P = {(g, π, g, π′) : g ∈ G, π ∈ P, π′ ∈ P }

is, by the Mackey decomposition theorem, isomorphic to

Ind_{G}^{ΔG×P ×P} (Hom_{k}(X′_{M}, X′_{M})),

where G is embedded as the subgroup {(sπ, π, sπ, π) : s ∈ S, π ∈ P } of ΔG × P × P. Since Hom_{k}(X′_{M}, X′_{M}) is split as a complex of kG-modules, Hom_{k}(X′_{N}, X′_{N}) is split as a complex of k[ΔG × P × P]-modules, and so its G-fixed point complex Hom_{kG}(X′_{N}, X′_{N}) is split as a complex of kP -bimodules.

By Theorem 2.1 it follows that X′_{N} is a splendid tilting complex. Alternatively, a similar calculation to the one above will show directly that Hom_{kP} (X′_{N}, X′_{N}) is split as a complex of kG-bimodules.

Remarks. (a) Of course, if Theorem 7.2 can be generalized to arbitrary p-groups, then Theorem 7.8 will also generalize.

(b) By Theorem 5.2, we can deduce that Theorem 7.8 is also true over O.

8. Finite Coxeter Groups

If W is a finite Coxeter group, then there is a very simple and well-known selfequivalence of the module category of RW given by tensoring with the sign representation R_{ϵ} of W. In fact, this equivalence may also be thought of as induced by an automorphism of the group ring, namely the map RW ⟶ RW that fixes elements of the alternating subgroup of W (i.e., those elements that are products of an even number of fundamental reflections) and multiplies by −1 those elements of W that are not in this subgroup.

The RW -bimodule that induces the equivalence is, considered as an R[W × W]- module, just Ind_{ΔW}^{W ×W} R_{ϵ}. In almost all situations, this bimodule is a splendid tilting complex; of course, the only property that needs to be checked is that it is a ppermutation R[W × W]-module. If p ≠ 2 then this is true, since then R_{ϵ} is a direct summand of the permutation module on the cosets of the alternating subgroup of W. If p = 2 and R = k then again it is true, since in this case R_{ϵ} ≅ R as an RW -module. However, in the one remaining case when p = 2 and R = O, it is just not true.

In this section, we shall show that there is a very natural splendid tilting complex that induces this self-equivalence in complete generality. In fact, our construction works over any coefficient ring, but to avoid further notation and terminology we shall stick with the rings R that we have used throughout this paper.

We shall consider a finite Coxeter group W generated by a set S of fundamental reflections. For a subset I of S, W_{I} will be the parabolic subgroup of W generated by I. By Σ = Σ(W, S) we shall denote the Coxeter complex of (W, S) (see for example [6, Section 1.15]): recall that this is a triangulation of the unit sphere in the natural representation of W as a real reflection group. Combinatorially, the poset of simplices of Σ is the set of all right cosets of all proper standard parabolic subgroups W_{I}, with the order relation given by reverse inclusion: a coset W_{I}g is a (|S|−|I|−1)-dimensional simplex.

We shall denote by C̃ the reduced simplicial cochain complex C̃^{∗}(Σ, R) of Σ with coefficients in R. The action of W on Σ gives each term C̃^{n}(Σ, R) the structure of a permutation RW -module. Since W acts on Σ so that the stabilizer of a simplex fixes every face of that vertex, the differentials in C̃ commute with the action of W, and so C̃ is a complex of RW -modules.

More generally, for every subset I of S we shall denote by Σ_{I} the Coxeter complex Σ(W_{I}, I) of W_{I} and by C̃_{I} the reduced simplicial cochain complex of Σ_{I}, which is a complex of permutation RW_{I}-modules.

Let us start by proving some lemmas about the complexes C̃_{I}.

Lemma 8.1. For any subset I ≠ ∅ of S, the geometric realization of Σ is W_{I}- equivariantly homeomorphic to the (|S| − |I|)-th unreduced suspension of the geometric realization of Σ_{I}. Hence, considered as a complex of RW_{I}-modules, C̃ is chain homotopy equivalent to C̃[|I| − |S|].

Proof. It is enough to consider the case |I| = |S| − 1, as then the general case follows by induction on |S| − |I|.

Let V be the standard reflection representation of W, and let V′ be the (|S| − 1)-dimensional subspace spanned by the orthogonal complements of the reflecting hyperplanes corresponding to I. Then W_{I} acts on V′ via its own standard reflection representation. Since W_{I} fixes pointwise the (1-dimensional) orthogonal complement to V′, the unit sphere in V is W_{I}-equivariantly homeomorphic to the suspension of the unit sphere in V′.

Remark. If we make the sensible convention that the suspension of the empty space is a 0-dimensional sphere S^{0}, then Lemma 8.1 is also true for I = ∅.

Lemma 8.2. Let W_{I} be a standard parabolic subgroup of W. Then the fixed point complex C̃^{W_{I}} is acyclic if I ≠ ∅ and is just C̃ if I = ∅.

Proof. Whenever a finite group G acts simplicially on a finite simplicial complex Δ in such a way that the quotient Δ/G is also a simplicial complex (which is true for the case of W_{I} acting on Σ), then the reduced simplicial cochain complex C̃^{∗}(Δ/G, R) is isomorphic to the fixed points of G acting on C̃^{∗}(Δ, R). Hence C̃^{W_{I}} is isomorphic to the reduced simplicial cochain complex of the quotient Σ/W_{I}. By Lemma 8.1, the geometric realization of Σ is W_{I}-equivariantly homeomorphic to an iterated suspension of Σ_{I}, and so the geometric realization of Σ/W_{I} is homeomorphic to an iterated suspension of the geometric realization of Σ_{I}/W_{I}. But if I ≠ ∅, then Σ_{I}/W_{I} is a single (|I| − 1)-dimensional simplex, and so is contractible, and so has trivial reduced cohomology.

If I = ∅, then of course W_{I} is the trivial group.

Lemma 8.3. Considering C̃ as an object of K^{b}(RW -mod),

Hom_{K^{b}(RW -mod)}(C̃, C̃[i]) ≅ \begin{cases}0 & if i ≠ 0, \\ R & if i = 0. \end{cases}

Proof. Since Hom_{K^{b}(RW −mod)}(C̃, C̃[i]) is the degree i homology of the total complex of the double complex Hom_{RW} (C̃, C̃), we can calculate this using a spectral sequence {E_{t}^{p,q}} with

E_{0}^{p,q} = Hom_{RW} (C̃^{−p}, C̃^{q}).

The (−p)-th row of the E_{1} page is the homology of the complex

Hom_{RW} (C̃^{p}, C̃).

Since every simplex of Σ of dimension p has stabilizer in W that is conjugate to a standard parabolic subgroup W_{I} of W where |I| = |S| − p − 1, C̃^{p} is isomorphic to a direct sum of permutation RW -modules on cosets of such subgroups. Thus Hom_{RW} (C̃^{p}, C̃) is isomorphic to a direct sum of fixed point complexes C̃^{W_{I}} for |I| = |S| − p − 1, and so it follows from Lemma 8.2 that for p ≠ |S| − 1, E_{1}^{p,q} = 0, and that for p = |S| − 1,

E_{1}^{p,q} ≅ \begin{cases}0 & if q ≠ |S| − 1, \\ R & if q = |S| − 1. \end{cases}

Hence the spectral sequence collapses at the E_{1} page, and so the homology of Hom_{RW} (C̃, C̃) is as claimed.

Proposition 8.4. The complex Hom_{R}(C̃, C̃) of RW -modules is split, with homology isomorphic to the trivial RW -module R concentrated in degree zero.

Proof. We just need to show that the natural chain map

R ⟶ Hom_{R}(C̃, C̃)

is a chain homotopy equivalence. Since the terms of C̃ are all direct sums of permutation modules on the cosets of standard parabolic subgroups, it is enough, by Lemma 1.1, to prove that the chain map induced on the W_{I}-fixed point complexes is a quasi-isomorphism for each I. This map is just the natural map

R ⟶ Hom_{RW_{I}}(C̃, C̃)

and so, since by Lemma 8.1 C̃ is isomorphic as a complex of RW_{I}-modules to a shifted version of C̃_{I}, we just need to show that the natural map

R ⟶ Hom_{RW_{I}}(C̃_{I}, C̃_{I})

is a quasi-isomorphism for each I. This follows from Lemma 8.3 applied to W_{I}.

Acknowledgements

I would like to thank Michel Broué for several conversations that inspired much of the work in this paper. I would also like to thank Lluis Puig for a remark that simplified the proof of Lemma 4.2.

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University of Bristol, School of Mathematics, University Walk, Bristol BS8 1TW, ENGLAND

E-mail address: J.Rickard@bristol.ac.uk