DERIVED EQUIVALENCES FOR GROUP ALGEBRAS

Miles Lee Holloway

School of Mathematics, University of Bristol, May 2001

A dissertation submitted to the University of Bristol in accordance with the requirements of the degree of Doctor of Philosophy in the Faculty of Science. ii

Abstract

Broué’s abelian defect conjecture suggests a deep link between the module categories of a block of a group algebra and its Brauer correspondent, viz that they should be derived equivalent.

Here we verify Broué’s conjecture for the non-principal block of kSL_{2}(p^{2}), the principal blocks of kSL_{2}(2^{n}), for 2 ≤ n ≤ 5, and kSL_{2}(27). We also verify the conjecture for the Hall-Janko group, and even its double cover, as well as for U_{3}(4) and Sp_{4}(4).

In all these cases we also verify Rickard’s refinement to Broué’s conjecture and show that the derived equivalence can be chosen to be a splendid equivalence for these examples. iv

Acknowledgement

I would like to express my thanks to my supervisor, Professor Jeremy Rickard, for his advice, encouragement and helpful answers to my numerous questions and also for persuading me of the benefits of a category theory viewpoint. I would also like to thank Dr Joseph Chuang for many useful discussions and for his encouragement.

Finally I would like to express my gratitude for the financial support I received from the Engineering and Physical Sciences Research Council.

Dedication

To Fiona Barron vi

Contents

1 Introduction 1 1.1 Assumptions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 3 2.1 Derived categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Finite-dimensional algebras . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.2 Self-injective algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.3 Symmetric Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Group algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.6 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Obtaining Stable Equivalences 19 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Stable equivalence of Morita type . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Stable equivalence of splendid type . . . . . . . . . . . . . . . . . . . . . . . 21

4 Obtaining Derived Equivalences 25 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Derived equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Computer calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4 From derived to splendid equivalence . . . . . . . . . . . . . . . . . . . . . . 28 4.5 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 SL_{2}(p^{n}) 33 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 The representation theory of kSL_{2}(p^{n}) . . . . . . . . . . . . . . . . . . . . . 34

5.2.1 Rectangular modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3 The non-principal block of kSL_{2}(p^{2}) . . . . . . . . . . . . . . . . . . . . . . 38 5.4 The principal block of kSL_{2}(2^{n}) for 2 ≤ n ≤ 5 . . . . . . . . . . . . . . . . . 53

5.5 The principal block of kSL_{2}(27) . . . . . . . . . . . . . . . . . . . . . . . . 60

6 The Hall-Janko group, its double cover, U_{3}(4) and Sp_{4}(4) 63 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 The representation theory of 2.J_{2}, U_{3}(4) and Sp_{4}(4) . . . . . . . . . . . . . 64 6.3 Stable equivalence of splendid type . . . . . . . . . . . . . . . . . . . . . . . 65 6.4 Splendid equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.5 Independence of choice of differentials . . . . . . . . . . . . . . . . . . . . . 70

A Loewy layers of Green correspondents 75

A.1 Principal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.1.1 G = U_{3}(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.1.2 G = J_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.1.3 G = Sp_{4}(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.2 Non-principal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 A.2.1 G = 2.J_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B Loewy layers of degree -1 terms of complexes S_{i} ⊗_{ekG} C 79

B.1 Principal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 B.1.1 G = U_{3}(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 B.1.2 G = J_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 B.1.3 G = Sp_{4}(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

B.2 Non-principal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 B.2.1 G = 2.J_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

C Images of simple modules under derived equivalence 83

C.1 Principal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 C.1.1 G = U_{3}(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 C.1.2 G = J_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 C.1.3 G = Sp_{4}(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

C.2 Non-principal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 C.2.1 G = 2.J_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

D Loewy layers of cohomology groups of complexes in Appendix C 85

D.1 Principal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 D.1.1 G = U_{3}(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 D.1.2 G = J_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 D.1.3 G = Sp_{4}(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

D.2 Non-principal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 D.2.1 G = 2.J_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

E MAGMA code computer programs 89

Chapter 1

Introduction

The representations of a finite group over an algebraically closed field of characteristic p > 0, dividing the order of the group, and the representations of its local subgroups, the normalizers of non-trivial p-subgroups, are related. Amongst the many attempts to tease out the relationship, Broué’s abelian defect conjecture (Conjecture 2.6.6 of Chapter 2, see also [4]) provides, albeit only for abelian defect groups, one of the most complete and deepest links to date.

Here we verify Broué’s conjecture for a number of specific blocks of groups, obtaining useful information about the equivalences that the conjecture predicts along the way. There is an important refinement to Broué’s conjecture that was first introduced, for principal blocks, by Rickard in [30]. For all the examples considered here where we verify Broué’s conjecture we’re also able to verify this stronger conjecture.

In Chapter 2 we collect the necessary background material on various categories and rings thus setting the stage for later chapters. We also include a section recalling several conjectures including Broué’s conjecture, thus placing it in some context.

Chapter 3 discusses various types of stable equivalences for group algebras and methods to construct such equivalences; an important first step towards verifying Broué’s conjecture.

Chapter 4 describes a method, due to Rickard (see [31]), for obtaining derived equivalences and is central in our approach to verifying Broué’s conjecture for our examples. This approach, common to all our examples, is outlined at the beginning of Section 4.4. We also discuss how, under certain circumstances, Rickard’s refinement to Broué’s conjecture may also be verified.

In Chapter 5 we apply the approach outlined in Chapter 4 to verify Broué’s conjecture for the non-principal block of kSL_{2}(p^{2}) , as well as for the principal blocks of kSL_{2}(2^{n}), for 2 ≤ n ≤ 5, and kSL_{2}(27). In Section 5.1 we describe how this work ties in with results due to Chuang [9] and subsequent work, using different methods to those here, verifying Broué’s conjecture for kSL_{2}(p^{n}) due to Okuyama [25].

Finally in Chapter 6 we verify Broué’s conjecture for the Hall-Janko group and its double cover as well as U_{3}(4) and Sp_{4}(4). For the examples of this chapter the establishment of a stable equivalence turns out to be more involved than for the examples considered in Chapter 5.

For the examples in Chapter 6 and the cases kSL_{2}(2^{n}) for 2 ≤ n ≤ 5, of Chapter 5 computer calculations were used to verify Broué’s conjecture for these cases. These calculations were all carried out using the computer algebra packages GAP and MAGMA. The main calculations using MAGMA are described in Section 4.3 of Chapter 4. Appendix E contains the code for routines, written in MAGMA, used during the computer calculations.

Appendices A, B, C and D contain various pieces of information needed for the verification of Broué’s conjecture for the examples in Chapter 6.

1.1 Assumptions and Notation

We describe here some standard assumptions and notation that we shall adopt. Further notation will be introduced in the text as needed.

All rings will be associative with a 1.

Given a ring R all R-modules will be unitary and, unless stated otherwise, will be assumed to be right R-modules.

We denote by R^{o} the opposite ring to R which has the same structure as R except for multiplication. If a, b ∈ R^{o} then ab in R^{o} is the element ba in R. We’ll write Z(R) for the centre of the ring R.

We denote by Mod- R the module category and mod- R the subcategory of finitely presented R-modules. We will also write Proj- R for the full subcategory of projective modules and proj- R for the full subcategory of finitely generated projective modules. We will simply write Hom_{R}(−, −) for Hom_{mod- R}(−, −). Complexes of R-modules, for a ring R, will always be taken to mean cochain complexes.

Given rings R and S an R-S-bimodule M will be a left R-module and right S-module such that, for r ∈ R, s ∈ S and m ∈ M, (rm)s = r(ms). We simply say an R-bimodule to mean an R-R-bimodule.

All groups will be assumed to be finite and with identity element written as 1. Where appropriate, mainly in Chapter 6, ATLAS notation (see [11]) will be used.

Throughout, unless stated otherwise, k will be an algebraically closed field of positive characteristic. The symbol ⊗ will always mean ⊗_{k}.

Given a pair of functors F and G we will speak of an adjoint pair (F, G) to mean F is a left adjoint to G, as in defined in [36].

Background

2.1 Derived categories

We briefly recall some of the basic facts about derived categories. Background information for the material in this section, together with proofs of all the statements made here, can be found in [16, 37].

If A denotes an abelian category, we can form the category of cochain complexes Ch(A). An object X^{•} in Ch(A) is a (cochain) complex

X^{•} = ⋯ → X^{i−1} ⟶^{d_{X}^{i−1}} X^{i} ⟶^{d_{X}^{i}} X^{i+1} → ⋯

where each X^{i}, d_{X}^{i} ∈ A and d_{X}^{i}d_{X}^{i−1} = 0. We will frequently drop the ^{•} and simply write X for such an object. We’ll also write d^{i} for d_{X}^{i}, as it should be clear from the context which X is intended. A (cochain) map between objects X and Y is a collection of maps {h^{i}} in A such that we have the commutative diagram:

\xymatrix{⋯ \ar[r] & X^{i} \ar[r]^{d^{i}} \ar[d]^{h^{i}} & X^{i+1} \ar[d]^{h^{i+1}} \ar[r] & ⋯ \\ ⋯ \ar[r] & Y^{i} \ar[r]^{d^{i}} & Y^{i+1} \ar[r] & ⋯}

It’s easy to see that Ch(A) is an abelian category. An object X ∈ Ch(A) is said to be bounded below (resp. bounded above) if X^{i} = 0 for all i ≤ N (resp. for all i ≥ N) for some integer N. X is said to be bounded if it’s both bounded above and below. With these definitions we can form the full subcategories: Ch^{+}(A) whose objects are bounded below, Ch^{−}(A) whose objects are bounded above and Ch^{b}(A) whose objects are bounded.

We have the important collection of functors {H^{i} : Ch(A) → A} called cohomology, where for X ∈ Ch(A) and f : X → Y in Ch(A) we have

H^{i}(X) := ker(d^{i})/ im(d^{i−1})

and, as f is a cochain map, it’s easy to see that it induces a map

H^{i}(f) := H^{i}(X) → H^{i}(Y)

One of the most important facts about the functors {H^{i}} is that if

0 → X → Y → Z → 0

is a short exact sequence in Ch(A), then the cohomology functors assemble to give a long exact sequence

⋯ → H^{n−1}(Z) → H^{n}(X) → H^{n}(Y) → H^{n}(Z) → H^{n+1}(X) → ⋯

We can introduce an equivalence relation ∼ on the maps in Ch(A) called homotopy. If f : X → Y is a cochain map, then f is said to be null homotopic, f ∼ 0 (here 0 is the zero cochain map), if there is a collection of maps (the homotopy) {s^{i}} in A that fit into the diagram

\xymatrix{⋯ \ar[r] & X^{i−1} \ar[r]^{d^{i−1}} \ar[d]_{f^{i−1}} & X^{i} \ar[dl]_{s^{i}} \ar[d]_{f^{i}} \ar[r]^{d^{i}} & X^{i+1} \ar[dl]_{s^{i+1}} \ar[r] \ar[d]^{f^{i+1}} & ⋯ \\ ⋯ \ar[r] & Y^{i−1} \ar[r]^{d^{i−1}} & Y^{i} \ar[r]^{d^{i}} & Y^{i+1} \ar[r] & ⋯}

such that

f^{i} = d^{i−1}s^{i} + s^{i+1}d^{i}  (2.1)

Given two maps f, g : X → Y, f ∼ g if f − g ∼ 0.

A complex X is said to be contractible if the identity map on X is null homotopic. It’s easy to check that if f ∼ g then fw ∼ gw and vf ∼ vg whenever the compositions make sense. We can thus form the quotient category K(A) called the homotopy category. It has the same objects as Ch(A) and if X, Y ∈ K(A) then

Hom_{K(A)}(X, Y) = Hom_{Ch(A)}(X, Y)/ ∼

Note that K(A) is no longer an abelian category (see Section 2.2). It’s easy to see, using (2.1), that the cohomology functors factor to give functors H^{i} : K(A) → A.

In many situations one is interested in the cohomology {H^{n}(X)} of a given complex X. However by throwing away the original complex a lot of useful information is lost. Part of the motivation for introducing the derived category was to try and retain this information but loosen the rigidity of the homotopy category in not identifying complexes that, as far as cohomology is concerned, should be the same.

For a general abelian category A the derived category D(A) may not exist, however it does exist for the case when A is the module category of a ring R. As this is the only case of interest to us, we will assume for the rest of this section that A = Mod- R.

A morphism f : X → Y in K(A) is called a quasi-isomorphism if the maps H^{i}(f) : H^{i}(X) → H^{i}(Y) are all isomorphisms. The class of morphisms that are quasi-isomorphisms will be denoted by Qis. Informally the derived category D(A) is obtained by inverting all quasi-isomorphisms. More formally Qis forms what is called a locally small multiplicative system on K(A). Under these circumstances there is a category D(A), the localisation of K(A) with respect to Qis, and a functor Q : K(A) → D(A) such that:

(1) Q(s) is an isomorphism for all s ∈ Qis.

(2) Any functor F : K(A) → D such that F (s) is an isomorphism for all s ∈ Qis factors uniquely through Q.

The objects of D(A) are the same as those of K(A). Given two objects X and Y

a diagram \xymatrix{X & X′ \ar[l]_{s} \ar[r]^{f} & Y}, with s ∈ Qis, represents a map in Hom_{D(A)}(X, Y). The structure of the multiplicative system Qis can be used to decide when two such diagrams define the same map in D(A).

The categories K^{∗}(A) (where ∗ = +, − or b) are so called localising subcategories of K(A) for Qis. This amounts to saying that we may localise K^{∗}(A) with respect to K^{∗}(A) ∩ Qis, so forming D^{∗}(A). The categories D^{∗}(A) are each a full subcategory of D(A).

One of the most useful properties of D(A) is the following

Proposition 2.1.1. If I is a bounded below cochain complex of injectives then

Hom_{D(A)}(X, I) = Hom_{K(A)}(X, I)

for every X. Dually if P is a bounded above cochain complex of projectives then

Hom_{D(A)}(P, X) = Hom_{K(A)}(P, X)

for every X.

An object X ∈ A can be regarded as a complex whose only non-zero term is X in degree zero. This complex is called the stalk complex of X. The following is a useful fact.

Proposition 2.1.2. The functor A → D^{∗}(A) sending an object X to its stalk complex induces an equivalence between A and the full subcategory of D^{∗}(A) consisting of objects X such that H^{i}(X) = 0 for i ≠ 0.

2.2 Triangulated categories

References for the material on triangulated categories in this section can be found in [16, 14].

Some of the categories of interest to us will not be abelian, for example K(A), D(A). Instead they have the structure of a triangulated category. The central part of an additive category C being triangulated is a collection T of triples (u, v, w) of maps called triangles that are to obey certain axioms. A triangle (u, v, w) fits into the following diagram

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} T X  (2.2)

Here T : C → C is an automorphism called the translation functor. One of the axioms states that if (u, v, w) and (u′, v′, w′) are both triples of maps as in (2.2) and they are isomorphic (in the obvious way) and (u, v, w) is a triangle then (u′, v′, w′) is a triangle. In practice when defining a triangulated structure on a category certain triangles are defined explicitly called strict triangles and then T is defined as the collection of all triples isomorphic to strict triangles. In the literature triangles are also written more suggestively (if a little inaccurately because w : Z → T X) as

\xymatrix{ & Z \ar[dl]_{w} \\ X \ar[rr]^{u} &  & Y \ar[ul]_{v}}

In many respects these triangles act as suitable replacements for short exact sequences in abelian categories.

An additive functor H : C → A where A is an abelian category is called a covariant cohomological functor if, given a triangle (2.2), we have the long exact sequence:

⋯ → H(T^{i}X) ⟶^{H(T^{i}u)} H(T^{i}Y) ⟶^{H(T^{i}v)} H(T^{i}Z) ⟶^{H(T^{i}w)} H(T^{i+1}X) → ⋯

H is called a contravariant cohomological functor if the arrows are reversed in the long exact sequence above.

An additive functor F : C → C′ between triangulated categories C and C′ is said to be exact if there’s an invertible natural transformation α : F T → T′F such that (F u, F v, F wα) is a triangle in C′ whenever (u, v, w) is a triangle in C. An equivalence of triangulated categories will always mean an exact equivalence.

A subcategory F of a triangulated category C is said to be a triangulated subcategory if F is a full subcategory and a triangulated category such that the inclusion functor F → C is an exact functor of triangulated categories.

A collection of objects S is said to generate the category C if the only triangulated subcategory of C that contains S is C itself.

A full triangulated subcategory E is called a thick subcategory of C if E is closed under taking direct summands. That is, if X ⊕ Y ∈ E then X ∈ E.

The triangulated structure on K(A) and D(A) can be described by giving the strict triangles which can be taken to be the same collection of triples for both categories. Given a map u : X → Y in Ch(A) we define T X = X[1] to be the cochain complex whose terms are X[1]^{n} = X^{n+1} and differential d[1]^{n} = −d^{n+1}. We also set T u = u[1] where u[1]^{n} = (−1)^{n}u^{n+1}. This clearly defines T as an automorphism of the categories Ch(A), K(A) and D(A).

We recall the mapping cone construction for a given map u; cone(u) is the complex whose terms are cone(u)^{n} = X^{n+1} ⊕ Y^{n} and the differential is

X^{n+1} ⊕ Y^{n} \underrightarrow{(\begin{matrix}−d^{n+1} & u^{n+1} \\ 0 & d^{n} \end{matrix} )} X^{n+2} ⊕ Y^{n+1}

The mapping cone of a map u fits into the following short exact sequence of cochain complexes in the obvious way:

0 → Y ⟶^{v} cone(u) ⟶^{w} X[1] → 0

The triple (u, v, w) then forms a strict triangle for K(A) and D(A). Furthermore the localising functor Q : K(A) → D(A) is an exact functor of triangulated categories.

2.3 Rings

If R and S are two rings then the well known Morita theory answers the question of when their module categories are equivalent. We recall the main results of the Morita theory (see for example [17]). Theorem 2.3.1. Let R and S be two rings. The following conditions are equivalent

(a) Mod- R and Mod- S are equivalent.

(b) There exists an R-S-bimodule M such that:

(i) The natural map R → End_{S}(M) is an isomorphism of rings. (ii) M as a left R-module or right S-module is a progenerator.

The question of when R and S have module categories that are derived equivalent, that is D^{∗}(Mod- R) and D^{∗}(Mod- S) are equivalent, was answered by Rickard in [28]. Theorem 2.3.2 (Rickard). Let R and S be two rings. The following conditions are equivalent:

(a) K^{−}(proj- R) and K^{−}(proj- S) are equivalent as triangulated categories;

(b) D^{−}(Mod- R) and D^{−}(Mod- S) are equivalent as triangulated categories;

(c) K^{b}(Proj- R) and K^{b}(Proj- S) are equivalent as triangulated categories;

(d) K^{b}(proj- R) and K^{b}(proj- S) are equivalent as triangulated categories;

(e) S is isomorphic to End(T), where T is an object of K^{b}(proj- R) satisfying (i) Hom(T, T [i]) = 0 for i ≠ 0,

(ii) add(T) generates K^{b}(proj- R) as a triangulated category.

An object T that satisfies conditions (i) and (ii) of Theorem 2.3.2(e) is called a tilting complex for R.

2.4 Algebras

We will be interested in some very specific types of rings, viz group algebras for finite groups. In this section we’ll describe four types of algebras, that successively focus on a narrower set of properties, ending with group algebras.

Given a field k we can consider an algebra Λ over k, or k-algebra, as a ring Λ together with an injective ring homomorphism k → Z(Λ) into the centre of Λ.

2.4.1 Finite-dimensional algebras

The finite dimensional k-algebras are k-algebras that are finite dimensional when regarded as k vector spaces. Given such a finite dimensional k-algebra Λ, the category of finitely generated Λ-modules, mod- Λ, is an abelian category with enough injectives as well as projectives.

The functor

D = Hom_{k}(−, k) : mod- Λ → mod- Λ^{o}

induces a duality of categories. We also have a functor Hom_{Λ}(−, Λ) : mod- Λ → mod- Λ^{o}, and given a module (or complex) M we’ll write M^{∨} for the module (or complex) Hom_{Λ}(M, Λ). The functor Hom_{Λ}(−, Λ) induces a duality:

Hom_{Λ}(−, Λ)|_{proj- Λ} : proj- Λ → proj- Λ^{o}

For finite-dimensional algebras the list in Theorem 2.3.2 can be added to (see [29]):

Corollary 2.4.1 (Rickard). If Λ and Γ are finite-dimensional k-algebras, then the following are equivalent:

(a) Λ and Γ are derived equivalent;

(b) K^{−}(proj- Λ) and K^{−}(proj- Γ) are equivalent as triangulated categories;

(c) D^{b}(mod- Λ) and D^{b}(mod- Γ) are equivalent as triangulated categories.

2.4.2 Self-injective algebras

The finite dimensional k-algebra Λ is said to be self-injective if Λ, viewed as a Λ-module, is injective as well as projective. For such algebras the stable module category carries the structure of a triangulated category.

Recall that, for Λ a finite dimensional k-algebra, the stable module category \underline{mod}- Λ has objects that are the same as mod- Λ, but for X, Y in \underline{mod}- Λ,

\underline{Hom}_{Λ}(X, Y) = Hom_{Λ}(X, Y)/ PHom_{Λ}(X, Y),

where PHom_{Λ}(X, Y) = {f ∈ Hom_{Λ}(X, Y) | f factors through a projective Λ-module}. Note that in \underline{mod}- Λ we have P ≅ 0 for P ∈ proj- Λ. Two finite-dimensional k-algebras Λ and Γ are said to be stably equivalent if they have equivalent stable module categories, \underline{mod}- Λ ≅ \underline{mod}- Γ.

Recall that for X ∈ \underline{mod}- Λ the Heller shift ΩX is defined by the short exact sequence

0 → ΩX ⟶ P_{X} ⟶^{p} X → 0

where p : P_{X} → X is the projective cover of X. This defines a functor

Ω : \underline{mod}- Λ → \underline{mod}- Λ

which is an automorphism if Λ is self-injective.

When Λ is self-injective we can describe the triangulated structure of \underline{mod}- Λ by describing the strict triangles on \underline{mod}- Λ. We take T = Ω^{−1} and if u : X → Y is in \underline{mod}- Λ, then with i : X → I_{X} an injective hull, we can construct Z via the pushout diagram:

[figure: X; i; u; I_{X}; Ω^{−1}X; Y; v; Z; w; Ω^{−1}X]

which defines the triple (u, v, w) as a strict triangle.

The triangulated structure on \underline{mod}- Λ is not unrelated to the triangulated structure on D^{b}(Λ). In fact we have the following result due to Rickard (see [32]).

Theorem 2.4.2 (Rickard). Let Λ be a self-injective algebra. The essential image of the natural embedding

K^{b}(proj- Λ) → D^{b}(mod- Λ)

(that is, the full subcategory of D^{b}(mod- Λ) consisting of all objects isomorphic to objects of K^{b}(proj- Λ)) is a thick subcategory. The quotient category

D^{b}(mod- Λ)/K^{b}(proj- Λ)

is equivalent as a triangulated category to the stable module category of Λ.

2.4.3 Symmetric Algebras

Amongst the self-injective algebras are the symmetric algebras which can be characterised in a number of ways: Proposition 2.4.3. The following conditions are equivalent for a finite dimensional kalgebra Λ to be symmetric:

(1) There exists a linear map λ : Λ → k such that (i) ker(λ) contains no left or right ideal. (ii) λ(ab) = λ(ba) for all a, b ∈ Λ.

(2) Λ ≅ DΛ as Λ-bimodules.

(3) D ≅ Hom_{Λ}(−, Λ) as functors from mod- Λ to mod- Λ^{o} or as functors from mod- Λ^{o} to mod- Λ.

(4) For P ∈ proj- Λ and M ∈ mod- Λ we have natural isomorphisms

Hom_{Λ}(P, M) ≅ D Hom_{Λ}(M, P)

An easy consequence of the definition is the following. Lemma 2.4.4. If Λ is a symmetric k-algebra and P an indecomposable projective Λ module then

soc(P) ≅ P/ rad(P)

Proof. By (4) of Proposition 2.4.3 we have Hom_{Λ}(P, soc(P)) ≅ D Hom_{Λ}(soc(P), P) ≅ k

The following definition (see [34]) will be useful. Definition 2.4.5. If Λ and Γ are symmetric algebras, then a Λ-Γ-bimodule M is said to be an exact bimodule, if M is projective as a left Λ-module and as a right Γ-module.

For a Λ-Γ-bimodule M, − ⊗_{Λ} M is a left adjoint to Hom_{Γ}(M, −). If M is projective as a right Γ-module then Hom_{Γ}(M, −) ≅ − ⊗_{Γ} M^{∨}. As Γ is symmetric then, by Proposition 2.4.3 (3), we also have that − ⊗ DM ≅ − ⊗ M^{∨} is a right adjoint to − ⊗_{Λ} M.

If M is an exact bimodule then, repeating the above argument with M^{∨}, we have that − ⊗_{Λ} M is both a left and right adjoint to − ⊗_{Γ} M^{∨}. Similarly if X is a bounded complex of exact Λ-Γ-bimodules then − ⊗_{Λ} X is a left and right adjoint of − ⊗_{Γ} X^{∨}.

For symmetric algebras Theorem 2.3.2 can be strengthened further still (see [29, 30]).

Corollary 2.4.6 (Rickard). If Λ is a symmetric algebra derived equivalent to Γ, then Γ is also a symmetric algebra. Furthermore there exists a bounded complex X of finitely generated exact Λ-Γ-bimodules, such that:

X^{∨} ⊗_{Λ} X ≅ Γ in K^{b}(Γ ⊗ Γ^{o}),
X ⊗_{Γ} X^{∨} ≅ Λ in K^{b}(Λ ⊗ Λ^{o})

So the following are equivalent for symmetric algebras Λ and Γ:

(a) Λ and Γ are derived equivalent.

(b) K^{b}(Λ) and K^{b}(Γ) are equivalent as triangulated categories.

Such a complex X is called a split endomorphism two-sided tilting complex.

2.5 Group algebras

Group algebras kG, where G is a group, will be our focus of interest from now on. Recall that we will always take a group to mean a finite group. If char(k) doesn’t divide the order of G then by Maschke’s theorem kG is a semi-simple algebra i.e. a product of matrix algebras. In that case the study of kG-modules reduces to the well established character theory for finite groups. Our interest will be in modular representations, that is the module category of kG when char(k) = p for some prime p dividing the order of the group G. Also, as we will not be interested in rationality questions, we will revert to our usual assumption that k is algebraically closed of characteristic p > 0.

In order to provide a link between the modular representations and character theory we will work with a p-modular system (K, O, k). That is O is a complete rank one discrete valuation ring with fraction field K and residue field k which will be taken to be algebraically closed. We thus have the following link between the various module categories:

\xymatrix{ & mod- OG \ar[dl]_{−⊗_{O}K} \ar[dr]^{−⊗_{O}k} \\ mod- KG &  & mod- kG}

Here mod- OG is taken to be the OG-modules which as O-modules are free of finite rank.

In general a group algebra kG is symmetric. In fact it is trivial to check that the k linear map λ : kG → k given by

λ(g) = \{ \begin{matrix}1 if g = 1, \\ 0 otherwise \end{matrix}

satisfies condition (1) of Proposition 2.4.3.

When dealing with finite-dimensional algebras it proves useful to break them up into their indecomposable algebra summands, or blocks. For group algebras these blocks have additional structure; there is a p-group associated to a given block, which is well defined up to conjugacy in G, called the defect group of that block. The defect group acts in some ways as a measure of the complexity of the block - a trivial defect group implies the block is semi-simple.

The blocks of an algebra Λ partition the indecomposable modules. If Λ = Λ_{1} ⊕⋯⊕Λ_{n} is a decomposition into blocks and M is an indecomposable Λ-module, then M ≅ Λ_{i}M for some unique i. The module M is said to belong to the block Λ_{i}.

For group algebras there is a canonical indecomposable module, viz the trivial module k. The block to which the trivial module belongs is called the principal block and this has as defect group a Sylow p-subgroup of G.

One of the tools providing a link between properties of a group G and its local subgroups is the Brauer construction. Given a group G and a p-subgroup Q, the Brauer functor is defined by

Br_{Q} : mod- kG → mod- k[N_{G}(Q)]

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

the quotient of the Q-fixed points by the sum over all proper subgroups R < Q, of the relative traces of the R-fixed points.

If M is a k-algebra with G acting via algebra automorphisms then M(Q) is also a k-algebra with N_{G}(Q) acting as algebra automorphisms.

If M is a permutation kG-module, say M = k[Ω] for some G-set Ω, then M(Q) ≅ k[Ω^{Q}].

We can consider kG as a k[G × G]-module with, for x ∈ kG and (g_{1}, g_{2}) ∈ G × G, group action

x.(g_{1}, g_{2}) = g_{2}^{−1}xg_{1}

By restriction kG becomes a k[ΔG]-module where ΔG = {(g, g) ∈ G × G |g ∈ G} is the diagonal embedding. In this case the above two observations show that the map

br_{Q} : (kG)^{ΔQ} → (kG)(ΔQ) = k[C_{G}(Q)]

is an algebra surjection, called the Brauer map. The Brauer map induces a not necessarily surjective map

br_{Q}|_{Z(kG)} : Z(kG) → Z(k[C_{G}(Q)])

With Q = D, a defect group of a block of kG, the Brauer map induces Brauer correspondence. If e is the central idempotent defining the principal block of kG and hence Q a Sylow p-subgroup of G, then br_{Q}(e) is the principal block idempotent for k[C_{G}(Q)].

2.6 Conjectures

In this section we recall some well known conjectures that motivate a lot of current research within modular representation theory. One of the oldest and still outstanding conjectures is Alperin’s weight conjecture, which has acted as a source of inspiration to subsequent conjectures of a more narrow and detailed kind.

Let us recall Alperin’s conjecture. If G is a finite group, a weight for G is represented by pair (Q, S) where Q is a p-subgroup and S is a simple kN_{G}(Q)-module that is projective as a k[N_{G}(Q)/Q]-module. The pairs (Q, S) and (Q′, S′) represent the same weight if Q′ is conjugate to Q and S ≅ S′. We then have (see [1])

Conjecture 2.6.1 (Alperin). The number of weights for G is equal to the number of simple kG-modules.

There is of course a block version of this conjecture. Given a weight represented by (Q, S), S must belong to some block of kN_{G}(Q) and this uniquely identifies a block of kG via Brauer correspondence. The weight (Q, S) is said to belong to this block.

Conjecture 2.6.2 (Alperin-Block version). The number of weights in a block of kG is equal to the number of simples in that block.

Let B = ekG be a block of kG with defect group D and let b = fkH be the Brauer correspondent, where H = N_{G}(D). If we write l(B) to denote the number of ordinary characters belonging to B and k(B) for the number of modular characters (i.e. the number of simple kG-modules), then a relatively easy consequence (see [1]) of Conjecture 2.6.2 is

Consequence 2.6.3. If D is abelian, then l(B) = l(b) and k(B) = k(b).

Attempts to explain, in this abelian defect case, how the above equalities could arise have led to a remarkable series of conjectures. In order to state some of the following conjectures as simply as possible we will, where appropriate, only state them for principal blocks.

We will need the following notation. Irr(G) denotes the set of ordinary irreducible characters. Irr(B) are those irreducible characters that belong to the block B. The abelian group of ℤ-linear sums of elements of Irr(G) will be denoted by ℤ Irr(G). For F = K or O, CF(G, F) = F ⊗_{ℤ}ℤ Irr(G) denotes the F -module of class functions on G. The submodule of class functions that vanish on p-singular elements will be denoted by CF_{p′}(G, F). Similarly we have CF(B, K) = K ⊗_{ℤ} ℤ Irr(B) and CF(B, O) = CF(G, O) ∩ CF(B, K).

We’ll write IBr(G) for the set of irreducible Brauer characters and IBr(B) for those that lie in the block B, adopting the convention that they belong to CF_{p′}(G, O) and CF_{p′}(B, O) respectively and are O-bases. The set of irreducible characters of projective OG -modules will be written as IPr(G), and the subset belonging to the block B will be written as IPr(B). IPr(G) is an O-basis for CF_{p′}^{pr}(G, O) the dual space to the subspace CF_{p′}(O) of CF(G, K) under the usual pairing ⟨ , ⟩.

With this set up we have, for x a p-element in G, the decomposition map

d_{G}^{x} : CF(G, O) → CF_{p′}(C_{G}(x), O)
(d_{G}^{x})(χ)(y) = \{ \begin{matrix}χ(xy) & if y is p-regular, \\ 0 & if y is p-singular \end{matrix}

As IBr(C_{G}(x)) is an O- basis for CF_{p′}(C_{G}(x), O) we can write the map d_{G}^{x} as a linear sum

(d_{G}^{x})(χ) = ∑_{ϕ∈IBr(C_{G}(X))} d_{χϕ}^{x}ϕ,

recovering the so called generalised decomposition numbers {d_{χϕ}^{x}}.

For any group G we’ll write B_{0}(G) for the principal block of G and P for its defect group, a Sylow p-subgroup of G. It follows from Brauer’s second and third main theorems that d_{G}^{x} : CF(B_{0}(G), O) → CF_{p′}(B_{0}(C_{G}(x)), O).

Having set up this notation we recall the definition of an isotypy (see [4, 5]).

Definition 2.6.4. An isotypy I between B_{0}(G) and B_{0}(H) is the datum, for every psubgroup Q of P, of a bijective isometry

I(Q) : ℤ Irr(B_{0}(C_{H}(Q))) ⟶^{∼} ℤ Irr(B_{0}(C_{G}(Q)))

such that the following conditions are fulfilled:

(Equivariance) For all h ∈ H, we have I(Q)^{h} = I(Q^{h}).

(Compatibility) For every subgroup Q of P and every x ∈ P, we still denote by

I(Q) : CF(B_{0}(C_{H}(Q)), K) ⟶^{∼} CF(B_{0}(C_{G}(Q)), K)

the bijective isometry defined by linear extension of I(Q). The following diagram is commutative:

\xymatrix{CF(B_{0}(C_{H}(Q)), K) \ar[d]_{d_{C_{H} (Q)}^{x}} \ar[r]^{I(Q)} & CF(B_{0}(C_{G}(Q)), K) \ar[d]^{d_{C_{G}(Q)}^{x}} \\ CF(B_{0}(C_{H}(Q⟨x⟩)), K) \ar[r]^{I(Q⟨x⟩)} & CF(B_{0}(C_{G}(Q⟨x⟩)), K)}

(Triviality) I(P) is the identity map.

Moreover the isotypy I is said to be normalised if I(Q)(1_{C_{H}(Q)}) = 1_{C_{G}(Q)} for all Q ⊆ P.

An easy consequence of the definition is that I(1) and its inverse R(1) define inverse isometries:

I(1) : CF_{p′}(B_{0}(H), O) → CF_{p′}(B_{0}(G), O)

Using the adjunction isomorphism coming from the pairing ⟨ , ⟩ it follows that

I(1) : CF_{p′}^{pr}(B_{0}(H), O) → CF_{p′}^{pr}(B_{0}(G), O)

It’s also not too hard to see that

I(1) : CF(B_{0}(H), O) → CF(B_{0}(G), O)

is also an isometry. These properties ensure that the isometry

I(1) : ℤ Irr(B_{0}(H)) → ℤ Irr(B_{0}(G))

is what is called a perfect isometry.

Associated to any linear map F : ℤ Irr(H) → ℤ Irr(G) is a character μ_{F} of H × G in the obvious way

μ_{F} = ∑ ζ ⊗ F (ζ)

It follows from definition 2.6.4 that the character μ associated to I(1) is what is called perfect in [4], indeed a map F that is an isometry and induced by a perfect character μ is the definition of a perfect isometry.

We can now state

Conjecture 2.6.5 (Broué-principal block version). If G is a finite group with abelian Sylow p-subgroup P and H = N_{G}(P), then there is a normalised isotypy between B_{0}(G) and B_{0}(H).

One very suggestive way to produce a perfect character is as the character of the bimodule K ⊗_{O} M where M is an OG-OH- bimodule that’s projective as a left and right module, such as would arise from lifting an exact kG-kH-bimodule. The character associated to an isotypy might then be expected to arise as an alternating sum of characters of such bimodules. Such an alternating sum suggests the presence of a complex of bimodules.

We are thus led to the following conjecture, called “Broué’s abelian defect conjecture”, which attempts to explain the presence of a perfect isometry as the “shadow” of some deeper structural connection. The verification of this conjecture for some specific examples will be our main objective.

Conjecture 2.6.6 (Broué). Let G be a finite group, k an algebraically closed field of characteristic p, and ekG a block of kG with abelian defect group D. Let H = N_{G}(D) and fkH be the Brauer correspondent of ekG, then ekG and fkH should be derived equivalent.

Currently this Conjecture is known to be true in the following general cases:

(1) D a cyclic group (see [32, 21, 35])

(2) D ≅ ℤ/2 × ℤ/2 (see [22, 34])

(3) G p-solvable (see [12, 15, 26])

(4) G a connected reductive algebraic group over F_{q} and p divides q − 1 but not the order of the Weyl group (see [27]).

(5) G the symmetric group and D ≅ (ℤ/p)^{r} for r ≤ 5 (see [8, 10], also unpublished work of Rickard)

(6) Principal blocks when D ≅ ℤ/3 × ℤ/3 (see [18])

(7) Principal block of kSL_{2}(p^{n}) (see [9] for n = 2, [33] for p^{n} = 8 and [25] in general)

As well as the above cases, the conjecture is also known to be true for a number of other groups, see [24, 19].

Throughout the rest of this paper, we will often have recourse to discuss blocks ekG, fkH that satisfy the assumptions of Conjecture 2.6.6. We will simply say “blocks as in Conjecture 2.6.6” to indicate these assumptions.

An isotypy is much more than just a perfect isometry between G and H. It provides a compatible family of perfect isometries between local subgroups. In an attempt to find the correct structural origin for isotypies Rickard was led to the notion of a splendid equivalence (see [30]).

Let us assume that ekG and fkH are blocks as in Conjecture 2.6.6, with common abelian defect group D. There are a number of different naming conventions in the literature for various complexes that induce various equivalences. In order to provide some consistency to this thesis we will make the following definition.

Definition 2.6.7. Suppose that a bounded complex X of ekG-fkH-bimodules has terms that, as k[G × H]-modules, are direct summands of relatively ΔD-projective permutation modules. Let η_{H} and η_{G} be the units of the adjoint pairs (− ⊗_{fkH} X^{∨}, − ⊗_{ekG} X) and (− ⊗_{ekG} X, − ⊗_{fkH} X^{∨}) respectively and let ϵ_{H} and ϵ_{G} be their respective counits. If the compositions:

fkH ⟶^{η_{H}} X^{∨} ⊗_{ekG} X ⟶^{ϵ_{H}} fkH,

ekG ⟶^{η_{G}} X ⊗_{fkH} X^{∨} ⟶^{ϵ_{G}} ekG

are isomorphisms of complexes of bimodules then the complex X is said to be a splendid complex.

It is immediate that the terms of a splendid complex are exact bimodules. This definition preserves most of the origin of the term splendid (see [30]) as “SPLit-ENDomorphism two-sided complex of summands of permutation modules Induced from Diagonal subgroups”.

For simplicity we’ll now assume for the rest of this section that G has an abelian Sylow p-subgroup so that in Conjecture 2.6.6 the groups G and H share a common Sylow p-subgroup. We’ll also assume that ekG and fkH are blocks as in Conjecture 2.6.6.

Definition 2.6.8. A splendid tilting complex of ekG-fkH-bimodules is a splendid complex X such that

X^{∨} ⊗_{ekG} X ≅ fkH in K^{b}(fkH ⊗ fkH^{o}),
X ⊗_{fkH} X^{∨} ≅ ekG in K^{b}(ekG ⊗ ekG^{o})

A derived equivalence between ekG and fkH that is induced by tensoring with a splendid tilting complex is called a splendid equivalence.

These are the definitions as given in [30]. Furthermore the definition of a splendid complex has the virtue that it’s compatible with the definition of a splendid stable equivalence as described in [34] (see Section 3.3).

The following Theorem due to Rickard (see [30]) provides the appropriate structural counterpart to an isotypy.

Theorem 2.6.9 (Rickard). Suppose G has an abelian Sylow p-subgroup, ekG is a block of kG with defect group D and fkH is its Brauer correspondent. If X is a splendid tilting complex of ekG-fkH-bimodules, then for every subgroup Q ≤ D, X(ΔQ) is a splendid tilting complex of br_{Q}(e)k[C_{G}(Q)]- br_{Q}(f)k[C_{H}(Q)]- bimodules

It should be noted that X(ΔQ) is actually a complex of kN_{G×H}(ΔQ)-modules, which by restriction is a complex of kC_{G}(Q)-kC_{H}(Q)-bimodules.

Finally we can state, for abelian Sylow p-subgroups, the strongest form of the current conjectures.

Conjecture 2.6.10 (Broué, Rickard). If G is a finite group with abelian Sylow psubgroup P and ekG is a block of kG with defect group D and fkH is its Brauer correspondent, where H = N_{G}(D), then there is a splendid equivalence between ekG and fkH.

18

Obtaining Stable Equivalences

3.1 Introduction

Given two finite dimensional k-algebras that are stably equivalent, it seems to be hard to describe the nature of the equivalence, as there is no obvious candidate to act as a progenerator imitating the classical Morita theory. In this chapter we’ll focus on stable equivalences with additional properties.

In Section 3.2 we describe stable equivalences of Morita type and describe a useful class of such equivalences for group algebras in the so called T.I. case.

In Section 3.3 we deal with blocks of group algebras as in Conjecture 2.6.6 and refine stable equivalences of Morita type to stable equivalences of splendid type. We’ll then briefly describe a construction of such an equivalence, due to Rouquier, when the defect groups of the blocks involved are elementary abelian of rank 2.

3.2 Stable equivalence of Morita type

Stable equivalences of Morita type were introduced by Broué (see for example [6]). Suppose Λ and Γ are finite-dimensional k-algebras.

Definition 3.2.1. A stable equivalence \underline{mod}- Λ ≅ \underline{mod}- Γ is said to be of Morita type if it is induced by an exact functor F : mod- Λ → mod- Γ.

Setting M = F Λ as a Λ-Γ-bimodule it’s easy to see that F is isomorphic to − ⊗_{Λ} M and exactness shows M is projective as a left Λ-module, whilst F Λ being projective shows M is projective as a right Γ-module. The functor − ⊗_{Γ} Hom_{Γ}(M, Γ) induces an inverse stable equivalence.

More generally suppose M is Λ-Γ-bimodule that is projective as a left and right module, and N = Hom_{Γ}(M, Γ). If F and G denote the functors induced by these bimodules:

F = − ⊗_{Λ}M : mod- Λ → mod- Γ,
G = − ⊗_{Γ}N : mod- Γ → mod- Λ

then F is a left adjoint to G.

If we now also assume that F and G are both left and right adjoints of each other and that Λ is a block then the following Lemma (see [30]) shows that, in checking whether F induces a stable equivalence of Morita type, we only have to check half of the equivalence.

Lemma 3.2.2. If G ∘ F ≅ Id in \underline{mod}- Λ then F induces a stable equivalence of Morita type.

Proof. We have to establish that F ∘ G ≅ Id in \underline{mod}- Γ.

Let η and ϵ be the unit and counit of the adjoint pair (F, G) and η′ and ϵ′ be the unit and counit of the adjoint pair (G, F), so we have

η : Id_{mod- Λ} → G ∘ F,  ϵ : F ∘ G → Id_{mod- Γ}
η′ : Id_{mod- Γ} → F ∘ G,  ϵ′ : G ∘ F → Id_{mod- Λ}

By Theorem IV.1.1 in [36] we have that Gϵ ∘ ηG and ϵ′G ∘ Gη′ are identities. In terms of bimodules then

ϵ ⊗ 1_{N} : N ⊗_{Λ} M ⊗_{Γ} N → N

is a split surjection and

η′ ⊗ 1_{N} : N → N ⊗_{Λ} M ⊗_{Γ} N

is a split injection.

We are assuming that G ∘ F ≅ Id_{\underline{mod}- Λ} or in terms of bimodules M ⊗_{Γ} N ≅ Λ ⊕ X for some projective Λ-bimodule X. So the maps induced by ϵ⊗1_{N} and η′ ⊗1_{N} in \underline{mod}- Λ⊗Γ^{o} are isomorphisms. As Γ is a block the map ϵη′ ∈ End_{mod- Γ⊗Γ^{o}}(Γ) is either an isomorphism or nilpotent. If ϵη′ is nilpotent then so is (ϵ⊗1_{N})∘(η′ ⊗1_{N}) = ϵη′ ⊗1_{N}, but this induces an isomorphism in \underline{mod}- Λ ⊗ Γ^{o}. Thus ϵη′ must be an isomorphism, or in terms of bimodules N ⊗_{Λ} M ≅ Γ ⊕ Y for some Γ-bimodule Y. It only remains to show that Y is a projective bimodule.

We have that

N ⊗_{Λ} M ⊗_{Γ} N ⊗_{Λ} M ≅ (Γ ⊕ Y) ⊗_{Γ} (Γ ⊕ Y) ≅ Γ ⊕ 2Y ⊕ Y ⊗_{Γ} Y

But we also have that

N ⊗_{Λ} M ⊗_{Γ} N ⊗_{Λ} M ≅ N ⊗_{Λ} (Λ ⊕ X) ⊗_{Λ} M ≅ Γ ⊕ Y ⊕ N ⊗_{Λ} X ⊗_{Λ} M

Here N ⊗_{Λ} X ⊗_{Λ} M is a projective Γ-bimodule and the Krull-Schmidt theorem shows Y is a summand of it.

If Λ and Γ are symmetric algebras then, by Proposition 2.4.3, N ≅ M^{∨} and F and G are both left and right adjoints of each other.

For blocks of group algebras Lemma 3.2.2 provides a quick way of establishing a stable equivalence of Morita type in the T.I. (Trivial Intersection) case.

A subgroup K ≤ G is said to be a trivial intersection (T.I.) subgroup if K^{g} ∩ K = {1} for g ∉ N_{G}(K).

If ekG and fkH are as in Conjecture 2.6.6 and the defect group D = P is a T.I. Sylow p-subgroup then setting M = fkGe, we have the following well known result.

Proposition 3.2.3. The functor − ⊗_{fkH} M induces a stable equivalence of Morita type between fkH and ekG.

Proof. Clearly M is an exact fkH-ekG-bimodule. By Lemma 3.2.2 we only need establish that M ⊗_{ekG} M^{∨} ≅ fkH ⊕ X where X is a projective fkH-bimodule.

Viewing kG as a k[G × G]-module as in Section 2.5, we can restrict kG to a k[H × H]- module and apply the Mackey decomposition formula

kG ≅ k ↑_{ΔG}^{G×G}↓_{H×H} ≅ kH ⊕ ⨁_{L} k ↑_{L}^{H×H}= kH ⊕ X′

where L ranges over the subgroups {H ∩ H^{g}|g ∈ H\G/H, g ∉ H}. As P is T.I. these subgroups are p′-subgroups and so X′ is a projective kH-bimodule. As e and f are Brauer correspondents (1 − e)f = 0 and so

M ⊗_{ekG} M^{∨} ≅ fkGf ≅ fkH ⊕ X

where X = fX′f is a projective fkH-bimodule.

3.3 Stable equivalence of splendid type

If ekG and fkH are as in Conjecture 2.6.6 then a consequence of the conjecture being true would be that ekG and fkH are stably equivalent of Morita type. If the stronger Conjecture 2.6.10 were true, then the terms of a complex X inducing the splendid equivalence are of a special kind, viz they are summands of permutation modules. However the bimodule that induces the stable equivalence will in general no longer be a summand of a permutation module.

From this perspective it would be more natural to look for a stable equivalence induced by an exact functor F : K^{b}(ekG) → K^{b}(fkH). We make the following definition.

Definition 3.3.1. ekG and fkH are said to be stably equivalent of splendid type, or splendid stably equivalent, if there exists a splendid complex X of ekG-fkH-bimodules such that in the decompositions:

X^{∨} ⊗_{ekG} X = fkH ⊕ Z
X ⊗_{fkH} X^{∨} = ekG ⊕ W

the complexes Z and W are homotopic to bounded complexes of projective bimodules.

Proposition 3.2.3 shows that in the T.I. Sylow p-subgroup case ekG and fkH are stably equivalent of splendid type. The splendid complex inducing the splendid stable equivalence being ekGf viewed as a complex concentrated in degree zero.

We now restrict to the situation where ekG and fkH are as in Conjecture 2.6.6 and the defect group D = P is a Sylow p-subgroup. The following theorem, due to Rouquier [34], provides the pivotal link between a global stable equivalence and local splendid equivalences.

Theorem 3.3.2. Let C be a splendid complex of ekG-fkH bimodules. The following are equivalent:

(i) C induces a splendid stable equivalence between ekG and fkH.

(ii) For every non-trivial subgroup Q of D, C(ΔQ) induces a splendid equivalence between e_{Q}kC_{G}(Q) and f_{Q}kC_{H}(Q).

Here e_{Q} (resp. f_{Q}) is the image of e (resp. f) under the Brauer map br_{Q} : Z(kG) → Z(kC_{G}(Q)) (resp. br_{Q} : Z(kH) → Z(kC_{H}(Q)). When e and f are principal block idempotents, so are e_{Q} and f_{Q}.

In [34] Rouquier succeeded in using the known splendid equivalences for blocks with cyclic defect to construct a complex C that satisfies condition (ii) of Theorem 3.3.2, hence induces a splendid stable equivalence, when the defect group P ≅ C_{p} × C_{p}.

Let us recall some of the details of Rouquier’s construction (see [34] for full details). For simplicity we’ll assume that e and f are principal blocks.

Given a complex C of ekG-fkH-bimodules as in Theorem 3.3.2 then C(ΔQ) is actually a complex of kN_{G×H}(ΔQ)-modules which is viewed, through restriction, as a complex of kC_{G}(Q)-kC_{H}(Q)-bimodules.

So the first stage in the construction is to construct a complex of kN_{G×H}(ΔQ)-modules whose restriction to C_{G}(Q) × C_{H}(Q) is a splendid tilting complex.

To do this we’ll use the known splendid equivalences for cyclic defect groups. Now e_{Q} and f_{Q} both have defect group P and are lifts of blocks e′_{Q} and f′_{Q}, where e′_{Q} and f′_{Q} are blocks of k[C_{G}(Q)/Q] and k[C_{H}(Q)/Q] respectively. The common defect group of e′_{Q} and f′_{Q} is P/Q, a cyclic p-group. So, by the cyclic defect case, we have a two term splendid tilting complex

C′ = 0 → N′_{Q} → e′_{Q}k[C_{G}(Q)/Q]f′_{Q} → 0

concentrated in degrees 0 and −1.

If we let X = N_{G}(Q)/Q and Y = N_{X}(P/Q) = N_{H}(Q)/Q then

X/(C_{G}(Q)/Q) = Y/(C_{H}(Q)/Q) = F

is a p′-group. Let the images in F of g ∈ X and h ∈ Y be ḡ and h̄ respectively and let Δ = {(g, h) ∈ X × Y |(ḡ, h̄) ∈ ΔF } then, by the construction of C′, it can be shown that C′ extends to a complex of Δ-modules.

If we now set X̃ = N_{G}(Q) and Ỹ equal to the inverse image in X̃ of Y so that Ỹ = N_{H}(Q), then X̃ × Ỹ /Q × Q = X×Y. The inverse image of Δ in X̃×Ỹ is Δ̃ = N_{G×H}(ΔQ).

It is possible to find a projective k[Δ̃/ΔQ]-module Ñ_{Q} such that k ⊗_{Q×Q} Ñ_{Q} = N′_{Q} and construct a complex of kΔ̃-modules

C̃_{Q} = Ñ_{Q} → e_{Q}kC_{G}(Q)f_{Q}

such that C_{Q} = C̃_{Q} ↓_{C_{G×H}(ΔQ)}^{N_{G×H}(ΔQ)} is a splendid tilting complex that induces a splendid equivalence between e_{Q}kC_{G}(Q) and f_{Q}kC_{H}(Q).

The final stage in the construction glues all these local complexes together to produce a complex C satisfying the conditions in Theorem 3.3.2:

C = ⨁_{Q} Ñ_{Q} ↑_{N_{G×H}(ΔQ)}^{G×H} ⟶^{d} ekGf  (3.1)

where the sum is over all cyclic p-subgroups Q < P, up to conjugacy in H.

24

Obtaining Derived Equivalences

4.1 Introduction

Suppose Λ and Γ are finite-dimensional symmetric k-algebras that are known to be stably equivalent of Morita type. In this chapter we’ll discuss how this stable equivalence can be lifted to a derived equivalence.

In Section 4.2 we state a theorem, due to Rickard, for lifting a stable equivalence of Morita type to a derived equivalence. This theorem is central to our approach to verifying Conjecture 2.6.6 for our examples. In Section 4.3 we describe the use of computer calculations in checking some of the conditions in Rickard’s theorem. In Section 4.4 we describe how, in applications of Rickard’s theorem to verifying Conjecture 2.6.6 for a specific example, knowing that the stable equivalence is of splendid type can strengthen the derived equivalence to a splendid equivalence, thus verifying Conjecture 2.6.10 too.

Finally in Section 4.5 we summarise the method we use for verifying Conjecture 2.6.6 (even Conjecture 2.6.10) for all our examples.

4.2 Derived equivalence

As described in Subsection 2.4.2, a derived equivalence between two finite-dimensional symmetric k-algebras Λ and Γ implies that they are stably equivalent. However not every stable equivalence, even of splendid type, comes from a derived equivalence.

If one suspects that there should be a derived equivalence, as in Conjecture 2.6.6, then lifting a stable equivalence to a derived equivalence is difficult.

The following theorem, due to Rickard [31] and based on a variation of a method due to Okuyama [24], provides a method to achieve this lifting.

Theorem 4.2.1 (Rickard). Let Λ and Γ be finite-dimensional symmetric k-algebras, let

F : mod-Γ → mod-Λ be an exact functor inducing a stable equivalence (i.e. a stable equivalence of Morita type), and let {S_{1}, …, S_{r}} be a set of representatives for the isomorphism classes of simple Γmodules.

If there are objects X_{1}, …, X_{r} of D^{b}(Λ) such that, for each 1 ≤ i ≤ r, X_{i} is stably isomorphic to F (S_{i}), and such that:

(a) Hom_{D^{b}(Λ)}(X_{i}, X_{j}[m]) = 0 for m < 0,

(b) Hom_{D^{b}(Λ)}(X_{i}, X_{j}) = \{ \begin{matrix}0 & if i ≠ j \\ k & if i = j \end{matrix} , and

(c) X_{1}, …, X_{r} generate D^{b}(Λ) as a triangulated category,

then D^{b}(Λ) and D^{b}(Γ) are equivalent as triangulated categories.

Once candidate objects {X_{i}} have been constructed the difficulty in applying Theorem 4.2.1 rests mainly in verifying conditions (a) and (b).

For some of our examples, such as the non-principal block of kSL_{2}(p^{2}) in Chapter 5, where we apply Theorem 4.2.1 to verifying Conjecture 2.6.6, we’re able to construct objects {X_{i}} that have a sufficiently simple structure to enable the calculations for parts (a) and (b) to be carried out directly in the derived category. However, in general, calculating homomorphisms in derived categories is quite hard and does not lend itself to implementation on a computer. Fortunately it will be sufficient to calculate homomorphisms in the homotopy category instead.

Lemma 4.2.2. Let Λ be a self-injective k-algebra and X, Y objects in K^{−}(Λ) (we’ll identify X, Y with their images in D^{b}(Λ)) such that the natural map Hom_{K^{−}(Λ)}(X, Y [m]) → Hom_{D^{b}(Λ)}(X, Y [m]) is an isomorphism for m ≤ 0 and injective for m = 1.

If X′ and Y′ are objects in K^{−}(Λ) that fit into triangles

X → X′ → Q → X[1]  (4.1)
Y → Y′ → P → Y [1]  (4.2)

where Q and P are objects in K^{b}(proj- Λ), then the natural map Hom_{K^{−}(Λ)}(X′, Y′[m]) → Hom_{D^{b}(Λ)}(X′, Y′[m]) is an isomorphism for m ≤ 0 and injective for m = 1.

Proof. To avoid unnecessary notational clutter, we’ll write Hom_{K^{−}(Λ)}(−, −) as _{K}(−, −), and Hom_{D^{b}(Λ)}(−, −) as _{D}(−, −).

Applying the contravariant functors _{K}(−, Y [m]) and _{D}(−, Y [m]) to triangle (4.1) we obtain the following ladder diagram with exact rows:

K(X[1], Y [m]) ⟶ _{K}(Q, Y [m]) ⟶ _{K}(X′, Y [m]) ⟶ _{K}(X, Y [m]) ⟶ _{K}(Q[−1], Y [m])
↓^{α}  ↓^{β}  ↓^{γ}  ↓^{δ}  ↓^{β}
D(X[1], Y [m]) ⟶ _{D}(Q, Y [m]) ⟶ _{D}(X′, Y [m]) ⟶ _{D}(X, Y [m]) ⟶ _{D}(Q[−1], Y [m])

The maps β are isomorphisms for all m by standard properties of derived categories, while for m ≤ 0, α and δ are isomorphisms by assumption. Hence by the 5-lemma γ is an isomorphism for m ≤ 0. When m = 1, α is an isomorphism and δ is injective by assumption and a diagram chase shows γ is injective.

We can now apply the covariant functors_{K}(X′, −) and_{D}(X′, −) to triangle (4.2) giving the ladder diagram with exact rows:

K(X′, P [m − 1]) ⟶ _{K}(X′, Y [m]) ⟶ _{K}(X′, Y′[m]) ⟶ _{K}(X′, P [m]) ⟶ _{K}(X′, Y [m + 1])
↓^{ζ}  ↓^{γ}  ↓^{η}  ↓^{ζ}  ↓^{γ}
D(X′, P [m − 1]) ⟶ _{D}(X′, Y [m]) ⟶ _{D}(X′, Y′[m]) ⟶ _{D}(X′, P [m]) ⟶ _{D}(X′, Y [m + 1])

As Λ is self-injective, it follows from standard properties of derived categories that the maps ζ are isomorphisms for all m. For m < 0 the γ are isomorphisms and hence by the 5-lemma so is η. When m = 0 we can replace _{D}(X′, Y [1]) by im(γ) and again apply the 5-lemma to conclude that η is an isomorphism. For m = 1 the injectivity of the γ on the left in the ladder diagram, together with a diagram chase, shows η is injective.

We will apply Lemma 4.2.2 to parts (a) and (b) of Theorem 4.2.1 when Γ = ekG and Λ = fkH are as in Conjecture 2.6.6 and the stable equivalence is of splendid type, induced by a complex C of ekG-fkH-bimodules. In the notation of Lemma 4.2.2 we’ll take X = S_{i} ⊗_{ekG} C and Y = S_{j} ⊗_{ekG} C.

In applying Lemma 4.2.2 to Theorem 4.2.1 one must be careful as to which choice of complex X_{i}, up to isomorphism in the homotopy category, one chooses as a representative from the isomorphism class of X_{i} in the derived category. In all our cases where computer calculations were used the complexes {X_{i}} we construct fit into triangles (4.1) and (4.2) with X′ = X_{i} and Y′ = X_{j} (see (3) of Section 4.5). That the remaining conditions of Lemma 4.2.2 hold follows from the commutative diagram

[figure: Hom_{K^{b}(fkH)}(S_{i} ⊗_{ekG} C, S_{j} ⊗_{ekG} C[m]); f; Hom_{D^{b}(fkH)}(S_{i} ⊗_{ekG} C, S_{j} ⊗_{ekG} C[m]); Hom_{K^{b}(ekG)}(S_{i} ⊗_{ekG} C ⊗_{fkH} C^{∨}, S_{j}[m]); g; Hom_{D^{b}(ekG)}(S_{i} ⊗_{ekG} C ⊗_{fkH} C^{∨}, S_{j}[m])]

The vertical maps are the adjunction isomorphisms and as C ⊗_{fkH} C^{∨} ≅ ekG ⊕ P in K^{b}(ekG ⊗ ekG^{o}), where P is a bounded complex of projective bimodules, g is trivially an isomorphism for m ≤ 0 and injective for m = 1 and so the same is true for f.

4.3 Computer calculations

For most of our examples, including all those in Chapter 6, the objects {X_{i}} that we construct will be sufficiently complicated that verifying (a) and (b) of Theorem 4.2.1 directly in the derived category will not be feasible. However thanks to Lemma 4.2.2 we can perform these calculations in the homotopy category. In addition we can use the fact that, for X and Y complexes of Λ-modules,

Hom_{K^{b}(Λ)}(X, Y [m]) = H^{m}(Hom_{Λ}^{•}(X, Y))

Together these observations provide an explicit computational approach to verifying conditions (a) and (b) of Theorem 4.2.1 that can be implemented on a computer.

All our computer calculations used to verify the conditions in Theorem 4.2.1 were performed using the computer algebra package MAGMA [23]. As well as using standard MAGMA functions, routines were written in the MAGMA code (see Appendix E) to take complexes X and Y of modules for a group G, stored as lists of differentials, and compute the double complex Hom_{kG}(X_{i}, Y_{j}), its total complex Hom_{kG}^{•}(X, Y) and the cohomology of this total complex.

In fact for all our examples relying upon computer calculations, the complexes {X_{i}} found satisfying the conditions in Theorem 4.2.1 also satisfied the stronger conditions

Hom_{K^{b}(kG)}(X_{i}, X_{j}[m]) = \{ \begin{matrix}k & if i = j and m = 0, \\ 0 & otherwise. \end{matrix}  (4.3)

This would suggest that we have found the images of simple modules under a splendid equivalence, or at least a homotopy equivalence, and as we’ll show in the next section this is indeed the case.

4.4 From derived to splendid equivalence

In all our examples where we verify Conjecture 2.6.6 by applying Theorem 4.2.1 we have an additional piece of information to start with; the stable equivalence is not just of Morita type but is of splendid type. This is sufficient by Theorem 4.4.2 to verify the stronger Conjecture 2.6.10 for all our examples as well.

Proposition 4.4.1. Suppose that Λ and Γ are finite-dimensional symmetric k-algebras and C is a complex of exact Λ-Γ-bimodules such that

C^{∨} ⊗_{Λ} C ≅ Γ ⊕ Z in K^{b}(Γ ⊗ Γ^{o}),
C ⊗_{Γ} C^{∨} ≅ Λ ⊕ W in K^{b}(Λ ⊗ Λ^{o})

where Z and W are bounded complexes of projective bimodules, so that − ⊗_{Λ} C induces a stable equivalence between Λ and Γ.

Suppose F : D^{b}(Λ) → D^{b}(Γ) is an equivalence that induces a stable equivalence that agrees on each simple Λ-module, up to isomorphism, with that induced by C, then there is complex X of Λ-Γ-bimodules that induces a homotopy equivalence between Λ and Γ and such that X^{i} = C^{i} ⊕ P^{i} for each i where P^{i} is a projective bimodule. Moreover, for each Λ-module Y, Y ⊗_{Λ} X ≅ F Y in D^{b}(Γ).

Proof. By the results in [29] we can replace the equivalence F by an equivalence induced by a standard two-sided tilting complex Δ′ and , by taking a truncated projective resolution if necessary, we can assume that

Δ′ = 0 → N → R^{−n+1} → R^{−n+2} → ⋯

where the R^{i} are projective bimodules and N is an exact bimodule. The exact functor − ⊗_{Λ} Ω^{−n}(N) induces a stable equivalence such that F Y is stably isomorphic to Y ⊗_{Λ} Ω^{−n}(N) for each Λ-module Y.

Let T be a projective bimodule resolution of C. Truncating T, for sufficiently large m we have the complex

Σ = 0 → M → T^{−m+1} → T^{−m+2} → ⋯

where M is an exact bimodule and the T^{i} are projective bimodules. The exact functor − ⊗_{Λ} Ω^{−m}(M) induces a stable equivalence isomorphic to that induced by C.

By taking m = n sufficiently large we have a stable auto-equivalence on \underline{mod}- Λ induced by − ⊗_{Λ} Ω^{−n}(M) ⊗_{Γ} Ω^{n}(N^{∨}). By assumption this sends simple Λ-modules to isomorphic copies of themselves. By a theorem due to Linckelmann (see [20]), this stable equivalence is actually a Morita auto-equivalence and so is induced by the Λ-bimodule _{ϕ}Λ, which is Λ with Λ action on the left twisted by ϕ ∈ Aut(Λ). So we have M ≅ _{ϕ}N and for each Λ-module Y, Y ⊗_{Λ} M ≅ Y ⊗_{Λ} N. Replacing Δ′ by the two-sided tilting complex Δ = _{ϕ}Λ ⊗_{Λ} Δ′, we have complexes

Δ = 0 → M → L^{−n+1} → L^{−n+2} → ⋯
Σ = 0 → M → T^{−n+1} → T^{−n+2} → ⋯

with L^{i} projective bimodules and for each Λ-module Y, F Y ≅ Y ⊗_{Λ} Δ.

As Σ is a truncated resolution of C, we have a map f : Σ → C in K^{b}(Γ ⊗ Λ^{o}). By the axioms of a triangulated category we have the following map of triangles

[figure: T̃; i; Σ; f; M[n]; h; T̃[1]; T̃; fi; C; R; T̃[1]]

Here R is the cone of the map fi and T̃ →^{i} Σ is the obvious map where

T̃ = 0 → T^{−n+1} → T^{−n+2} → ⋯

Viewed as a map of triangles in D^{b}(Γ ⊗ Λ^{o}), f is an isomorphism and, again by the axioms of a triangulated category, so is h. In other words R[−n] is a right resolution of M.

If we set

L̃ = 0 → L^{−n+1} → L^{−n+2} → ⋯

we have a complex of injective bimodules. The identity map M → M lifts to a map r : R[−1] → L̃ in K^{b}(Γ ⊗ Λ^{o}) which we can complete to a triangle:

R[−1] →^{r} L̃ → X → R

defining an object X in K^{b}(Γ ⊗ Λ^{o}). By construction X^{n} = C^{n} ⊕ P^{n} for some projective bimodule P^{n}, for each n and X ≅ C in K^{b}(Γ ⊗ Λ^{o})/K^{b}(proj- Γ ⊗ Λ^{o}).

We have to check that X induces a homotopy equivalence between Λ and Γ. Taking − ⊗_{Γ} X^{∨} as left adjoint to − ⊗_{Λ} X we can consider the cone V of the counit ϵ,

X^{∨} ⊗_{Λ} X →^{ϵ} Γ → V  (4.4)

In K^{b}(Γ ⊗ Γ^{o})/K^{b}(proj- Γ ⊗ Γ^{o}) we have X^{∨} ⊗_{Λ} X ≅ C^{∨} ⊗_{Λ} C ≅ Γ and so V is homotopic to a complex of projective Γ-bimodules. So we only have to show V is acyclic, but X ≅ Δ in D^{b}(Γ ⊗ Λ^{o}) and so the long exact sequence of cohomology applied to (4.4) now shows V is acyclic as required.

We are particularly interested in the situation where Λ = ekG and Γ = fkH are as in Conjecture 2.6.6 and for which we have a splendid stable equivalence. Combining Theorem 4.2.1 with Proposition 4.4.1 we have:

Theorem 4.4.2. Suppose that C is a complex of ekG-fkH-bimodules that induces a splendid stable equivalence between ekG and fkH and let {S_{1}, …, S_{r}} be a set of representatives for the isomorphism classes of simple ekG-modules. If there are objects X_{1}, …, X_{r} of D^{b}(fkH) such that, for each 1 ≤ i ≤ r, X_{i} is stably isomorphic to S_{i} ⊗_{ekG} C, and such that:

(a) Hom_{D^{b}(fkH)}(X_{i}, X_{j}[m]) = 0 for m < 0,

(b) Hom_{D^{b}(fkH)}(X_{i}, X_{j}) = \{ \begin{matrix}0 & if i ≠ j \\ k & if i = j \end{matrix} , and

(c) X_{1}, …, X_{r} generate D^{b}(fkH) as a triangulated category,

then there is a splendid tilting complex X that lifts C and induces a splendid equivalence between ekG and fkH such that, for each 1 ≤ i ≤ r, S_{i} ⊗_{ekG} X ≅ X_{i} in D^{b}(fkH).

For our examples relying upon computer calculations the construction of objects {X_{i}} satisfying the conditions of Theorem 4.4.2 will be such that each X_{i} will not only be stably isomorphic to S_{i} ⊗_{ekG} C but also isomorphic in K^{b}(fkH)/K^{b}(proj- fkH), see (2) of Section 4.5. Under these circumstances we have the following.

Corollary 4.4.3. With the notation of Theorem 4.4.2, let objects {X_{i}} satisfy the conditions of Theorem 4.4.2. If X_{i} ≅ S_{i} ⊗_{ekG} C in K^{b}(fkH)/K^{b}(proj- fkH) then S_{i} ⊗_{ekG} X ≅ X_{i} in K^{b}(fkH).

Proof. We’re given that:

(a) X_{i} ≅ S_{i} ⊗_{ekG} X in D^{b}(fkH),

(b) X_{i} ≅ S_{i} ⊗_{ekG} C in K^{b}(fkH)/K^{b}(proj- fkH)

Setting Y_{i} = X_{i} ⊗_{fkH} X^{∨} and using X ≅ C in K^{b}(fkH ⊗ ekG^{o})/K^{b}(proj- fkH ⊗ ekG^{o}) these become:

(a′) Y_{i} ≅ S_{i} in D^{b}(ekG),

(b′) Y_{i} ≅ S_{i} in K^{b}(ekG)/K^{b}(proj- ekG)

The latter condition says that we have, in K^{b}(ekG), triangles:

Z → Y_{i} → P → Z[1],

Z → S_{i} ⟶^{u} Q → Z[1]

where P and Q are bounded complexes of projectives. The second triangle shows that Z ≅ cone(u)[−1]. The first triangle now shows that Y_{i}, as a complex, has projective modules in non-zero degrees. It now follows from (a′) that Y_{i} ≅ S_{i} in K^{b}(ekG) and hence X_{i} ≅ S_{i} ⊗_{ekG} X in K^{b}(fkH).

4.5 The method

In this section we pull together earlier results to outline the method we will use to verify Broué’s Conjecture 2.6.6 and even its refinement Conjecture 2.6.10 for all our examples.

We suppose that ekG and fkH are blocks as in Conjecture 2.6.6 and we suppose that {S_{1}, …, S_{r}} is a complete list of representatives for the isomorphism classes of simple ekG-modules.

We also assume that we have, thanks to the results in Chapter 3, a complex C of ekGfkH-bimodules that induces a stable equivalence of splendid type. We don’t necessarily need C to be calculated explicitly.

We can summarise the steps we use to verify Conjecture 2.6.6 and Conjecture 2.6.10 for our examples as:

(1) For each 1 ≤ i ≤ r, calculate explicitly S_{i} ⊗_{ekG} C as a complex of fkH-modules.

(2) For each 1 ≤ i ≤ r, define a complex X_{i} of fkH-modules such that X_{i} is stably isomorphic to S_{i} ⊗_{ekG} C.

In fact for all our examples each X_{i} will fit into a triangle, in K^{b}(fkH),

P_{i} → S_{i} ⊗_{ekG} C → X_{i} → P_{i}[1]  (4.5)

where P_{i} is a bounded complex of projective fkH-modules. After completing step (3) it will follow from Corollary 4.4.3 that the objects {X_{i}} are images in the homotopy category of simple modules under a splendid equivalence.

However for the non-principal block of kSL_{2}(p^{2}) we will take as our definition of {X_{i}} objects that are isomorphic in D^{b}(fkH) to objects defined by triangles (4.5). This will simplify the calculations for step (3).

It is of course immediate from triangle (4.5) that X_{i} is stably isomorphic to S_{i}⊗_{ekG}C.

So this stage essentially involves, for each 1 ≤ i ≤ r, a careful choice of complex P_{i} and map P_{i} → S_{i} ⊗_{ekG} C so that stage (3) will be successful.

(3) Verify conditions (a), (b) and (c) of Theorem 4.4.2 for the objects {X_{i}}.

SL_{2}(p^{n})

5.1 Introduction

In this chapter we discuss an important example where Conjecture 2.6.6 is applicable, viz the group algebra kSL_{2}(p^{n}) in defining characteristic. We verify Conjecture 2.6.6 for the non-principal blocks of kSL_{2}(p^{2}) as well as for the principal blocks of kSL_{2}(32) and kSL_{2}(27).

Prior to the work described here, Chuang in [9] verified Conjecture 2.6.10 for the principal blocks of kSL_{2}(p^{2}). His approach for the principal blocks of kSL_{2}(p^{2}) has acted as a guide to ours for the non-principal blocks of kSL_{2}(p^{2}), and our result both mirrors and complements that in [9].

After the work described in this chapter was finished, Okuyama in [25] succeeded in verifying Conjecture 2.6.10 for the principal blocks of kSL_{2}(p^{n}). His approach uses different methods to ours and whilst ingenious, it seems hard to obtain concrete information about the derived equivalences he arrives at.

Although our approach will not extend to cover kSL_{2}(p^{n}) in general, it does have the merit of obtaining such concrete information about the derived equivalences, viz an explicit description of the images of simple modules under the equivalence.

Before proceeding further let us recall some elementary facts. The special linear group G = SL_{2}(p^{n}) has an elementary abelian p-group of rank n as a Sylow p-subgroup, which we’ll write as P ≅ C_{p}^{n}. We can realise P as consisting of the upper triangular matrices with both diagonal entries equal to 1. The normaliser H = N_{G}(P) ≅ P ⋊C_{p^{n}−1}, consists of all the upper triangular matrices in G.

Throughout this section and the next we assume char(k) = p. If p is odd then the group algebra kG has three blocks: a simple block whose unique irreducible character is the Steinberg character, the principal block, and another block with defect group P. If p = 2 then kG has only two blocks: the principal block and a simple block to which the Steinberg character belongs.

The group algebra kH has two blocks if p is odd and a single block if p = 2, and Conjecture 2.6.6 predicts that these should have derived categories equivalent to those of the corresponding blocks of kG under Brauer correspondence.

As P is a T.I. subgroup, we have by Proposition 3.2.3 that the corresponding blocks of kG and kH of full defect are stably equivalent of Morita type, indeed are stably equivalent of splendid type.

In Section 5.3 we verify Conjecture 2.6.10 for the non-principal blocks of kSL_{2}(p^{2}). In Section 5.4 we do the same for the principal block of kSL_{2}(32), in fact our approach provides a uniform analysis for the principal blocks of kSL_{2}(2^{n}) for 2 ≤ n ≤ 5 and so for completeness we discuss all these blocks. Finally in Section 5.5 we verify Conjecture 2.6.10 for the principal block of kSL_{2}(27). In Section 5.2 we provide background information on the representation theory of kG and kH that we’ll need as well as establishing some of the notation used throughout this Chapter.

5.2 The representation theory of kSL_{2}(p^{n})

The representation theory of kH and the structure of the restrictions of the simple kGmodules to H are well-understood [2, 7]. Let α be a generator of the multiplicative group 𝔽_{p^{n}}^{×}, and let

y = ( \begin{matrix}α^{−1} & 0 \\ 0 & α \end{matrix} ) ∈ H.

The one-dimensional simple kH-modules will be indexed by elements of ℤ/(p^{n}−1)ℤ, where the simple module indexed by i has y acting as multiplication by α^{i}. We shall denote this simple as U_{i} or simply by i. The reason for the dual notation is that we shall want to apply U_{i} ⊗ − and also consider diagrams of kH-modules (see below) and wish to avoid a plethora of U’s appearing. We make one exception to the indexing in that the trivial module, U_{0}, will always be written as k. This avoids possible confusion with the zero module in diagrams. If p is odd, then precisely the simples indexed by even integers are in the principal block.

The simple modules for kG are given by the Steinberg Tensor Product Theorem. Let V_{1} be the natural two-dimensional module, let σ : 𝔽_{p^{n}} → 𝔽_{p^{n}} be the Frobenius automorphism, and let V_{i} = σ^{i−1}(V_{1}) for 1 ≤ i ≤ n (note that σ^{n} is the identity). Let V_{i}^{(t)} be the tth symmetric power of V_{i}, which is a (t + 1)-dimensional module. Then as b = (b_{1}, …, b_{n}) runs over all n-tuples with entries in {0, …, p − 1},

S_{b} = S(b_{1}, …, b_{n}) := V_{1}^{(b_{1})} ⊗ ⋯ ⊗ V_{n}^{(b_{n})}

runs over distinct representatives of the isomorphism classes of simple kG-modules. If b = (p − 1, …, p − 1) then S_{b} is the Steinberg module which is the unique projective simple kG-module.

The set of n-tuples indexing the simple kG-modules are in 1-1 correspondence with the set of integers b, 0 ≤ b ≤ p^{n} − 1 via their p-adic expansions

b ⟷ (b_{1}, …, b_{n}), b = ∑_{i=1}^{n} b_{i}p^{i−1}

Thus the set of non-projective simple kG-modules are also indexed by elements of ℤ/(p^{n} − 1)ℤ and in the sequel we’ll identify i ∈ ℤ/(p^{n} − 1)ℤ with the n-tuple associated to it without comment.

We’ll write M_{b} = M(b_{1}, …, b_{n}) for the restriction of S_{b} to kH. If b = p^{n} − 1, then M_{b} is the projective cover of the trivial kH-module. The other indecomposable projective kH-modules are the tensor products U_{i} ⊗ M_{b} of M_{b} with the simple kH-modules.

The Steinberg module forms a block by itself and if p is odd, then the remaining simple kG-modules divide into two blocks, with S_{b} in the principal block if and only if b_{1} +⋯+b_{n} is even. The restrictions M_{b} are all indecomposable, and divide into blocks in the same way.

The eigenvalues of y acting by conjugation (x ↦ yxy^{−1}) on rad(kP)/ rad^{2}(kP) are α^{−2}, α^{−2p}, …, α^{−2p^{n−1}}. For 1 ≤ i ≤ n, let z_{i} ∈ rad(kP) \ rad^{2}(kP) be an eigenvector with eigenvalue α^{−2p^{i−1}}.

Every kH-module N that we shall consider here, including every module M_{b} and every projective module, has a structure that is very easy to visualise. There is a basis B for N such that every element β ∈ B is an eigenvector for the action of y, and such that for every β ∈ B and z_{i}, either z_{i}β = 0 or z_{i}β is another element of B. There is a partial order on B induced by the relations z_{i}β < β for each β ∈ B and z_{i} for which z_{i}β ≠ 0. The value of i is determined by the eigenvalues of y on β and z_{i}β, so the structure of N is precisely determined by the collection of basis elements, together with their corresponding eigenvalues, and the partial order on B. We shall give diagrams that give this data for such modules, with the eigenvalues of the basis elements described by the label of the corresponding simple kH-module, and the partial order described by the position of the integers on the page. For example, the projective cover of the trivial module for p = 3, n = 2 is described by the following diagram.

k
6  2
4 k 4
6  2
k

The maps between modules of this kind that take basis elements to basis elements or zero are very easy to visualise, and their kernels and cokernels are easily calculated modules of the same kind.

The Frobenius automorphism σ induces autoequivalences on the module categories of the blocks of kH and hence also on the bounded homotopy and bounded derived categories of blocks of kH. We will simply write σ and refer to the Frobenius automorphism for all of these maps. The action of σ on the simple kH-modules is to map the simple labelled by i ∈ ℤ/(p^{n} − 1)ℤ to pi. Again we’ll use σ to denote multiplication by p on ℤ/(p^{n} − 1)ℤ.

5.2.1 Rectangular modules

For kSL_{2}(p^{2}), we’ll be particularly interested in the class of modules that are rectangular in shape. Such a module can be specified by giving the simple kH-module t that is its top, and the length of its sides 1 + a, 1 + b (that is the number of simple kH-modules that form the sides). Such a module will be written as Rect(a, b, t) and is pictured in Figure 5.1. With this notation M_{b} = M(b_{1}, b_{2}) = Rect(b_{1}, b_{2}, b_{1} + pb_{2}) and P_{b} = Rect(p − 1, p − 1, b).

[figure: 1 + b; 1 + a; t]

Figure 5.1: Rect(a, b, t)

Due to the frequency with which it recurs in this Chapter we will define ŝ:= p − 2 − s, where 0 ≤ s ≤ p − 2. We’ll also frequently use the modules Z(s) := Rect(s, p − 1, s + pŝ) (and their Frobenius conjugates σ(Z(s)) = Rect(p−1, s, ŝ+ps)), where 0 ≤ s ≤ p−2. Note that Z(s) = U_{−p(1+s)} ⊗M(s, p−1) and these modules have periodic projective resolutions. In fact:

Ω^{h}(Z(s)) = \{ \begin{matrix}U_{−ph} ⊗ Z(s) & if h is even, \\ U_{−ph+1−2p(1+s)} ⊗ Z(ŝ) & if h is odd. \end{matrix}

The following Lemma describes the properties of the rectangular modules that we shall need. This simple minded approach helps avoid the appeal to more complicated calculations of Ext groups contained in [7].

Lemma 5.2.1. With the notation above:

(1) dim_{k} Hom_{kH}(Rect(a, b, t), Rect(c, d, s)) is equal to the number of ordered pairs (k_{1}, k_{2}) that satisfy:

(1.1) max(0, c − a) ≤ k_{1} ≤ c, (1.2) max(0, d − b) ≤ k_{2} ≤ d, (1.3) 2k_{1} + 2pk_{2} + t − s ≡ 0 mod p^{2} − 1.

(2) dim_{k} \underline{Hom}_{kH}(Rect(a, b, t), Rect(c, d, s)) is equal to the number of ordered pairs (k_{1}, k_{2}) that satisfy:

(2.1) max(0, c − a) ≤ k_{1} ≤ c, (2.2) max(0, d − b) ≤ k_{2} ≤ d, (2.3) (k_{1}, k_{2}) doesn’t satisfy k_{1} ≥ (1 + â) and k_{2} ≥ (1 + b̂), (2.4) 2k_{1} + 2pk_{2} + t − s ≡ 0 mod p^{2} − 1.

(3) dim_{k} Ext_{kH}^{1}(Rect(a, b, t), Rect(c, d, s)) is equal to the number of ordered pairs (k_{1}, k_{2}) that satisfy at least one of conditions (3.1) or (3.2), where:

(3.1)

(3.1.1) max(0, c − a) ≤ k_{1} ≤ c,

(3.1.2) max(0, d̂− b) ≤ k_{2} ≤ min(b̂, d̂),

(3.1.3) 2k_{1} − 2 + 2p(k_{2} + d + 1) + t − s ≡ 0 mod p^{2} − 1. (3.2)

(3.2.1) max(0, ĉ − a) ≤ k_{1} ≤ min(ĉ, â),

(3.2.2) max(0, d − b) ≤ k_{2} ≤ d,

(3.2.3) 2k_{1} + 2c + 2 + 2p(k_{2} − 1) + t − s ≡ 0 mod p^{2} − 1.

Proof. Using the structure of kG modules described above, we can proceed as follows.

For (1), it suffices to show that the number of composition factors of Rect(c, d, s) equal to t = head(Rect(a, b, t)) is given by the number of ordered pairs (k_{1}, k_{2}) satisfying (1.1)-(1.3), but this is immediate once (k_{1}, k_{2}) are interpreted as in the Figure 5.2.

[figure: 1 + c; 1 + d; t; s; k_{2}; k_{1}] Figure 5.2:

It’s clear that (2) follows from (1) once the additional condition (2.3) is interpreted as ruling out those composition factors of Rect(c, d, s) equal to t, which have soc(Rect(a, b, t)) lying sufficiently high above, as these correspond to maps that factor through the injective envelope of Rect(a, b, t).

For (3) we can use the following isomorphisms for kH-modules M and N,

Ext_{kH}^{1}(M, N) ≅ \underline{Hom}_{kH}(Ω(M), N) ≅ \underline{Hom}_{kH}(M, Ω^{−1}(N))

to re-interpret (3) as calculating dim \underline{Hom}_{kH}(Rect(a, b, t), Ω^{−1}(Rect(c, d, s))). We can then apply the arguments used in (2). If we set r = soc(Rect(c, d, s)), then we have the diagram of Ω^{−1}(Rect(c, d, s)) as shown in Figure 5.3. In Figure 5.3 we have broken

[figure: 1 + ĉ; 1 + d̂; A; C; B; r]
Figure 5.3: Ω^{−1}(Rect(c, d, s))

Ω^{−1}(Rect(c, d, s)) up into regions A, B and C. It’s easy to see that any non-zero map from Rect(a, b, t) into region B would factor through the injective envelope of Rect(a, b, t). So we need only consider maps from Rect(a, b, t) to either the region A which corresponds to the conditions in (3.1), or to region C which corresponds to the conditions in (3.2).

For (3.1), noting that region A is the module Rect(c, d̂, h) where h = r − 2(1 + ĉ), we can use (1) to obtain conditions (3.1.1),(3.1.3), the composition factors in Rect(c, d̂, h) that we must avoid are those which have soc(Rect(a, b, t)) lying sufficiently high above them in the region A ∪ B, as they correspond to maps factoring through the injective envelope of Rect(a, b, t)), excluding these is easily seen, together with the conditions in (1), to lead to condition (3.1.2). Exactly the same argument applied to maps whose images lie in region C gives the conditions (3.2).

5.3 The non-principal block of kSL_{2}(p^{2})

In this section we concentrate on the non-principal block of kSL_{2}(p^{2}), with char(k) = p > 2. Let A and B denote the non-principal blocks, of full defect, of kSL_{2}(p^{2}) and kH respectively. Throughout this section we will simply write Hom(−, −) for Hom_{D^{b}(B)}(−, −) and make use without comment of the full embedding of mod- B in D^{b}(B), see Section 2.1.

Our main result in this section is

Theorem 5.3.1. Conjecture 2.6.10 is true for A and B.

To prove this theorem we’ll follow the approach outlined in Section 4.5. As already noted, P is a T.I. Sylow p-subgroup and so the stalk complex C = A induces a stable equivalence of splendid type and stage (1) of Section 4.5 has already been described in Section 5.2.

Recall that the simple A-modules {S_{b}|b ∈ I} and their Green correspondents {M_{b}|b ∈ I} are indexed by I = {(b_{1}, b_{2})|0 ≤ b_{1}, b_{2} ≤ p − 1, and b_{1} + b_{2} ≡ 1 mod 2}. It will prove useful to partition this set as I = I_{<} ∪ I_{p−1} ∪ I_{>}, where

I_{<} = {b ∈ I | b_{1} + b_{2} < p − 1},
I_{p−1} = {b ∈ I | either b_{1} = p − 1 or b_{2} = p − 1},
I_{>} = {b ∈ I | b_{1} + b_{2} > p − 1, and b_{1} ≠ p − 1, b_{2} ≠ p − 1}

The following notation will be useful.

Definition 5.3.2. If b ∈ I_{>} with b = (b_{1}, b_{2}), define

h = h(b) := \{ \begin{matrix}b_{1} − b̂_{2} & if b_{1} < (p − 2)/2 \\ b̂_{1} − b̂_{2} & if b_{1} > (p − 2)/2 \end{matrix}

We now turn to stage (2) of Section 4.5 and define objects {X_{i}}.

Definition 5.3.3. For b ∈ I we define X_{b} ∈ D^{b}(B) as follows:

(1) If b ∈ I_{<}, take X_{b} = M_{b} viewed as a complex concentrated in degree zero.

(2) If b ∈ I_{p−1}, with say b = (b_{1}, p − 1) (so b_{1} ≤ p − 2, and b_{1} is odd), take

X_{b} = \{ \begin{matrix}Z(b_{1})[1 + b_{1}] & if b_{1} < (p − 2)/2. \\ Z(b̂_{1})[1 + b̂_{1}] & if b_{1} > (p − 2)/2. \end{matrix}

If b = (p − 1, b_{2}), then we take X_{b} = σ(X_{(b_{2},p−1)}).

(3) If b ∈ I_{>}, with b = (b_{1}, b_{2}) and b_{2} > b_{1}, then we have the following short exact sequence

0 → W → Ω^{1}(M_{b}) → U_{−1} ⊗ M(b_{1}, b̂_{2}) → 0

where W = U_{p(b_{2}−b_{1}−1)} ⊗ Z(b̂_{1}). Let Y denote a minimal projective resolution of W and let σ_{>−h+1}Y denote the brutal truncation ( (σ_{>−h+1}Y)^{i} = Y^{i} if i > −h + 1 and (σ_{>−h+1}Y)^{i} = 0 otherwise). We take u to be the composite of the cochain maps σ_{>−h+1}Y → W and W → Ω^{1}(M_{b}) and define X_{b} = cone(u)[1]. If b = (b_{1}, b_{2}) and b_{1} > b_{2}, then we take X^{b} = σ(X^{(b_{2},b_{1})}). Note that if h = h(b) = 1 then the construction gives X_{b} = Ω^{1}(M_{b})[1].

Clearly the {X_{i}} in definition 5.3.3 are invariant under the Frobenius automorphism σ. When b ∈ I_{>} and h = h(b) > 1, X_{b} has non-zero cohomology concentrated in two degrees and it’s easy to see that

H^{−1}(X_{b}) = U_{−1} ⊗ M(b_{1}, b̂_{2}), H^{−h}(X_{b}) = \{ \begin{matrix}Z(b_{1}) & if b_{1} < (p − 2)/2 \\ Z(b̂_{1}) & if b_{1} > (p − 2)/2 \end{matrix}

An alternative description of the objects {X_{i}} that is perhaps more illuminating is based upon certain n-tuple complexes that are described in [7]. The cohomology of simple kSL_{2}(p^{n})-modules is discussed in [7], which also describes minimal projective resolutions of the {M_{b}}. These minimal resolutions are total complexes of certain n-tuple complexes.

When n = 2 we can describe truncations of these double complexes. The bounded total complexes {Y_{i}} that result are complexes of projective modules and we have a natural map of complexes Y_{b} → M_{b}. We can complete this to a triangle:

Y_{b} → M_{b} → X_{b} → Y_{b}[1]

thus defining objects {X_{i}} that are isomorphic, in D^{b}(B), to those in Definition 5.3.3.

The truncation of the double complexes is easy to describe. Replace any projective module in the double complex that has a path of maps to a projective module belonging to a certain “excluded” set by the zero module. The “excluded” set of projectives is {P_{b}|b ∈ I_{<}}. The set I_{<}, that indexes this “excluded” set, can also be described as the maximal subset of I such that {M_{b}|b ∈ I_{<}} is invariant under the Frobenius action and for which conditions (a) and (b) of Theorem 4.4.2 already hold with X_{b} = M_{b}.

To illustrate this construction we’ll consider the B-module M(2, 5) for p = 7. It’s easy to see that I_{<} = {1, 3, 5, 7, 9, 11, 15, 17, 21, 23, 29, 35}. Using the description in [7] the double complex, whose total complex yields the minimal projective resolution of M(2, 5), is

\xymatrix{⋮ \ar[d] & ⋮ \ar[d] & ⋮ \ar[d] \\ P_{35} \ar[d] & P_{29} \ar[l] \ar[d] & P_{21} \ar[l] \ar[d] & … \ar[l] \\ P_{1} \ar[d] & P_{43} \ar[l] \ar[d] & P_{35} \ar[l] \ar[d] & … \ar[l] \\ P_{37} & P_{31} \ar[l] & P_{23} \ar[l] & … \ar[l]}

It’s clear from this, together with the “excluded” set I_{<}, that the total complex of the truncated complex is

Y (2, 5) = ⋯ → 0 → P_{31} → P_{37} → 0 → ⋯

The complex X(2, 5) is then

X(2, 5) = ⋯ → 0 → P_{31} → P_{37} → M(2, 5) → 0 → ⋯

This alternative construction of the {X_{i}} also works for the principal blocks of kSL_{2}(2^{n}) (char(k) = 2 and n ≤ 5) and kSL_{2}(27) (char(k) = 3).

By using the description of the double complexes from [7] it’s fairly easy to see that the two descriptions of the {X_{i}} are isomorphic in D^{b}(B).

This alternative description shows that we can choose representatives from the isomorphism classes of the X_{i} such that the triangles (4.5) of Section 4.5 hold. Thus steps (1) and (2) of Section 4.5 are complete.

We now turn to step (3) of Section 4.5. Our main task in proving Theorem 5.3.1 will be the verification of conditions (a) and (b) of Theorem 4.4.2 for the objects {X_{i}} of definition 5.3.3. In order to do this we will need the following three technical Lemmas. The proofs will all use Lemma 5.2.1. In order to represent them efficiently, each statement will be reinterpreted using the notation of Lemma 5.2.1, followed by the conditions imposed on pairs (k_{1}, k_{2}) of positive integers. Occasionally it will prove useful to write the conditions for k_{i} in terms of η_{i} := p − 1 − k_{i} (for i = 1, 2) which we will do without comment. Throughout δ will represent a particular coset representative from the congruence class required to be zero in Lemma 5.2.1 ((1.3),(2.4),(3.1.3) or (3.2.3) of the Lemma). These conditions will then be shown to have either no solution or precisely one solution as required.

Lemma 5.3.4. If a, b ∈ I_{<} and 0 ≤ s, t ≤ (p − 3)/2, then:

(a) Hom(M_{a}, M_{b}) = \{ \begin{matrix}k & if a = b, \\ 0 & otherwise. \end{matrix}

(b) Hom(Z(s), Z(t)) = \{ \begin{matrix}k & if s = t, \\ 0 & otherwise. \end{matrix}

(c) Hom(Z(s), σ(Z(t))) = 0.

(d) Hom(Z(s), Z(t)[r]) = 0 for r = 1, … t − s.

(e) Hom(Z(s), σ(Z(t))[r]) = 0 for r ≥ 1.

Proof. Proof of (a):

dim Hom_{B}(Rect(a_{1}, a_{2}, a_{1} + pa_{2}), Rect(b_{1}, b_{2}, b_{1} + pb_{2}))
max(0, b_{1} − a_{1}) ≤ k_{1} ≤ b_{1}, max(0, b_{2} − a_{2}) ≤ k_{2} ≤ b_{2}
δ = 2k_{1} − (b_{1} − a_{1}) + p(2k_{2} − (b_{2} − a_{2}))

We have a_{1} + b_{1} + a_{2} + b_{2} < 2(p − 1), so either a_{1} + b_{1} < p − 1 or a_{2} + b_{2} < p − 1. If a_{1} + b_{1} < p − 1 then for pδ = 2k_{2} − (b_{2} − a_{2}) + p(2k_{1} − (b_{1} − a_{1})), pδ ≥ 0 with equality only if a = b and pδ ≤ a_{2} + b_{2} + p(a_{1} + b_{1}) < p^{2} − 1. If a_{2} + b_{2} < p − 1 then exactly the same argument works with δ.

Proof of (b):

dim Hom_{B}(Rect(s, p − 1, s + pŝ), Rect(t, p − 1, t + pt̂))
max(0, t − s) ≤ k_{1} ≤ t, 0 ≤ k_{2} ≤ p − 1
pδ = 2k_{2} + (t − s) + p(2k_{1} − (t − s))

As s + t ≤ p − 3, pδ ≤ p − 1 + t − s + p(t + s) < p^{2} − 1 and, as t ≥ s, pδ ≥ 0 with equality only if s = t.

Proof of (c):

dim Hom_{B}(Rect(s, p − 1, s + pŝ), Rect(p − 1, t, t̂+ pt))
0 ≤ η_{1} ≤ s, 0 ≤ k_{2} ≤ t
pδ = 2k_{2} − s − t − 3 + p(1 + s + t − 2η_{1})

pδ ≥ −1 − t − s + p(1 + t − s) > 0 as t ≥ s and pδ ≤ −(3 + t + s) + p(3 + t + s) < p^{2} − 1. Proof of (d): We use Hom(Z(s), Z(t)[r]) ≅ \underline{Hom}_{B}(Ω^{r}(Z(s)), Z(t)) and split into two

cases:

\underline{case1: r even}, Ω^{r}(Z(s)) = Rect(s, p − 1, s + p(1 + r̂)).

dim \underline{Hom}_{B}(Rect(s, p − 1, s + p(1 + r̂)), Rect(t, p − 1, t + pt̂))
t − s ≤ k_{1} ≤ min(ŝ, t), 0 ≤ k_{2} ≤ p − 1,
pδ = 2k_{2} + (t − s − r) + p(2k_{1} − (t − s))

where t > s and r = 1, … t − s. Clearly 0 < pδ < p^{2} − 1. \underline{case2: r odd}, Ω^{r}(Z(s) = Rect(ŝ, p − 1, ŝ + 1 + p(p − 1 − r − 2(1 + s))).

dim \underline{Hom}_{B}(Rect(s, p − 1, ŝ + 1 + p(p − 1 − r − 2(1 + s)), Rect(t, p − 1, t + pt̂))

0 ≤ k_{1} ≤ min(s, t), 0 ≤ k_{2} ≤ p − 1,

pδ = 2k_{2} + t − s − r + 1 + p(2k_{1} − (s + t + 2))δ = 2k_{1} − (s + t + 2) + p(2k_{2} + t − s − r + 1)

where t > s and r = 1, … t − s. Clearly −(p^{2} − 1) < pδ < 0.

Proof of (e): Using Hom(Z(s), σ(Z(t))[r]) ≅ \underline{Hom}_{B}(Ω^{r}(Z(s)), σ(Z(t))).

dim \underline{Hom}_{B}(Rect(p − 1, t,?), Rect(s, p − 1,?))
1 + ŝ ≤ k_{1} ≤ p − 1, 0 ≤ k_{2} ≤ t,
(k_{1}, k_{2}) doesn’t satisfy k_{1} ≥ 1 + ŝ, k_{2} ≥ 0

We do not care what ? is as clearly no solution exists.

Lemma 5.3.5. If c ∈ I_{>} with c = (c_{1}, c_{2}), c_{2} > c_{1}, a ∈ I_{<} and 0 ≤ s ≤ (p − 3)/2 then:

(a) Hom(M_{a}, Z(s)[r]) = 0 for 0 ≤ r ≤ 1 + s.

(b) Hom(M_{a}, U_{−1} ⊗ M(c_{1}, ĉ_{2})[r]) = 0 for r = 0, 1.

(c) If 1 + h ≤ r ≤ 1 + s then:

(i) Hom(M_{c}, Z(s)[r]) = 0

(ii) Hom(M_{c}, σ(Z(s))[r]) = 0

(d) If 0 ≤ r ≤ min(h, s) then:

(i) Hom(U_{−1} ⊗ M(c_{1}, ĉ_{2}), Z(s)[r]) = \{ \begin{matrix}k & if r is even and (s, r) = (c_{1}, h), \\ k & if r is odd and (s, r) = (ĉ_{1}, h), \\ 0 & otherwise. \end{matrix}

(ii) Hom(U_{−1} ⊗ M(c_{1}, ĉ_{2}), σ(Z(s))[r]) = 0

Proof. Proof of (a): If r = 0 then:

dim Hom_{B}(Rect(a_{1}, a_{2}, a_{1} + pa_{2}), Rect(s, p − 1, s + pŝ))
max(0, s − a_{1}) ≤ k_{1} ≤ s, 1 + â_{2} ≤ k_{2} ≤ p − 1,
δ = 2k_{1} − (s − a_{1}) + p(s + a_{2} − 2η_{2})

If s < a_{2}, then s+a_{1} < a_{2} +a_{1} < p−1 and with pδ = s+a_{2} −2η_{2} +p(2k_{1} −(s−a_{1})+1), pδ > s−a_{2}+p > 0 and pδ ≤ s+a_{2}+p(s+a_{1}+1) < p^{2}−1. If s ≥ a_{2}, then s+a_{2} ≤ 2s ≤ p−3 and δ ≥ 1 + p(s − a_{2}) > 0 and δ ≤ s + a_{1} + 1 + p(p − 3).

If r > 0 then we have Hom(M_{a}, Z(s)[r]) ≅ \underline{Hom}_{B}(M_{a}, Ω^{−r}(Z(s))). We split this into two cases.

\underline{case1: r even} Ω^{−r}(Z(s)) = Rect(s, p − 1, s + p(r + ŝ)).

dim Hom_{B}(Rect(a_{1}, a_{2}, a_{1} + pa_{2}), Rect(s, p − 1, s + p(r + ŝ)))
max(0, s − a_{1}) ≤ k_{1} min(s, â_{1}), 0 ≤ η_{2} ≤ a_{2},
pδ = s + a_{2} − r − 2η_{2} + p(2k_{1} − (s − a_{1}) + 1)

where 1 ≤ r ≤ 1 + s. If s ≤ â_{1}, then a_{1} + s ≤ p − 2 and pδ ≥ −a_{2} − 1 + p > 0 and pδ ≤ s+a_{2}−1+p(s+a_{1}+1) ≤ p^{2}−1, with equality only if a_{1}+s = p−2 and s+a_{2} = p, as a ∈ I_{<}, equality is impossible. If s > â_{1} then pδ > 0 and pδ ≤ s+a_{2}−1+p(p−1−(s−â_{1})) < p^{2}−1.

\underline{case2: r odd} Ω^{−r}(Z(s)) = Rect(ŝ, p − 1, ŝ + p(r + ŝ)).

dim Hom_{B}(Rect(a_{1}, a_{2}, a_{1} + pa_{2}), Rect(ŝ, p − 1, ŝ + p(r + ŝ)))
max(0, ŝ − a_{1}) ≤ k_{1} min(ŝ, â_{1}), 0 ≤ η_{2} ≤ a_{2},
pδ = a_{2} + s − r − 2η_{2} + p(2k_{1} − (ŝ − a_{1}) + 1)

where 1 ≤ r ≤ 1 + s. Clearly pδ > p − a_{2} − 1 > 0 and if ŝ > â_{1} then pδ ≤ a_{2} + s − 1 + p(ŝ + a_{1} + 1) < p^{2} − 1. If ŝ > â_{1} then pδ ≤ a_{2} + s − 1 + p(p − 1 − (ŝ − â_{1})) < p^{2} − 1.

Proof of (b): We treat the cases r = 0 and r = 1 separately:

\underline{case1: r = 0}

dim Hom_{B}(Rect(a_{1}, a_{2}, a_{1} + pa_{2}), Rect(c_{1}, ĉ_{2}, c_{1} − 1 + pĉ_{2}))
max(0, c_{1} − a_{1}) ≤ k_{1} ≤ c_{1}, max(0, ĉ_{2} − a_{2}) ≤ k_{2} ≤ ĉ_{2},
δ = 2k_{1} − (c_{1} − a_{1}) + 1 + p(2k_{2} − (ĉ_{2} − a_{2}))

We have c_{1} + a_{1} + ĉ_{2} + a_{2} < 2(p − 2), so either ĉ_{2} + a_{2} < p − 2 or c_{1} + a_{1} < p − 2, suppose the former, then δ > 0 and δ ≤ c_{1} + a_{1} + 1 + p(ĉ_{2} + a_{2}) < p^{2} − 1. If c_{1} + a_{1} < p − 2, then with pδ = 2k_{2} − (ĉ_{2} − a_{2}) + p(2k_{1} − (c_{1} − a_{1}) + 1), pδ ≥ p > 0 and pδ < p^{2} − 1.

\underline{case2: r = 1},

dim Ext_{B}^{1}(Rect(a_{1}, a_{2}, a_{1} + pa_{2}), Rect(c_{1}, ĉ_{2}, c_{1} − 1 + pĉ_{2}))

Suppose (k_{1}, k_{2}) satisfy (3.1) of Lemma 5.2.1 then:

max(0, c_{1} − a_{1}) ≤ k_{1} ≤ c_{1}, max(0, c_{2} − a_{2}) ≤ k_{2} ≤ min(c_{2}, â_{2}),
δ = 2k_{1} − (c_{1} − a_{1}) + p(2k_{2} − (c_{2} − a_{2}))

Clearly δ ≥ 0 with equality only if a_{1} = c_{1} and a_{2} = c_{2} which is impossible. If c_{2} ≤ â_{2} then δ ≤ c_{1} + a_{1} + p(c_{2} + a_{2}) < p^{2} − 1. If c_{2} > â_{2} then δ ≤ c_{1} + a_{1} + p(2â_{2} − c_{2} + a_{2}) = c_{1} + a_{1} + p(â_{2} + ĉ_{2}) < p^{2} − 1.

If (k_{1}, k_{2}) satisfy (3.2) of Lemma 5.2.1 then:

max(0, ĉ_{1} − a_{1}) ≤ k_{1} ≤ min(ĉ_{1}, â_{1}), max(0, ĉ_{2} − a_{2}) ≤ k_{2} ≤ ĉ_{2},
δ = 2k_{1} + 1 − (ĉ_{1} − a_{1}) + p(2k_{2} − (ĉ_{2} − a_{2}) − 1)

With pδ = 2k_{2} − (ĉ_{2} − a_{2}) − 1 + p(2k_{1} + 1 − (ĉ_{1} − a_{1})) we clearly have pδ > 0 and if ĉ_{1} ≥ â_{1} then k_{1} ≤ â_{1} and pδ ≤ ĉ_{2} + a_{2} − 1 + p(p − 1 − (ĉ_{1} − â_{1})) ≤ p^{2} − 1 with equality only if ĉ_{2} + a_{2} = p and ĉ_{1} = â_{1} which is impossible as a ∈ I_{<} and c ∈ I_{>}. If ĉ_{1} < â_{1}, so k_{1} ≤ ĉ_{1} then pδ ≤ ĉ_{2} + a_{2} − 1 + p(ĉ_{1} + a_{1} − 1) < p^{2} − 1 as ĉ_{1} + a_{1} < p − 2.

Proof of (c): We combine (i) and (ii), using the Frobenius automorphism, by showing that Hom(M_{c}, Z(s)[r]) ≅ \underline{Hom}_{B}(M_{c}, Ω^{−r}(Z(s))) = 0 for c ∈ I_{>} (with no other restriction on c), 0 ≤ s ≤ (p − 3)/2 and 1 + h ≤ r ≤ 1 + s. We are in a similar situation to (a) and we again split up the calculations into two cases. We can apply the restrictions we had in (a) by replacing a with c, but now we have 1 + h ≤ r ≤ 1 + s.

\underline{case1: r even}

max(0, s − c_{1}) ≤ k_{1} ≤ min(ŝ, ĉ_{1}), 0 ≤ η_{2} ≤ c_{2},
pδ = s + c_{2} − r − 2η_{2} + p(2k_{1} − (s − c_{1}) + 1)

Again pδ > 0 and if s ≤ ĉ_{1} then pδ ≤ s+c_{2} −1−h+p(s+c_{1} +1) ≤ p^{2} −1 with equality only if s+c_{1} = p−2 (so c_{1} > (p−2)/2) and s+c_{2} −h = p, but h = ĉ_{1} −ĉ_{2} = c_{2} −c_{1} and so this cannot happen. If s > ĉ_{1} then k_{1} ≤ ĉ_{1} and pδ ≤ s+a_{2} −1+p(p−1−(s−â_{1})) < p^{2} −1.

\underline{case2: r odd}

max(0, ŝ − c_{1}) ≤ k_{1} min(ŝ, ĉ_{1}), 0 ≤ η_{2} ≤ c_{2},
pδ = c_{2} + s − r − 2η_{2} + p(2k_{1} − (ŝ − c_{1}) + 1)

Clearly pδ > 0 and if ŝ ≥ ĉ_{1}, so k_{1} ≤ ĉ_{1}, then pδ ≤ c_{2}+s−1−h+p(p−1−(ŝ−ĉ_{1})) ≤ p^{2}−1 with equality only if ŝ = ĉ_{1} (so c_{1} < (p − 2)/2) and c_{2} + s − h = p, but h = c_{1} − ĉ_{2} and so this is impossible. If ŝ < ĉ_{1}, so k_{1} ≤ ŝ, then pδ ≤ c_{2} + s − 1 − h + p(ŝ + c_{1} + 1) < p^{2} − 1.

Proof of (d): Using the Frobenius automorphism (ii) becomes Hom(U_{−p}⊗M(ĉ_{2}, c_{1}), Z(s)) = 0. First we will deal with the case r = 0 for (i) and (ii):

(i)

dim Hom_{B}(Rect(c_{1}, ĉ_{2}, c_{1} + 1 + pĉ_{2}), Rect(s, p − 1, s + pŝ))
max(0, s − c_{1}) ≤ k_{1} ≤ s, 0 ≤ η_{2} ≤ ĉ_{2},
δ = 2k_{1} + 2 − (s − c_{1}) + p(s + ĉ_{2} − 2η_{2})

If s ≥ ĉ_{2} then δ ≥ 2 + p(s − ĉ_{2}) > 0 and δ ≤ s + c_{1} + 2 + p(s + ĉ_{2}) < p^{2} − 1. With pδ = s + ĉ_{2} − 2η_{2} + p(2k_{1} + 2 − (s − c_{1})) and s < ĉ_{2} then pδ ≥ s + ĉ_{2} + 2p > 0 and pδ < s + ĉ_{2} + p(s + c_{1} + 2) < p^{2} − 1.

(ii)

dim Hom_{B}(Rect(ĉ_{2}, c_{1}, ĉ_{2} + p(c_{1} + 1)), Rect(s, p − 1, s + pŝ))
max(0, s − ĉ_{2}) ≤ k_{1} ≤ s, 0 ≤ η_{2} ≤ c_{1},
pδ = 1 + s + c_{1} − 2η_{2} + p(2k_{1} + 1 − (s − ĉ_{2}))

Clearly pδ > 0 and pδ ≤ 1+s+c_{1}+p(s+ĉ_{2}+1). If s ≥ ĉ_{2} then s+ĉ_{2}+1 ≤ 2s+1 < p−1 and so pδ < p^{2} − 1 while if s < ĉ_{2} then s + ĉ_{2} + 1 ≤ 2ĉ_{2} + 1 < p − 1, as c ∈ I_{>} with c_{2} > c_{1} so necessarily c_{2} > (p − 2)/2, and again pδ < p^{2} − 1.

For r > 0, we can use Hom(U_{−1}⊗M(c_{1}, ĉ_{2}), Z(s)) ≅ \underline{Hom}_{B}(U_{−1}⊗M(c_{1}, ĉ_{2}), Ω^{−r}(Z(s))) and again treat the cases r even and r odd separately:

(i) \underline{case1: r even}

dim \underline{Hom}_{B}(Rect(c_{1}, ĉ_{2}, c_{1} − 1 + pĉ_{2}), Rect(s, p − 1, s + p(ŝ + r)))
max(0, s − c_{1}) ≤ k_{1} ≤ min(s, ĉ_{1}), 0 ≤ η_{2} ≤ ĉ_{2},
pδ = ĉ_{2} + s − r − 2η_{2} + p(2k_{1} − (s − c_{1}))

where 0 ≤ r ≤ min(s, h). Now pδ ≥ s−ĉ_{2}−r_{max}+p(2k_{1}−(s−c_{1})) and the only way that this expression can be ≤ 0 is if s = c_{1} (so h = c_{1}−ĉ_{2}) and s−ĉ_{2}−r_{max} ≤ 0. As h < s, r_{max} = h, so pδ ≥ 0 with equality only if (s, r) = (c_{1}, h) as required. pδ ≤ ĉ_{2} + s + p(2k_{1} − (s − c_{1})) and if s ≤ ĉ_{1} then pδ ≤ ĉ_{2} + s + p(s + c_{1}) < p^{2} − 1 as s + c_{1} < p − 1, while if s > ĉ_{1} then pδ ≤ ĉ_{2} + s + p(p − 1 − (s − ĉ_{1}) < p^{2} − 1.

\underline{case2: r odd}

dim \underline{Hom}_{B}(Rect(c_{1}, ĉ_{2}, c_{1} − 1 + pĉ_{2}), Rect(ŝ, p − 1, ŝ + p(ŝ + r)))
max(0, ŝ − c_{1}) ≤ k_{1} ≤ min(ŝ, ĉ_{1}), 0 ≤ η_{2} ≤ ĉ_{2},
pδ = ĉ_{2} + s − r − 2η_{2} + p(2k_{1} − (ŝ − c_{1}))

where 0 ≤ r ≤ min(s, h). Now pδ ≥ s−ĉ_{2}−r_{max}+p(2k_{1}−(ŝ−c_{1})) and the only way that this expression can be ≤ 0 is if ŝ = c_{1} (so h = ĉ_{1}−ĉ_{2}) and ŝ−ĉ_{2}−r_{max} ≤ 0. As h < s, r_{max} = h, so pδ ≥ 0 with equality only if (s, r) = (ĉ_{1}, h) as required. pδ ≤ ĉ_{2} + s + p(2k_{1} − (ŝ − c_{1})) and if ŝ ≥ ĉ_{1} then pδ ≤ ĉ_{2} + s + p(p − 1 − (ŝ − ĉ_{1})) < p^{2} − 1 as s + ĉ_{2} < p − 1, while if ŝ < ĉ_{1} then pδ ≤ ĉ_{2} + s + p(ŝ + c_{1}) < p^{2} − 1.

(ii)

\underline{case1: r even}

dim \underline{Hom}_{B}(Rect(ĉ_{2}, c_{1}, ĉ_{2} + p(c_{1} − 1)), Rect(s, p − 1, s + p(ŝ + r)))
max(0, s − ĉ_{2}) ≤ k_{1} ≤ min(s, c_{2}), 0 ≤ η_{2} ≤ c_{1},
pδ = −1 + c_{1} + s − r − 2η_{2} + p(2k_{1} − (s − ĉ_{2}) + 1)

Clearly pδ ≥ p − 1 + s − c_{1} − r_{max} > 0 as r_{max} ≤ s. Now as c ∈ I_{>} with c_{2} > c_{1}, s < c_{2} and pδ ≤ −1 + c_{1} + s + p(s + ĉ_{2} + 1) < p^{2} − 1.

\underline{case2: r odd}

dim \underline{Hom}_{B}(Rect(ĉ_{2}, c_{1}, ĉ_{2} + p(c_{1} − 1)), Rect(ŝ, p − 1, ŝ + p(r + ŝ))
max(0, ŝ − ĉ_{2}) ≤ k_{1} ≤ min(ŝ, c_{2}), 0 ≤ η_{2} ≤ c_{1},
pδ = s + c_{1} − 1 − r − 2η_{2} + p(2k_{1} − (ŝ − ĉ_{2}) + 1)

Clearly pδ ≥ p + s − c_{1} − 1 − r_{max} > 0 as r_{max} ≤ s. If ŝ ≥ c_{2}, so k_{1} ≤ c_{2}, then pδ ≤ s+c_{1}−1+p(p−1−(ŝ−c_{2})) < p^{2}−1, while if ŝ < c_{2} then pδ ≤ s+c_{1}−1+p(ŝ+ĉ_{2}+1) < p^{2}−1.

Lemma 5.3.6. If 0 ≤ s ≤ (p − 3)/2 and c, d ∈ I_{>} with c = (c_{1}, c_{2}), c_{2} > c_{1} and d = (d_{1}, d_{2}), d_{2} > d_{1}, then:

(a) Hom(U_{−1} ⊗ M(c_{1}, ĉ_{2}), U_{−1} ⊗ M(d_{1}, d̂_{2})) = \{ \begin{matrix}k & if c = d, \\ 0 & otherwise \end{matrix}

(b) Hom(U_{−1} ⊗ M(c_{1}, ĉ_{2}), U_{−p} ⊗ M(d̂_{1}, d_{2})) = 0

(c) Hom(Z(s), U_{−1} ⊗ M(c_{1}, ĉ_{2})) = Hom(σ(Z(s)), U_{−1} ⊗ M(c_{1}, ĉ_{2})) = 0

Proof. Proof of (a):

dim Hom_{B}(Rect(c_{1}, ĉ_{2}, c_{1} − 1 + pĉ_{2}), Rect(d_{1}, d̂_{2}, d_{1} − 1 + pd̂_{2}))
max(0, d_{1} − c_{1}) ≤ k_{1} ≤ d_{1}, max(0, d̂_{2} − ĉ_{2}) ≤ k_{2} ≤ d̂_{2},
δ = 2k_{1} − (d_{1} − c_{1}) + p(2k_{2} − (d̂_{2} − ĉ_{2}))

We have c_{1} + d_{1} + d̂_{2} + ĉ_{2} < 2(p − 1) so either d̂_{2} + ĉ_{2} < p − 1 or c_{1} + d_{1} < p − 1. Suppose the former, then δ ≤ c_{1} + d_{1} + p(d̂_{2} + ĉ_{2}) < p^{2} − 1. Also δ ≥ 0 with equality only if c = d. If c_{1} + d_{1} < p − 1 then exactly the same argument works for pδ = 2k_{2} − (d̂_{2} − ĉ_{2}) + p(2k_{1} − (d_{1} − c_{1})).

Proof of (b):

dim Hom_{B}(Rect(c_{1}, ĉ_{2}, c_{1} − 1 + pĉ_{2}), Rect(d̂_{2}, d_{1}, d̂_{2} + p(d_{1} − 1))
max(0, d̂_{2} − c_{1}) ≤ k_{1} ≤ d̂_{2}, max(0, d_{1} − ĉ_{2}) ≤ k_{2} ≤ d_{1},
δ = 2k_{1} − (d̂_{2} − c_{1}) − 1 + p(2k_{2} − (d_{1} − ĉ_{2}) + 1)

If d_{2} < c_{2} then d_{1} < d_{2} < c_{2} and d_{1} + ĉ_{2} + 1 = p − 1 − (c_{2} − d_{1}) < p − 2. δ ≥ −1 + p > 0 and δ ≤ d̂_{2} + c_{1} − 1 + p(d_{1} + ĉ_{2} + 1) < p^{2} − 1. If d_{2} ≥ c_{2} then d̂_{2} ≤ ĉ_{2} < c_{1} and with pδ = 2k_{2} − (d_{1} − ĉ_{2}) + 1 + p(2k_{1} − (d̂_{2} − c_{1}) − 1), pδ ≥ 1 + p(c_{1} − d̂_{1} − 1) > 0 and pδ ≤ d_{1} + ĉ_{2} + 1 + p(d̂_{2} + c_{1} − 1) < p^{2} − 1, as d̂_{2} + c_{1} ≤ ĉ_{2} + c_{1} < p − 2.

Proof of (c): We’ll deal first with the Hom space involving Z(s):

dim Hom_{B}(Rect(s, p − 1, s + pŝ), Rect(c_{1}, ĉ_{2}, c_{1} − 1 + pĉ_{2}))
max(0, c_{1} − s) ≤ k_{1} ≤ c_{1}, 0 ≤ k_{2} ≤ ĉ_{2},
δ = 2k_{1} − (c_{1} − s) + 1 + p(2k_{2} − (ĉ_{2} − ŝ))

We have ŝ > ĉ_{2} and so δ > 0. If s ≥ ĉ_{2} then δ ≤ c_{1} + s + 1 + p(ĉ_{2} + ŝ) < p^{2} − 1 as ĉ_{2} + ŝ ≤ p − 2. If s < ĉ_{2}, then c_{1} + s + 1 < c_{1} + ĉ_{2} + 1 < p − 2. With pδ = 2k_{2} − (ĉ_{2} − ŝ) + p(2k_{1} − (c_{1} − s) + 1) we have 0 < pδ < p^{2} − 1.

Now we treat the Hom space involving σ(Z(s)):

dim Hom_{B}(Rect(s, p − 1, s + pŝ), Rect(c_{1}, ĉ_{2}, c_{1} − 1 + pĉ_{2}))
0 ≤ k_{1} ≤ c_{1}, max(0, ĉ_{2} − s) ≤ k_{2} ≤ ĉ_{2},
δ = 2k_{1} − c_{1} + ŝ + 1 + p(2k_{2} − (ĉ_{2} − s))

As ĉ_{2} + ŝ < p − 2, δ ≤ c_{1} + ŝ + 1 + p(ĉ_{2} + s) < p^{2} − 1. If s ≤ ĉ_{2} then ŝ ≥ c_{2} > c_{1} and δ ≥ ŝ − c_{1} + 1 > 0. If s > ĉ_{2} then δ ≥ ŝ − c_{1} + 1 + p(s − ĉ_{2}).

We now focus on establishing conditions (a) and (b) of Theorem 4.4.2. We can simplify matters by partitioning {X_{b}} = A ∪ B ∪ C, where A = {X_{b}|b ∈ I_{<}}, B = {X_{b}|b ∈ I_{p−1}} and C = {X_{b}|b ∈ I_{>}}. A useful picture of the complexes {X_{b}} in definition 5.3.3 is shown in Figure 5.4, where • represents a non-zero module.

• ; • ; ⋮ ; • ; • } A

\begin{matrix}& & & & & & • \\ & & & & & · & • \\ & & & & · \\ & & & · \\ & & · \\ • & · \\ • \end{matrix} \} B

• ; • ; ⋮ ; • → ⋯ → • → • ; • → ⋯ → • → • } C

Figure 5.4: {X_{b}}

In order to break up the checking of conditions (a) and (b) of Theorem 4.4.2 according to our partition we’ll make the following definition. Given a finite set S of objects in D^{b}(B), let V(S) denote that the following equations hold for X, Y ∈ S:

dim Hom(X, Y [−r]) =0 for r > 0,
dim Hom(X, Y) =\{ \begin{matrix}1 if X = Y, \\ 0 otherwise \end{matrix}

Moreover for finite sets S, T of objects in D^{b}(B), let V(S, T ) denote that the following equations hold for X ∈ S and Y ∈ T :

dim Hom(X, Y [−r]) = 0 for r ≥ 0.

So conditions (a) and (b) of Theorem 4.4.2 are V({X_{b}}).

We’ll want to exploit the fact that for X_{c} ∈ C, H^{−h}(X_{c}) = N where N = Z(s) or σ(Z(s)) for some s. In fact if I denotes a minimal injective resolution of N, then the truncated complex J, where J^{i} = I[h]^{i} for i ≤ −2 and J^{i} = 0 for i ≥ −1, is isomorphic to the complex obtained from (X^{c}, d) by replacing the degree −1 term with im(d^{−2}). Consequently if Y is a bounded complex with cohomology concentrated in degrees ≤ t then

Hom(Y [r], X_{c}) = Hom(Y [r], I[h])

for r + t ≤ −2. As I[h] ≅ X_{b}[−l] for some X_{b} ∈ B and l > 0 we have, for r + t ≤ −2,

Hom(Y [r], X_{c}) = Hom(Y [r], X_{b}[−l])

We’ll also make use of invariance under the Frobenius automorphism, σ, to cut down on the amount of checking needed.

Having established this observation and notation we now employ Lemmas 5.3.4, 5.3.5 and 5.3.6 to prove:

Proposition 5.3.7. V(A), V(B), V(C), V(A, B), V(A, C), V(B, C) and V(C, B) all hold.

Proof. That V(A) and V(B) hold follows immediately from Lemma 5.3.4. While V(A, B) holding is just Lemma 5.3.5 (a).

Proof of V(A, C):

By the observation preceding the proposition together with V(A, B) holding, it suffices to check that Hom(M_{a}, X_{c}[−r]) = 0 for r = 0, 1. Applying the covariant cohomological functor Hom(M_{a}, −) to the triangle defining X_{c} and again using V(A, B) we’re reduced to Lemma 5.3.5 (b).

Proof of V(B, C):

Again applying the observation preceding this proposition together with V(B) most of V(B, C) follows except possibly for Hom(X_{b}, X_{c}) when X_{b} is concentrated in degree −1, i.e. X_{b} = Z(0)[1] (or X_{b} = σ(Z(0))[1]). For this case we can apply the covariant cohomological functor Hom(X_{b}, −) to the triangle defining X_{c} and the required condition is easily seen to follow from Lemma 5.3.6.

Proof of V(C, B):

We have to establish Hom(X_{c}, X_{b}[−r]) = 0 for r ≥ 0 where say c = (c_{1}, c_{2}) with c_{2} > c_{1}. There are two triangles in D^{b}(B) that can define X_{c}:

U[h] → X_{c} → V [1] → U[h + 1]
L → M_{c} → X_{c} → L[1]

where U = H^{−h}(X_{c}) and V = H^{−1}(X_{c}) are the two non-zero cohomology groups and L is a bounded complex of projectives (arising as the truncation of the double complex associated to M_{c}). Now X_{b} = N[1 + s] where N = Z(s) or σ(Z(s)) for some suitable s. We can apply the contravariant cohomological functor Hom(−, X_{b}[−r]) to the first triangle yielding the exact sequence:

Hom(L, N[s − r]) → Hom(X_{c}, X_{b}[−r]) → Hom(M_{c}, N[1 + s − r]) → Hom(L, N[1 + s − r])

As L^{i} = 0 for i ≤ −h to conclude Hom(X_{c}, X_{b}[−r]) = 0 for r ≤ s − h we only need Hom(M_{c}, N[t]) = 0 for t = 1 + h, …, 1 + s. which is Lemma 5.3.5 (c). For r > s − h we’ll apply the functor Hom(−, X_{b}[−r]) to the second triangle giving the exact sequence:

Hom(U[h + 1], N[1 + s − r]) → Hom(V [1], N[1 + s − r]) → Hom(X_{c}, X_{b}[−r]) → Hom(U[h], N[1 + s − r]) → Hom(V, N[1 + s − r])

For r > 1 + s − h we only need Hom(V, N[t]) = 0 for 0 ≤ t < h − 1. While for r = 1 + s − h we need Hom(V, N[h − 1]) = 0 and Hom(U, N) → Hom(V, N[h]) to be 0 → 0 or k →^{=} k depending on whether U ≠ N or U = N. All these requirements follow from Lemma 5.3.4 (a),(b) and Lemma 5.3.5 (d).

Proof of V(C):

The observation preceding the proposition together with V(C, B) show that we have only to prove for c, c′ ∈ I_{>} that:

Hom(X_{c′}, X_{c}) = \{ \begin{matrix}k & if c′ = c, \\ 0 & otherwise \end{matrix}

We have the following triangles in D^{b}(B):

U[h] → X_{c} → V [1] → U[h + 1]
Ũ[h̃] → X_{c′} → Ṽ [1] → Ũ[h̃ + 1]

Applying the functor Hom(X_{c′}, −) to the first triangle gives the exact sequence:

Hom(X_{c′}, U[h]) → Hom(X_{c′}, X_{c}) → Hom(X_{c′}, V [1])

As U[h] = X_{b}[−d] for some d > 0, it follows from V(C, B) that we require:

Hom(X_{c′}, V [1]) = \{ \begin{matrix}k & if c′ = c, \\ 0 & otherwise \end{matrix}

Applying the functor Hom(−, V [1]) to the second triangle we have the exact sequence:

Hom(Ũ[1 + h̃], V [1]) → Hom(Ṽ [1], V [1]) → Hom(Ũ[h̃], V [1])

As h̃ ≥ 1 and we require that:

Hom(Ũ, V) =0
Hom(Ṽ , V) =\{ \begin{matrix}k & if c′ = c, \\ 0 & otherwise \end{matrix}

These both follow from Lemma 5.3.6.

We now turn to establishing part (c) of Theorem 4.4.2. By using triangles, for X_{c} ∈ C and h = h(c) > 1, of the form

H^{−h}(X_{c})[h] → X_{c} → H^{−1}(X_{c})[1] → H^{−h}(X_{c})[h + 1]

and noting that H^{−h}(X_{c}) (as a complex concentrated in degree 0) is X_{b}[r] for some r and X_{b} ∈ B, then it’s clear that the cohomology modules {H^{−1}(X_{c})} are contained in the triangulated category generated by the {X_{i}}. If h = h(c) = 1 then we can use the short exact sequence in (3) of definition 5.3.3 and, noting that in this case W is X_{b}[r] for some r and X_{b} ∈ B, we obtain that U_{−1} ⊗(c_{1}, ĉ_{2}) is contained in the triangulated category generated by the {X_{i}}.

Proposition 5.3.8. The {X_{i}} generate D^{b}(B) as a triangulated category.

Proof. Let T be the triangulated sub-category generated by the {X_{b}}. We’ll prove T = D^{b}(B) by showing that T contains every simple B-module. By the remarks preceding the proposition and the earlier description of the cohomology modules it’s clear that T contains the following modules; {M_{a} | a ∈ I_{<}}, {Z(s), σ(Z(s)) | 0 ≤ s ≤ (p − 3)/2}, {U_{−1} ⊗ M(c_{1}, ĉ_{2}) | c ∈ I_{>}, c_{2} > c_{1}} and {U_{−p} ⊗ M(ĉ_{1}, c_{2}) | c ∈ I_{>}, c_{1} > c_{2}}.

The triangles

U_{−1} ⊗ M(s, s) → Z(s) → M(s, ŝ) → U_{−1} ⊗ M(s, s)[1]
U_{−p} ⊗ M(s, s) → Z(s) → M(ŝ, s) → U_{−p} ⊗ M(s, s)[1]

show that T also contains the modules {U_{−1} ⊗ M(s, s), U_{−p} ⊗ M(s, s) | 0 ≤ s ≤ (p − 3)/2}. Let Y = {M_{a} | a = (a_{1}, a_{2}) ∈ I_{<} with a_{1} > a_{2}} ∪ {U_{−1} ⊗ M(c_{1}, ĉ_{2}) | c = (c_{1}, c_{2}) ∈ I_{>} with c_{2} > c_{1}} ∪ {U_{−1} ⊗ M(s, s) | 0 ≤ s ≤ (p − 3)/2}. So T contains the modules in Y and σ(Y). We order the elements in Y as follows (reading from left to right and top to bottom):

U_{−1} ⊗ M(0, 0), M(1, 0), …, U_{−1} ⊗ M(2i, 0), M(2i + 1, 0), …, M(p − 2, 0), U_{−1} ⊗ M(1, 1), M(2, 1), …, U_{−1} ⊗ M(1 + 2i, 1), M(1 + 2i + 1, 1), …, M(p − 3, 1),

⋮ U_{−1} ⊗ M(s, s), M(s + 1, s), …, U_{−1} ⊗ M(s + 2i, s), M(s + 2i + 1, s), …, M(p − 2 − s, s),

⋮ U_{−1} ⊗ M((p − 3)/2, (p − 3)/2), M((p − 1)/2, (p − 3)/2)

Similarly order the elements of σ(Y). The last module to appear in Y is M((p − 1)/2, (p − 3)/2) whose composition factors are the simple modules indexed by {(1 + (p − 3)/2 − 2i) + p((p − 3)/2 − 2j) | 0 ≤ i ≤ (p − 1)/2, 0 ≤ j ≤ (p − 3)/2}. Similarly the last module in σ(Y) is M((p − 3)/2, (p − 1)/2) whose composition factors are indexed by {((p − 3)/2 − 2i) + p(1 + (p − 3)/2 − 2j) | 0 ≤ i ≤ (p − 3)/2, 0 ≤ j ≤ (p − 1)/2}. As every odd integer n, 0 ≤ n < p^{2} − 1 has a representative in ℤ/(p^{2} − 1) of the form ±x ± py with 0 ≤ x, y ≤ (p − 1)/2 and not x = y = (p − 1)/2, we see that every simple B-module appears once as a composition factor of the last element in Y or σ(Y).

We’ll now show that each module in Y has all its composition factors except for one appearing as composition factors of earlier modules in Y. As the first entry in Y is a simple B-module, this will show that every composition factor of the last module in Y is contained in T . Repeating with σ(Y) will then show T = D^{b}(B).

We have the following maps between adjacent modules in Y

M(s + 2i − 1, s) ⟵^{p} U_{−1} ⊗ M(s + 2i, s) ⟶^{i} M(s + 2i + 1, s)

where i is injective with coker(i) = U_{s+2i+1} ⊗ M(0, s) and p is surjective with ker(p) = U_{−(s+2i+1)} ⊗ M(0, s).

Using the obvious triangles arising from p and i we can replace our list Y by the following list of modules contained in T U_{−1} ⊗ M(0, 0), U_{1} ⊗ M(0, 0), …, U_{−(2i+1)} ⊗ M(0, 0), U_{(2i+1)} ⊗ M(0, 0), …, U_{(p−2)} ⊗ M(0, 0),

⋮ U_{−(1+s)} ⊗ M(0, s), U_{(1+s)} ⊗ M(0, s), …, U_{−(s+2i+1)} ⊗ M(0, s), U_{(s+2i+1)} ⊗ M(0, s), …, Uŝ ⊗ M(0, s),

⋮ U_{−(p−1)/2} ⊗ M(0, (p − 3)/2), U_{(p−1)/2} ⊗ M(0, (p − 3)/2)

So it will be sufficient to show that each module in this list has all but one of its composition factors contained as composition factors of earlier modules. This is immediate as

\frac{U_{−(s+2i+1)} ⊗ M(0, s)}{soc(U_{−(s+2i+1)} ⊗ M(0, s))} ≅ U_{s−1+2j+1} ⊗ M(0, s − 1) where j = (p − 1)/2 − s − i

and

rad(U_{s+2i+1} ⊗ M(0, s)) ≅ U_{s−1+2j+1} ⊗ M(0, s − 1) where j = (p − 1)/2 − s − i.

All the stages in Section 4.5 have now been completed and so Theorem 5.3.1 has been proved.

5.4 The principal block of kSL_{2}(2^{n}) for 2 ≤ n ≤ 5

In this section we verify Conjecture 2.6.10 for the principal blocks of kSL_{2}(2^{n}), 2 ≤ n ≤ 5, with the aid of computer calculations. Throughout this section char(k) = 2. Although Conjecture 2.6.10 for kSL_{2}(2^{n}) in the cases n ≤ 4 was known to be true prior to this work (n = 2 due to Rickard, n = 3 due to Rouquier and n = 4 due to Chuang) we include these cases here, together with n = 5, because we’re able to present a uniform approach to all these cases. Unfortunately this approach breaks down for n = 6.

If we denote by A the principal block of kSL_{2}(2^{n}), with 2 ≤ n ≤ 5, and B the Brauer correspondent then our main result in this section is:

Theorem 5.4.1. Conjecture 2.6.10 is true for A and B.

We’ll follow the method outlined in Section 4.5 to prove it.

As before stage (1) of Section 4.5 has already been described in Section 5.2 and so we start with stage (2) of Section 4.5 . To define the objects {X_{i}} we mimic the alternative construction used in Section 5.3. If I is equal to the set of coefficients of 2-adic expansions of integers i, 0 ≤ i ≤ 2^{n} − 1, then I indexes the set of simple B-modules. Recall that in [7] certain n-tuple complexes are described whose total complexes yield minimal projective resolutions of the Green correspondents {M_{b}}, b ∈ I. For the n-tuple complex associated to M_{b}, b ∈ I, we can truncate this as in Section 5.3 and form the total complex Y_{b}. For the “excluded” subset used to truncate the n-tuple complexes we take the maximal subset of I, invariant under the Frobenius map σ, which indexes those M_{b} which, viewed as complexes concentrated in degree zero, already satisfy conditions (a) and (b) of Theorem 4.4.2. It’s easy to see, using Lemma 5.2.1, that

I_{<} = {(b_{1}, b_{2}, …, b_{n})|b_{1} + b_{2} + … + b_{n} < (n/2)}

We define X_{b} exactly as in Section 5.3 by completing the natural map Y_{b} → M_{b} to a triangle:

Y_{b} → M_{b} → X_{b} → Y_{b}[1]

Although we might expect to form truncations of n-tuple complexes in constructing the {X_{i}}, in practice at most only double complexes arise.

We now turn to stage (3) of Section 4.5. By using the results in Section 4.3, verifying conditions (a) and (b) of Theorem 4.4.2 is reduced to calculating the cohomology of total complexes of the double complexes Hom_{kH}(X_{b}^{i}, X_{c}^{j}) As each X_{b} is a total complex of a double complex, we have in effect to compute a 4-tuple complex W^{••••} = Hom_{B}(X_{b}_{••}, X_{c}^{••}).

By using duality and invariance of the {X_{b}} under the Frobenius map σ, we can reduce the number of calculations needed.

Typical terms of W^{••••} will be Hom_{B}(P, Q), Hom_{B}(M_{b}, P), Hom_{B}(Q, M_{c}) and Hom_{B}(M_{b}, M_{c}), where P, Q are indecomposable projective modules. As vector spaces these terms are either k^{2}, k or 0.

We can apply the duality functor D = Hom(−, k) to W^{••••}, when b ≠ c, and using the natural isomorphism, for P a projective module,

D(Hom_{B}(P, −)) ≅ Hom_{B}(−, P)

we almost obtain the 4-tuple complex associated with Hom_{K^{b}(B)}(X_{c}, X_{b}[m]). The only possible discrepancy involves the term Hom_{B}(M_{b}, M_{c}) and maps to and from it. In general we have

⋯ → P → M_{b} → 0
⋯ → Q → M_{c} → 0

where P and Q are projective covers of M_{b} and M_{c} respectively. The appropriate part of W^{••••} is:

\xymatrix{ & Hom_{B}(M_{b}, Q) \ar[d] \\ Hom_{B}(P, M_{c}) & Hom_{B}(M_{b}, M_{c}) \ar[d] \ar[l] & 0 \ar[l] \\  & 0}

The vertical map is onto as \underline{Hom}_{B}(M_{b}, M_{c}) = 0. The dual picture is:

\xymatrix{ & Hom_{B}(M_{c}, P) \ar[d] \\ Hom_{B}(Q, M_{b}) & D Hom_{B}(M_{b}, M_{c}) \ar[d] \ar[l] & 0 \ar[l] \\  & 0}

So we have to check that

ker(Hom_{B}(M_{c}, P) → D Hom_{B}(M_{b}, M_{c})),
im(D Hom_{B}(M_{b}, M_{c}) → Hom_{B}(Q, M_{b}))

remain unchanged if D Hom_{B}(M_{b}, M_{c}) is replaced by Hom_{B}(M_{c}, M_{b}). As D(Hom_{B}(M_{b}, M_{c})) = Hom_{B}(M_{c}, M_{b}) as vector spaces, this is trivial if Hom_{B}(M_{b}, M_{c}) = 0. The only other alternative is Hom_{B}(M_{b}, M_{c}) = D Hom_{B}(M_{b}, M_{c}) = Hom_{B}(M_{c}, M_{b}) = k with all arrows isomorphisms, and again the image and kernel spaces remain unchanged.

So we have, for b ≠ c, as vector spaces

Hom_{K^{b}(B)}(X_{c}, X_{b}[−m]) = Hom_{K^{b}(B)}(X_{b}, X_{c}[m])

This result together with the invariance of the {X_{b}} under σ and, by construction, the fact that {X_{b}|b ∈ I_{<}} already satisfy conditions (a) and (b) of Theorem 4.4.2 reduce the number of calculations needed. For n = 5 we only have 62 calculations instead of a potential 930 such calculations.

As described in Section 4.3, the calculations were carried out using routines written in MAGMA (see Appendix E).

All that remains for stage (3) of Section 4.5 is to verify condition (c) of Theorem 4.4.2. That is that the {X_{i}} generate D^{b}(B) as a triangulated category. This can be done by showing that the triangulated category T generated by {X_{i}} contains all the simple B-modules.

For the sake of completeness we will check this for n = 2, 3 and 4 as well as n = 5. We will exploit the fact that the {X_{i}} are invariant under the Frobenius automorphism σ of order n, thus if a certain module (viewed as a complex concentrated in degree zero) is contained in T , then so are all its Frobenius conjugates.

For each n, 2 ≤ n ≤ 5, we will choose representatives, up to the action of σ, from the {X_{i}} and list the terms of these complexes together with their non-zero cohomology. We will then write down triangles that show all the simple B-modules belong to T .

\underline{n = 2}

X_{0} = M_{0},  H^{0}(X_{0}) = k
X_{1} = 0 → P_{1} → M_{1},  H^{−1}(X_{1}) = \begin{matrix}k \\ 1 \end{matrix}

It is immediate that T contains all the simple B-modules.

\underline{n = 3}

X_{0} = M_{0},  H^{0}(X_{0}) = k
X_{1} = M_{1},  H^{0}(X_{1}) = {1 \atop 6}

X_{3} = 0 → P_{3} → M_{3},  H^{−1}(X_{3}) = \begin{aligned}& 2 \\ k & & 5 \\ & 3 \end{aligned}

Using triangles such as:

\begin{matrix}k \\ 3 \end{matrix} → H^{−1}(X_{3}) → H^{0}(X_{2}) → \begin{matrix}k \\ 3 \end{matrix} [1]

it’s easy to see T = D^{b}(B).

\underline{n = 4}

X_{0} = M_{0},  H^{0}(X_{0}) = k, X_{1} = M_{1},  H^{0}(X_{1}) = {1 \atop 14} ,

X_{3} = 0 → P_{10} → P_{3} → M_{3}, H^{−2}(X_{3}) =[figure: 10; 11,; 13; 12; 14; 1; k; 2]

H^{−1}(X_{3}) = \begin{aligned}& 2 \\ k & & 13 \\ & 11 \end{aligned} ,

X_{10} = 0 → P_{10} → M_{10}, H^{−1}(X_{10}) = [figure: 4; 3; 7; 8; k; 14; 10; 11; 1; 12; 13; 2]

X_{11} = 0 → P_{10} → P_{3} → P_{11} → M_{11}, H^{−3}(X_{11}) ≅ H^{−2}(X_{3})

As H^{−2}(X_{3}) ≅ H^{−3}(X_{11}) ∈ T , we see that H^{−1}(X_{3}) ∈ T and the triangle

\begin{matrix}k \\ 11 \end{matrix} → H^{−1}(X_{3}) → X_{2} → \begin{matrix}k \\ 11 \end{matrix} [1]

shows 11 ∈ T (hence 7, 14, 13 ∈ T ). As X_{1} ∈ T , so 1 ∈ T (hence 2, 4, 8 ∈ T ). The triangle

H^{−3}(X_{11}) → H^{−1}(X_{10}) → 8 7  4 3 → H^{−3}(X_{11})[1]

shows 3 ∈ T (hence 6, 12, 9 ∈ T ). Finally as H^{−1}(X_{10}) ∈ T , we get 5 ∈ T (hence also 10 ∈ T ).

X_{0} = M_{0}, H^{0}(X_{0}) = k, X_{1} = M_{1}, H^{0}(X_{1}) = {1 \atop 30} ,

X_{6} = M_{6}, H^{0}(X_{6}) = 6 29  2 25 , X_{18} = M_{18},  H^{0}(X_{18}) = 18 17  14 13 ,

X_{7} = 0 → P_{22} → P_{7} → M_{7}, H^{−1}(X_{7}) =[figure: 23; 25,; 29; 27; k; 4; 2; 6] H^{−2}(X_{7})

X_{11} = 0 → P_{11} → M_{11}, H^{−1}(X_{11})

X_{23} = 0 → P_{22} → P_{7} → P_{23} → M_{23}, H^{−3}(X_{23}) ≅ H^{−2}(X_{7})

Due to the difficulty of drawing modules with dimensions greater than 3, we have chosen to just write down the non-zero cohomology where appropriate. However it is not difficult to write down, for those modules not illustrated, the simple B-modules together and their coordinates that describe them.

As H^{−3}(X_{23}) ∈ T the triangle

H^{−3}(X_{23})[2] → X_{7} → H^{−1}(X_{7})[1] → H^{−3}(X_{23})[3]

shows H^{−1}(X_{7}) ∈ T . Denoting by L the module \begin{aligned}& 4 \\ k & & 2 \\ & 23 \end{aligned} , then triangles

L → H^{−1}(X_{7}) → X_{6} → L[1],

\begin{matrix}k \\ 23 \end{matrix} → L → X_{4} → \begin{matrix}k \\ 23 \end{matrix} [1]

show that 23 ∈ T (hence 15, 30, 29, 27 ∈ T ). We can picture the top Loewy layer structure of H^{−1}(X_{11}) ∈ T as:

[figure: 10; 3]

It’s fairly easy to calculate that the submodule whose top is the 3 indicated is isomorphic to H^{−3}(X_{27}) ∈ T , and so we have a triangle

H^{−3}(X_{27}) → H^{−1}(X_{11}) → G → H^{−3}(X_{27})[1]

where G is the module:

[figure: 19; 21; 25; 23; 4; 8; 6; 10]

Denoting by J the module 8 4  23 19 , then triangles

J → G → X_{10} → J[1],

{23 \atop 19} → J → X_{8} → \begin{matrix}23 \\ 19 \end{matrix} [1]

show 19 ∈ T (hence 7, 14, 28, 25 ∈ T ). We have all but one of the composition factors of X_{3} in T and so we must have 3 ∈ T (hence 6, 12, 24, 17 ∈ T ). We now only need 5 and 11 and their conjugates belong to T . As H^{−3}(X_{15}), 3 =Top(H^{−3}(X_{15})) ∈ T , we have R = rad(H^{−3}(X_{15})) ∈ T . We may picture the top Loewy layer structure of R as:

[figure: 18; 2; 30; 1]

We have the triangle

T → R → {1 ⊕ 30 ⊕ 2} → T [1]

where the top Loewy layer structure of T looks like:

[figure: 15; 27; 29; 12; 28; k; 16; 13; 17; 14; 18]

So we can form the triangle:

F → T → {29 ⊕ 28 ⊕ k} → F [1]

where F ∈ T is the module:

[figure: 11; 13; 14; 12; 15; 16; 17; 18; 27]

We now have the triangle

W → F → X_{18} → W [1]

and all but one of the composition factors of W ∈ T are already known to lie in T and so we have 11 ∈ T (hence 22, 13, 26, 21 ∈ T ). Similarly X^{18}, rad(X_{18}) ∈ T and so 18 ∈ T (hence 5, 10, 20, 9 ∈ T ) and so T = D^{b}(B).

All the stages of Section 4.5 have now been completed, thus finishing the proof of Theorem 5.4.1.

5.5 The principal block of kSL_{2}(27)

In this section we verify Conjecture 2.6.10 for the principal block of kSL_{2}(27). Throughout this section char(k) = 3. Denoting by A and B the principal blocks of kSL_{2}(27) and kH respectively, our main result is:

Theorem 5.5.1. Conjecture 2.6.10 is true for A and B.

Again we’ll prove this by following the method described in Section 4.5.

As stage (1) of Section 4.5 is described in Section 5.2 we start with stage (2). To define the objects {X_{i}} we repeat the construction used in Sections 5.3 and 5.4. With I equal to the set of coefficients of the 3-adic expansions of integers i, 0 ≤ i < 27, the excluded set I_{<} is easily seen, using Lemma 5.2.1, to be

I_{<} = {(b_{1}, b_{2}, b_{3})|b_{1} + b_{2} + b_{3} < 3}

Representatives, up to the action of σ, from {X_{i}} are:

X_{0} = M(0, 0, 0),  X_{12} = M(0, 1, 1),  X_{18} = M(0, 0, 2),
X_{22} = Ω(M(1, 1, 2))[1], X_{24} = Ω^{2}(M(0, 2, 2))[2]

We now turn to stage (3) of Section 4.5. It is very easy to see that conditions (a) and (b) of Theorem 4.4.2 amount to checking that the following hold:

End_{B}(M(0, 0, 0)) = End_{B}(M(1, 1, 0)) = End_{B}(ΩM(1, 1, 2)) = End_{B}(Ω^{2}M(0, 2, 2)) = k

Hom_{B}(ΩM(1, 1, 2), Ω^{2}M(0, 2, 2)) = Hom_{B}(ΩM(1, 1, 2), Ω^{2}M(2, 0, 2)) = 0

Hom_{B}(ΩM(1, 1, 2), Ω^{2}M(2, 2, 0)) = Hom_{B}(M(0, 0, 0), Ω^{2}M(0, 2, 2)) = 0 Hom_{B}(M(0, 0, 0), ΩM(1, 1, 2)) = 0

The remaining conditions can be obtained by applying σ to one of the above. That these conditions do hold follows trivially from the diagrams of the modules involved, see Figure 5.4.

24 1416 6 8  8 14 1824  20 22 6 8 24 16 2 20  14 12 16 10  8 18 1622 18 3 4 6 k 20 Ω^{2}M(2, 2, 0) = , 18 10 2 k 24 Ω^{2}M(2, 0, 2) =  16 22 2  k 6 12 20 4 6 k 20 ΩM(1, 2, 1) =  14 16 18  k 2 4 24 10 2 k 24 ΩM(2, 1, 1) = , 18 8 M(1, 0, 1) =  6 20 M(0, 1, 1) = k M(0, 0, 2) = , 20 6 k M(0, 2, 0) = , 2214 2  6 2012 18 k 8 Ω^{2}M(0, 2, 2) =  , 22 14 6  k 18 10 8 12 18 k 8 , 2224 4 2  , 24 2 k M(2, 0, 0) = , k ΩM(1, 1, 2) = M(1, 1, 0) = M(0, 0, 0) =

Figure 5.4:

Only condition (c) of Theorem 4.4.2 remains to be checked. Let T be the triangulated subcategory generated by the {X_{i}}. We’ll show that all 13 simple B-modules lie in T . Clearly all the modules listed in Figure 5.4 lie in T . Letting Y = coker(Ω^{2}M(0, 2, 2) → ΩM(1, 1, 2)), we have triangles:

Ω^{2}M(0, 2, 2) → ΩM(1, 1, 2) → Y → Ω^{2}M(0, 2, 2)[1]

\begin{matrix}k \\ 24 \end{matrix} → Y → M(1, 0, 1) → \begin{matrix}k \\ 24 \end{matrix} [1]

As k ∈ T , we also have 24 ∈ T (hence 20, 8 ∈ T ). As M(2, 0, 0), M(0, 2, 0), M(0, 0, 2) ∈ T we get 2, 6, 18 ∈ T .

Now let Y = ker(Ω^{2}M(2, 0, 2) → M(2, 0, 0)), and form triangles

Y → Ω^{2}M(2, 0, 2) → M(2, 0, 0) → Y [1],

{6 \atop 14} → Y → M(1, 0, 1) → \begin{matrix}6 \\ 14 \end{matrix} [1]

So we have 14 ∈ T (hence 16, 22 ∈ T ). Finally, as M(1, 1, 0), M(0, 1, 1), M(1, 0, 1) ∈ T we have 4, 10, 12 ∈ T . So T = D^{b}(B).

Hence all the stages of Section 4.5 have been checked and so Theorem 5.5.1 is proved.

The Hall-Janko group, its double cover, U_{3}(4) and Sp_{4}(4)

6.1 Introduction

In this chapter we verify Conjecture 2.6.6 applied to the groups: 2.J_{2} (we subsume the applicable cases of the Hall-Janko group, J_{2}, within those of its double cover 2.J_{2}), U_{3}(4) and Sp_{4}(4).

For what primes and blocks can Conjecture 2.6.6 apply to the groups 2.J_{2}, U_{3}(4) and Sp_{4}(4)? Apart from those cases for which the conjecture is already known to be true, see Section 2.6, the only remaining non-trivial application of Conjecture 2.6.6 to the examples considered here is when p = 5.

When p = 5 the defect groups of blocks of k[2.J_{2}], kU_{3}(4) and kSp_{4}(4) that are abelian and non-cyclic are all elementary abelian of order 25; in fact the defect groups are all Sylow 5-subgroups so that Conjecture 2.6.10 is applicable.

Let us introduce some notation for this chapter. Unless otherwise stated we’ll take char(k) = 5 (in fact as far as the computer calculations are concerned, GF(5) is a suitable splitting field for our groups). We’ll write ekG and fkH to denote blocks of full defect as in Conjecture 2.6.6, with G being one of our groups. Which blocks and which group should be clear from the context.

Our main result for this chapter is:

Theorem 6.1.1. Conjecture 2.6.10 is true for ekG and fkH.

We’ll prove this by following the method of Section 4.5.

In Section 6.2 we’ll record some relevant information about our groups and the blocks for which Conjecture 2.6.10 is applicable.

In Section 6.3 we carry out stage (1) of Section 4.5 with the aid of computer calculations and describe the stable equivalence of splendid type referred to in Section 3.3 and described in [34]. In particular we will calculate the images of simple modules under tensoring with the splendid complex C of Section 3.3.

In Section 6.4 we turn to stages (2) and (3) of Section 4.5 and describe the computer calculations used to try and find the objects {X_{i}} as well as the computer calculations used to establish conditions (a), (b) and (c) of Theorem 4.4.2.

Finally in Section 6.5 we show, with the aid of further computer calculations, that information about the complexes {X_{i}} contained in Appendices C and D is sufficient to determine the complexes, up to isomorphism, in the homotopy category.

At least as far as Section 6.2, the defect group of our blocks will be written P, a Sylow 5-subgroup of G. The simple ekG-modules will be denoted by {S_{i}}, while simple fkHmodules will be indexed by integers except for the trivial module which will always be written as k. The ordering used can be determined with reference to the Loewy structure of the Green correspondents listed in Appendix A.

6.2 The representation theory of 2.J_{2}, U_{3}(4) and Sp_{4}(4)

\underline{G = U_{3}(4)}

|G| = 62400 = 2^{6}.3.5^{2}.13. The principal block ekG is the only block of full defect of kG. H = P : Σ_{3}. Up to conjugacy in H there are two cyclic 5-subgroups of P. Only one, Q, has distinct centralizers in G and H, with C_{G}(Q) = Q × A_{5} and C_{H}(Q) = Q × D_{10}.

simple ekG-modules S_{1} = k S_{2} S_{3}
dimension 1 12 39
simple fkH-modules k  2 3
dimension  1  1 2

\underline{G = 2.J_{2}}

|G| = 1209600 = 2^{8}.3^{3}.5^{2}.17. There are two blocks of full defect. The principal block is of course the same as the principal block of kJ_{2} and we’ll always view it as such in the sequel.

For the principal block ekG with G = J_{2} and H = P : D_{12} there are, up to conjugacy in H, two cyclic 5-subgroups. Only one, Q, has distinct centralizers in G and H, with C_{G}(Q) = Q × A_{5} and C_{H}(Q) = Q × D_{10}.

simple ekG-modules S_{1} = k S_{2} S_{3} S_{4} S_{5} S_{6}
dimension 1 14 21 41 85 189
simple fkH-modules k  2 3 4 5 6
dimension  1  1 1 1 2 2

For the non-principal block ekG with G = 2.J_{2} and H = P : (2.D_{12}), there is only one cyclic 5-subgroup Q, up to conjugacy in H, with distinct centralizers. C_{G}(Q) = Q×SL_{2}(5) and C_{H}(Q) = Q × (5 : 4).

simple ekG-modules S_{1} S_{2} S_{3} S_{4} S_{5} S_{6}
dimension  6 14 56 64 190 202
simple fkH-modules 1 2 3 4 5  6
dimension  1 1 1 1 2  2

\underline{G = Sp_{4}(4)}

|G| = 979200 = 2^{8}.3^{2}.5^{2}.17. The principal block ekG is the only block of full defect and H = P : D_{8}. Up to conjugacy in H there are only two cyclic 5-subgroups with distinct centralizers in G and H. Denoting these two groups as Q_{1} and Q_{2}, we have C_{G}(Q_{i}) = Q_{i} × A_{5} and C_{H}(Q_{i}) = Q_{i} × D_{10} for i = 1, 2.

simple ekG-modules S_{1} = k S_{2} S_{3} S_{4} S_{5}
dimension 1 18 33 33 153
simple fkH-modules k  2 3 4 5
dimension  1  1 1 1 2

6.3 Stable equivalence of splendid type

We now turn to establishing stage (1) of Section 4.5. For the examples considered in this Chapter, restriction does not induce a stable equivalence between ekG and fkH. However, as the defect groups are all Sylow 5-subgroups isomorphic to C_{5} × C_{5}, we can use Rouquier’s construction of a 2-term complex C outlined in Section 3.3 of Chapter 3 in order to obtain a stable equivalence of splendid type.

Actually all we’ll need to know are the images of the simple ekG-modules under tensoring with C. We’ll start by describing what the terms of C are for our examples. Then we’ll describe how the terms of S_{i} ⊗_{ekG} C were calculated explicitly using MAGMA and finally we describe how the differential of S_{i} ⊗_{ekG} C was computed.

We now describe the modules Ñ_{Q} that appear in the degree −1 term of the complex C (see (3.1) of Chapter 3) for our examples. To keep matters simple we’ll start with the principal blocks only. For principal blocks the information from Section 6.2 shows that the local situation in each of our cases is essentially the same. If Q denotes a cyclic 5-subgroup with distinct centralizers in G and H, then we have C_{G}(Q) = Q × A_{5} and C_{H}(Q) = Q × D_{10}.

A splendid equivalence between kQ and kQ is induced by kQ viewed as a kQ-kQ-bimodule. A splendid equivalence between the principal blocks e_{Q}kA_{5} and f_{Q}kD_{10} can be calculated using [35]. It is induced by the splendid tilting complex:

P_{1} ⊗ R_{1} → e_{Q}kA_{5}f_{Q},

where P_{1} (resp. R_{1}) is the unique indecomposable projective module of e_{Q}kA_{5} (resp. f_{Q}kD_{10}) that is not the projective cover of the trivial module. So the required local splendid equivalence between e_{Q}kC_{G}(Q) and f_{Q}kC_{H}(Q) is induced by the splendid tilting complex:

kQ ⊗ P_{1} ⊗ R_{1} → e_{Q}kC_{G}(Q)f_{Q}

We need to lift this to a complex of kN_{G×H}(ΔQ)-modules. For G = U_{3}(4), N_{G×H}(ΔQ) = C_{G×H}(ΔQ), and so it is already a complex of kN_{G×H}(ΔQ)-modules. For the other cases, we always have N_{G×H}(ΔQ) = C_{G×H}(ΔQ).2 and so we need to lift kQ ⊗ P_{1} ⊗ R_{1} through a cyclic 2-group. In fact N_{G×H}(ΔQ) = A_{5} × (Q × C_{H}(Q)).2, so that P_{1} already lifts to a kN_{G×H}(ΔQ)-module. To lift R_{1} and kQ we have that:

Q × Q ◁ (Q × C_{H}(Q)).2 with (Q × C_{H}(Q)).2/(Q × Q) ≅ D_{10}.2
1 × D_{10} ◁ (Q × C_{H}(Q)).2 with (Q × C_{H}(Q)).2/(1 × D_{10}) ≅ (Q × Q).2

So it is enough to lift R_{1} to a k[D_{10}.2]-module and kQ to a k(Q × Q).2-module. Now k[D_{10}.2] has 2 blocks each containing two indecomposable projective modules that are lifts of the 2 indecomposable projective kD_{10}-modules. We want the projective module R̄_{1} lying in the principal block of kD_{10}.2 that lifts R_{1}. For kQ we have kQ = k ↑_{ΔQ}^{Q×Q}, as a

kQ-kQ-bimodule. It follows easily from the Mackey decomposition formula that k ↑_{ΔQ.2}^{(Q×Q).2}

is the required lift. So the module Ñ_{Q} = k ↑_{ΔQ.2}^{(Q×Q).2} ⊗P_{1} ⊗ R̄_{1}. Note that in the case of G = Sp_{4}(4) we have two non-zero contributions Ñ_{Q_{1}} ↑^{G×H} and Ñ_{Q_{2}} ↑^{G×H} to the second term of the complex C.

For the non-principal block ekG ( G = 2.J_{2}) the local splendid tilting complex is given by:

kQ ⊗ P_{2} ⊗ R_{2} → e_{Q}kC_{G}(Q)f_{Q}

where P_{2} is the projective lying in the non-principal block of kSL_{2}(5) that is the projective cover of the 4 dimensional simple module T. R_{2} is the projective cover of the top of T ↓_{2.D_{10}}. Again N_{G×H}(ΔQ) = C_{G×H}(ΔQ).2 and as in the principal block case, P_{2} already lifts. We take R̄_{2} to be the lift of R_{2} lying in the non-principal block of k[2.(D_{10}.2)]. So Ñ_{Q} = k ↑_{ΔQ.2}^{(Q×Q).2} ⊗P_{2} ⊗ R̄_{2}.

Our next step is to calculate the terms of the complexes {S_{i} ⊗_{ekG} C}. To help with the calculations, the character of the complex C was computed using GAP. More precisely the character of each term of C was calculated and so the composition factors of each term of the complexes {S_{i} ⊗_{ekG} C} were calculated (this was not done for the non-principal block of k[2.J_{2}] due to limitations in memory space available to GAP).

We’ll want to calculate the terms of the complexes {S_{i} ⊗_{ekG} C}, using functions in MAGMA’s toolkit. By way of illustration we only consider the principal block cases and the contribution from only one cyclic 5-subgroup Q. The degree zero term is easy as this is just restriction S_{i} ↓_{H}^{G}. The degree −1 term is

S_{i} ⊗_{ekG} Ñ_{Q} ↑_{N_{G×H^{∘}}(ΔQ)}^{G×H}= S_{i} ⊗_{e_{Q}kN_{G}(Q)} (Ñ_{Q} ↑_{N_{G×H^{∘}}(ΔQ)}^{N_{G}(Q)×N_{H}(Q)}) ⊗_{f_{Q}kN_{H}(Q)} kHf

= S_{i} ⊗_{e_{Q}kN_{G}(Q)} P_{1} ⊗ k(Q.2) ⊗_{k[(ΔQ).2]} k(Q.2) ⊗ R̄_{1} ⊗_{f_{Q}kN_{H}(Q)} kfH

In order to use MAGMA’s functions, we interpret this term as a summand of S_{i} ↓_{Q.2}^{G}↑_{Q.2}^{H}. Although we cannot determine the summand of S_{i} ↓_{Q.2}^{G} we want, we can at least calculate its dimension as dim Hom_{kA_{5}}(P_{1}, S_{i} ↓_{A_{5}}^{G}). Once this summand has been identified, to calculate the summand of ? ↑_{Q.2}^{H} required, one should calculate ? ↑_{Q.2}^{Q.2×D_{10}.2}↓_{N_{H}(Q)}^{Q.2×D_{10}.2}

taking only that summand of ? ↑_{Q.2}^{Q.2×D_{10}.2} that restricts to D_{10}.2 to give nR̄_{1} (for some n). As Q.2 × D_{10}.2 ≰ H, one would have to carefully construct this group in addition to the others, but it’s simpler to proceed as follows. We can split ? ↑_{Q.2}^{N_{H}(Q)} into 2 summands say M_{0}, M_{1}, where M_{1} ↓_{D_{10}} is nR_{1} (for some n) and M_{0} restricts to give a sum of copies of the other projective D_{10}-module. Let M be an indecomposable summand of M_{1}, and let M ↑^{Q.2×D_{10}.2}= M̄ ⊕ ¯ M̄ where M̄ ↓_{D_{10}.2} is nR̄_{1} (for some n) and ¯ M̄ restricts to give a sum of copies of the other lift of R_{1}. As the identity functor is a summand of ↑_{N_{H}(Q)}^{Q.2×D_{10}.2}↓_{N_{H}(Q)} we have M|(M̄ ↓_{N_{H}(Q)} ⊕ ¯M̄ ↓_{N_{H}(Q)}). There are two possibilities; either M̄ ↓_{N_{H}(Q)}≅ ¯ ¯M̸ ↓_{N_{H}(Q)}, in which case we would expect M ↓_{D_{10}.2} to be a sum of copies of R̄_{1} or a sum of copies of the other lift, or M̄ ↓_{N_{H}(Q)}≅ ¯ M̄ ↓_{N_{H}(Q)}, in which case M ↓_{D_{10}.2} will be a sum possibly involving both lifts of R_{1}. In our calculations this latter situation always arose and so the correct summand we want, viz M̄ ↓_{N_{H}(Q)}≅ M, can be identified by examining the summands of ? ↑_{Q.2}^{N_{H}(Q)}.

These calculations were carried through for our examples and the Loewy structures of the non-projective summand of the degree −1 terms of the complexes {S_{i} ⊗_{ekG} C} are listed in Appendix B. Where ambiguities arose as to which summand of ? ↓_{Q.2} to take, most were resolved with reference to the composition factors of the module required (for G = J_{2} and S_{6} there were still two possible candidates and one was eliminated as no suitable differential exists, see below).

Finally, having determined the terms of the complexes {S_{i} ⊗_{ekG} C} we now need to calculate the differentials. Although the differential in C is explicitly described in [34], that description does not readily lend itself to our calculations. In fact it will prove sufficient to know that such a differential exists for which C induces a stable equivalence in order to calculate the differentials of the complexes {S_{i} ⊗_{ekG} C} using MAGMA. The general situation we’re faced with is having a complex

Z = 0 ⟶ X ⟶^{d} Y ⟶ 0

with X and Y non-projective modules, such that the image of Z in \underline{mod}- fkH is the image of a simple ekG-module. In particular \underline{End}_{fkH}(Z) ≅ k. Clearly we must have a choice of d that is non-zero in \underline{Hom}_{fkH}(X, Y). In most of our cases dim \underline{Hom}_{fkH}(X, Y) = 2. By careful choice of coset representatives, maps f_{1}, f_{2} ∈ Hom_{fkH}(X, Y) were found whose images formed a basis for \underline{Hom}_{fkH}(X, Y), and such that f_{2} = ϕf_{1} with ϕ ∈ End_{fkH}(X) a nilpotent endomorphism. Up to a scalar multiple an element of Hom_{fkH}(X, Y) spanned by f_{1}, f_{2} will be of the form f_{1} + λϕf_{1} (λ ∈ k) or ϕf_{1}. The following isomorphism in Ch(mod- fkH)

X ⟶^{f_{1}+λϕf_{1}} Y
↓^{1+λϕ}  ∥
X ⟶^{f_{1}} Y

shows that up to isomorphism there are only two choices for the differential, viz d = f_{1} or d = ϕf_{1}. For each choice dim \underline{End}_{fkH}(Z) was calculated and was equal to 1 only for d = f_{1}. To calculate dim \underline{End}_{fkH}(Z) using MAGMA functions, it is much easier to work with the module in \underline{mod}- fkH represented by Z. This module can be calculated by noting that we have the triangle

X ⟶^{d} Y ⟶ Z ⟶ X[1]

in D^{b}(fkH) and hence the same triangle in \underline{mod}- fkH, whose triangulated structure (see Section 2.4.2) can be used to calculate the module representing Z.

6.4 Splendid equivalence

Having completed stage (1) of Section 4.5 in the previous section, we now finish the proof of Theorem 6.1.1 by describing the completion of stages (2) and (3).

We start by describing the computer calculations employed to construct the objects {X_{i}} for stage (2). In describing the construction of X_{i}, we’ll use the fact that it fits into a triangle (4.5). So we’ll describe the construction of the complex P_{i} and map P_{i} → S_{i}⊗_{ekG}C that fits into triangle (4.5).

In all cases the complex P_{i} and map P_{i} → S_{i} ⊗_{ekG} C were computed as a trimmed projective resolution of S_{i} ⊗_{ekG} C. That is, having already constructed the degree −n + 1 term, the degree −n term, P_{i}^{−n}, will be a certain summand i : P_{i}^{−n} → P C where θ : P C → K_{i}^{−n+1} is the projective cover of K_{i}^{−n+1} = ker(d^{−n+1}). The differential d^{−n} will then be the composite

P_{i}^{−n} ⟶^{i} P C ⟶^{θ} K_{i}^{−n+1} ⟶^{j} P_{i}^{−n+1}

where j is the natural inclusion. There will of course be no unique choice for θ, see Section 6.5.

In seeking the correct terms of P_{i} for our examples, character calculations using GAP again provided a useful guide.

As described in Section 6.3 the character χ_{C} of the complex C was calculated. In addition an isotypy, see Section 2.6, for each of our cases was calculated. As described in Section 2.6 a perfect isometry

I : CF ((G, e), K) → CF ((H, f), K)

coming from the isotypy, where (K, O, k) is a p modular system, is suspected to be the map induced by the character, χ, of a complex K ⊗_{O} X, where X is a splendid tilting complex of eOG-fOH-bimodules.

Using decomposition matrices, the map on Grothendieck groups K_{0}(mod- ekG) → K_{0}(mod- fkH) induced by the character χ of k ⊗_{O} X can be calculated. Assuming the splendid tilting complex X lifts our complex C, then we would have a triangle

Q → C → k ⊗_{O} X → Q[1]

where Q is a bounded complex of projective ekG-fkH-bimodules. If this is the case then χ − χ_{C} should induce a map K_{0}(mod- ekG) → K_{0}(proj- fkH). The image of the simple ekG module S_{i} will then be the image in K_{0}(proj- fkH) of the complex P_{i} required to define X_{i} and in this way can act as a guide in choosing the terms of P_{i}.

After suitable candidate objects {X_{i}} were found that agreed, in the Grothendieck group, with our character calculations, the checking of conditions (a) and (b) of Theorem 4.4.2 for stage (3) of Section 4.5 was done. These conditions were checked using MAGMA, as described in Section 4.3. The complexes {X_{i}}, whose terms are described in Appendix C and whose cohomology is described in Appendix D, were found to satisfy conditions (a) and (b) of Theorem 4.4.2.

Finally we need check condition (c) of Theorem 4.4.2 for our objects {X_{i}}. In doing so we’ll make use of the results in Appendix D without explicit reference. Appendix D describes the Loewy layers of the non-zero cohomology groups of the X_{i}, together with some useful short exact sequences. The short exact sequences together with the isomorphisms of some of the cohomology modules were verified using standard functions in MAGMA.

Proof of condition (c) of Theorem 4.4.2. If we let T denote the full triangulated subcategory of D^{b}(fkH) generated by the {X_{i}} then, for each of our examples, we’ll show that T = D^{b}(fkH) by showing that T contains all the simple fkH-modules.

\underline{G = U_{3}(4)} Clearly k ∈ T and as H^{−1}(X_{2}) = k, we must have H^{−2}(X_{2}) ∈ T and hence 3 ∈ T . Finally as H^{−3}(X_{3}) ∈ T and has 2 occuring as a composition factor only once, 2 ∈ T .

All simple fkH-modules have now been shown to be lie in T .

\underline{G = J_{2}}

We have H^{0}(X_{1}) = k ∈ T . As k = H^{−1}(X_{3}) ∈ T we must have H^{−2}(X_{3}) ∈ T . Also, as X_{5} only has one non-zero cohomology module, H^{−4}(X_{5}) ≅ H^{−3}(X_{4}) ∈ T , and hence 2 = H^{−2}(X_{4}) ∈ T .

The short exact sequence (D.2) now shows {2 \atop 5} ∈ T and hence 5 ∈ T . The short exact

sequence (D.3) now implies H^{−3}(X_{6}) ∈ T . As k = H^{−2}(X_{6}) the only other non-zero cohomology for X_{6}, viz 3 = H^{−4}(X_{6}), must be in T .

Finally as H^{−3}(X_{2}) ∈ T and has only one composition factor isomorphic to 6, with multiplicity one, 6 ∈ T . So all simple fkH-modules belong to T .

\underline{G = Sp_{4}(4)}

Again we have k = H^{0}(X_{1}) ∈ T . As H^{−2}(X_{2}) = k, we must have H^{−3}(X_{2}) ∈ T and thus 5 ∈ T . Similarly, as H^{−1}(X_{3}) = H^{−1}(X_{4}) = k, we also have H^{−3}(X_{3}) = 2 ∈ T and H^{−3}(X_{4}) = 4 ∈ T .

Finally X_{5} has only one non-zero cohomology and so H^{−4}(X_{5}) ∈ T . The simple fkHmodule 3 occurs in the composition factors of H^{−4}(X_{5}) with multiplicity one, so 3 ∈ T . All simple fkH-modules have now been shown to be in T .

G = 2.J_{2}, non-principal block

As X_{2} and X_{5} both only have one non-zero cohomology module, 2 = H^{−3}(X_{2}) ∈ T and H^{−5}(X_{5}) ∈ T . The short exact sequence (D.4) shows H^{−2}(X_{6}) ∈ T and so we must have H^{−1}(X_{6}) ∈ T . The short exact sequence (D.5) now implies that 1 ∈ T .

As H^{−3}(X_{3}) ≅ H^{−4}(X_{3}) ≅ H^{−5}(X_{5}) ∈ T , the only other non-zero cohomology module for X_{3}, viz H^{−2}(X_{3}), must belong to T . The short exact sequence (D.6) now implies that H^{−2}(X_{4}) ∈ T and hence 6 ∈ T and H^{−3}(X_{4}) ∈ T . Using H^{−2}(X_{1}) ≅ H^{−3}(X_{4}) ∈ T , we must have H^{−1}(X_{1}) ∈ T and hence 4 ∈ T .

The simple fkH-module 5 occurs only once in the Loewy structure of H^{−3}(X_{4}) ∈ T , thus 5 ∈ T . Similarly the simple fkH-module 3 occurs only once in the Loewy structure of H^{−2}(X_{3}) ∈ T and so 3 ∈ T . All simple fkH-modules have now been shown to belong to T .

So all the stages of Section 4.5 have been completed, thus proving Theorem 6.1.1.

6.5 Independence of choice of differentials

In this section we’ll show that, given the information in Appendices C and D, the complexes {X_{i}} for the examples in this chapter are independent, up to isomorphism in K^{b}(fkH), of the choice of differentials.

Suppose we have constructed two trimmed resolutions of S_{i} ⊗_{ekG} C, both consistent with the information in Appendices C and D, up to degree −n + 1. Suppose also that we have a cochain isomorphism between the two complexes, which we may as well take to be an identification.

In constructing the degree −n differential suppose we make two choices; ψ, ψ′ ∈ Hom_{fkH}(P_{i}^{−n}, K_{i}^{−n+1}), where K_{i}^{−n+1} = ker(d^{−n+1}), and hence two choices for the degree −n differential. Can we find a cochain isomorphism between the two new complexes?

We’ll show that we can for all the X_{i} described in Appendices C and D except for one. However, for that exceptional case we’ll still be able to show, with the help of computer calculations, that it is independent, up to isomorphism in K^{b}(fkH), of the choice of differentials.

Returning to our two partially constructed trimmed resolutions, we have the following diagram,

[figure: P_{i}^{−n}; ψ; P_{i}^{−n+1}; P_{i}^{−n+2}; …; K_{i}^{−n+1}; j; P_{i}^{−n}; ψ′; P_{i}^{−n+1}; P_{i}^{−n+2}; …; K_{i}^{−n+1}; j]

where j : K^{−n+1} → P_{i}^{−n+1} is the natural inclusion.

If Hom_{fkH}(P_{i}^{−n}, H^{−n+1}(X_{i})) = 0, then im(ψ) can be described as the largest submodule of K_{i}^{−n+1} for which there are no maps from P_{i}^{−n} to the quotient. Hence im(ψ) = im(ψ′) and so we have an isomorphism ϕ such that

[figure: P_{i}^{−n}; ϕ; d^{−n}=jψ; P_{i}^{−n+1}; P_{i}^{−n+2}; …; P_{i}^{−n}; d′^{−n}=jψ′; P_{i}^{−n+1}; P_{i}^{−n+2}; …]

is a cochain map isomorphism.

From the information in Appendices C and D the only complexes not covered by the above argument are: X_{6} (n = 4) for J_{2}, X_{6} (n = 2) and X_{3} (n = 3, 4) for 2.J_{2}, nonprincipal block. For these cases Hom_{fkH}(P_{i}^{−n}, H^{−n+1}(X_{i})) ≠ 0.

We now fix a map ψ, with q : K^{−n+1} → K^{−n+1}/ im(ψ) = H^{−n+1}(X_{i}) the natural quotient, and suppose there is a map η : P_{i}^{−n+1} → K_{i}^{−n+1} such that qη ≠ 0. A choice of map ψ′ not covered by the above argument is to take ψ′ = ψ + η. If the map η factors as η = γd^{−n} = γjψ, where γ : P_{i}^{−n+1} → K_{i}^{−n} then we can adjust our degree −n + 1 identification to an isomorphism, 1 + jγ, and so extend the cochain isomorphism to

[figure: P_{i}^{−n}; d^{−n}=jψ; P_{i}^{−n+1}; 1+jγ; P_{i}^{−n+2}; …; P_{i}^{−n}; d′^{−n}=j(ψ+η); P_{i}^{−n+1}; P_{i}^{−n+2}; …]

By using standard MAGMA routines it was checked that, in the cases X_{6} for J_{2} and X_{6} for 2.J_{2}, all possible such maps η do factor as described above. So these complexes are independent, up to isomorphism, of the choice of differential.

We’re left with X_{3} (n = 3, 4) for 2.J_{2}, non-principal block. In constructing the degree −3 differential, there is a map η : X^{−3} = P_{3}^{−3} ⟶^{p} P_{5} → K_{3}^{−2}, with qη ≠ 0 where p : P_{3}^{−3} → P_{5} is the projection onto P_{5}. From Appendix C it’s clear that η cannot factor through d^{−3} and so a choice of ψ′ = ψ + λη, λ ∈GF(5)^{∗}, cannot be dealt with as above. The map η was constructed in MAGMA and coker(ψ + λη) computed, for λ ∈GF(5)^{∗}. In all cases, the cokernel was found to have dimension less than dim H^{−2}(X_{3}) i.e. a choice ψ′ = ψ + λη is inconsistent with the information about X_{3} in Appendices C and D. We can therefore only change ψ by adding η : X^{−3} ⟶^{p} P_{3} → K_{3}^{−2}, where qη ≠ 0 and p is the projection onto P_{3}. Using MAGMA, this map η was found to factor through X_{3}^{−2} and so a choice of ψ′ = ψ + λη can be dealt with as above.

All that remains is the construction of the degree −4 differential. There are two maps:

η_{1} : X_{3}^{−4} ⟶^{p_{1}} P_{1} → K_{3}^{−3} and η_{3} : X_{3}^{−4} ⟶^{p_{3}} P_{3} → K_{3}^{−3}, where p_{1} and p_{3} are the natural

projections and qη_{1} ≠ 0 and qη_{3} ≠ 0. Again, using standard routines in MAGMA, both maps can be shown to factor through X_{3}^{−4}. In general, we may choose ψ′ = λ_{1}η_{1}+λ_{3}η_{3}, λ_{1}, λ_{3} ∈GF(5). By choosing to factorise say η_{1} through X_{3}^{−4} and using the above arguments we can assume that λ_{1} = 0, but now η_{3} will no longer factorise through X_{3}^{−4}.

So there are, up to isomorphism, potentially four different choices of degree −4 differential and hence four different complexes: X_{3_{1}}, X_{3_{2}}, X_{3_{3}} and X_{3_{4}}, corresponding to each λ ∈GF(5)^{∗}, in addition to the complex X_{3} already constructed for Section 6.4.

These four complexes were constructed and, using MAGMA as described in Section 4.3, the following calculations, for 1 ≤ r ≤ 4, were verified.

Hom_{K^{b}(fkH)}(X_{3_{r}}, X_{j}[m]) = Hom_{K^{b}(fkH)}(X_{j}, X_{3_{r}}[m]) = 0  for m ≠ 0, 1 ≤ j ≤ 6,
Hom_{K^{b}(fkH)}(X_{3_{r}}, X_{j}) = Hom_{K^{b}(fkH)}(X_{j}, X_{3_{r}}) = 0  for j ∈ {1, 2, 4, 5, 6},

Hom_{K^{b}(fkH)}(X_{3_{r}}, X_{3}) = Hom_{K^{b}(fkH)}(X_{3}, X_{3_{r}}) = k

Under the derived equivalence between ekG and fkH from Section 6.4 it follows from the above calculations that the images Y_{r} in D^{b}(ekG) of X_{3_{r}}, for 1 ≤ r ≤ 4, satisfy

Hom_{D^{b}(ekG)}(Y_{r}, S_{j}[m]) = Hom_{D^{b}(ekG)}(S_{j}, Y_{r}[m]) = 0  for m < 0, 1 ≤ j ≤ 6,
Hom_{D^{b}(ekG)}(Y_{r}, S_{j}) = Hom_{D^{b}(ekG)}(S_{j}, Y_{r}) = 0  for j ∈ {1, 2, 4, 5, 6},
Hom_{D^{b}(ekG)}(Y_{r}, S_{3}) = Hom_{D^{b}(ekG)}(S_{3}, Y_{r}) = k

The first of these shows that each Y_{r} has non-zero cohomology only in degree 0. The remaining conditions now show Y_{r} ≅ S_{3} in D^{b}(ekG) and so, for 1 ≤ r ≤ 4, X_{3_{r}} ≅ X_{3} in D^{b}(fkH). By construction X_{3_{r}} ≅ S_{3} ⊗_{ekG} C in K^{b}(fkH)/K^{b}(proj- fkH) and hence, by the same argument as used in the proof of Corollary 4.4.3, we must have, for 1 ≤ r ≤ 4, X_{3_{r}} ≅ S_{3} ⊗_{ekG} X ≅ X_{3} in K^{b}(fkH).

74

Appendix A

Loewy layers of Green correspondents

We let R_{i} = S_{i} ↓_{H} and denote by RG_{i} the Green correspondent of S_{i}. P_{i} denotes the projective cover of the simple fkH-module indexed by i.

A.1 Principal blocks

A.1.1 G = U_{3}(4)

R_{1} = RG_{1},  R_{2} = RG_{2},  R_{3} = RG_{3} ⊕ P_{1},
RG_{1} = k,  RG_{2} = 2 3 2  3 2  3 3 2 ,  RG_{3} = \begin{aligned}& & 3 \\ & k & & 3 \\ k & & 2 & & 3 \\ & k & & 3 \\ & & 3 \end{aligned}

A.1.2 G = J_{2}

R_{1} = RG_{1},  R_{2} = RG_{2},  R_{3} = RG_{3},
R_{4} = RG_{4} ⊕ P_{1},  R_{5} = RG_{5} ⊕ P_{1} ⊕ P_{5},  R_{6} = RG_{6} ⊕ 2P_{6} ⊕ P_{5},

RG_{1} = k,  RG_{2} = \begin{aligned}& & 5 \\ & 2 & & 6 \\ k & & 4 & & 5 \\ & 2 & & 6 \\ & & 5 \end{aligned} ,  RG_{3} = \begin{aligned}& 3 & & 6 \\ & k & & 5 \\ & 3 & & 6 \\ k & & 4 & & 5 \\ 6 & & 3 & & 6 \\ & & 5 \\ & & 3 \end{aligned} ,

RG_{4} = \begin{aligned}& & 3 \\ & & 5 \\ & 3 & & 6 \\ 4 & & k & & 5 \\ & 3 & & 6 \\ & & 5 \\ & & 3 \end{aligned} , RG_{5} = \begin{aligned}& 5 \\ & 6 \\ k & & 4 \\ & 6 \\ & 5 \end{aligned} ,

RG_{6} = \begin{aligned}& & & 4 & & 3 & & 2 \\ & & & 5 & & 6 & & 5 \\ & 4 & & 6 & & 2 & & 6 & & 3 \\ 5 & & 4 & & k & & k & & 4 & & 5 \\ & 6 & & 3 & & 6 & & 2 & & 6 \\ & & & 5 & & 4 & & 5 \\ & & & & 2 & & 3 \end{aligned}

A.1.3 G = Sp_{4}(4)

R_{1} = RG_{1},  R_{2} = RG_{2},  R_{3} = RG_{3} ⊕ P_{4},
R_{4} = RG_{4} ⊕ P_{2},  R_{5} = RG_{5} ⊕ P_{1} ⊕ 2P_{5},

RG_{1} = k,  RG_{2} = 3  3 5 4  3  2 5  5 3  4  2  3 5 3 ,  RG_{3} = \begin{aligned}& 4 \\ & 5 \\ k & & 4 \\ & 5 \\ & 4 \end{aligned} ,

RG_{4} = \begin{aligned}& 2 \\ & 5 \\ k & & 2 \\ & 5 \\ & 2 \end{aligned} ,  RG_{5} = \begin{aligned}& & 5 & & 5 \\ k & & 2 & & 4 & & k \\ & & 5 & & 5 \\ k & & 2 & & 3 & & k \\ & 5 & & 5 & & 5 \\ & 2 & & 4 & & k \\ & & & 5 \end{aligned}

A.2 Non-principal blocks

A.2.1 G = 2.J_{2}

R_{1} = RG_{1},  R_{2} = RG_{2},  R_{3} = RG_{3}
R_{4} = RG_{4},  R_{5} = RG_{5} ⊕ P_{1} ⊕ P_{2} ⊕ P_{5} ⊕ P_{6},
R_{6} = RG_{6} ⊕ P_{3} ⊕ P_{4} ⊕ P_{5} ⊕ P_{6}, RG_{1} = 6 5 3 1 ,  RG_{2} = 3  4 6 4  5 3  6 4  5 3

RG_{3} = 2 1 5 3 6 6 1 2 2 4 5 5 5 1 1 3 6 6 6 6 2 2 2 4 4 5 5 5 1 1 3 6 6 2 5 5 1 6 , RG_{4} = 5 6 1 2 3 4 5 6 1 2 3 4 5 5 6 6 2 4 5 5 5 6 6 1 1 2 3 3 4 6 6 6 2 4 5 5 5 1 3 6 6 5 ,

RG_{5} = 6 2 4 5 1 3 6 6 2 4 5 5 5 1 3 6 6 6 2 4 5 5 1 3 6 5 ,  RG_{6} = 3 5 6 4 5 5 1 3 3 6 6 2 4 4 5 5 5 1 3 3 6 6 6 6 2 4 4 5 5 3 6 6 4 5

Appendix B

Loewy layers of degree -1 terms of complexes S_{i} ⊗_{ekG} C

Let M_{i} denote the module in degree -1 in the complex S_{i} ⊗_{ekG} C. Let Mst_{i} denote the non-projective summand of M_{i}. P_{i} denotes the projective cover of the simple fkH-module indexed by i.

B.1 Principal blocks

B.1.1 G = U_{3}(4)

M_{1} = 0,  M_{2} = Mst_{2},  M_{3} = Mst_{3},

Mst_{2} = \begin{matrix}2 & 3 \\ k & 2 & 3 & 3 \\ k & 2 & 2 & 3 & 3 & 3 \\ k & k & 2 & 2 & 3 & 3 & 3 & 3 \\ k & k & 2 & 2 & 3 & 3 & 3 & 3 \\ k & 2 & 2 & 3 & 3 & 3 \\ k & 2 & 3 & 3 \\ 2 & 3 \end{matrix} , Mst_{3} = \begin{matrix}2 & 3 \\ k & 2 & 3 & 3 \\ k & 2 & 2 & 3 & 3 & 3 \\ k & k & 2 & 3 & 3 & 3 \\ k & 2 & 2 & 3 & 3 & 3 \\ k & 2 & 3 & 3 \\ 2 & 3 \end{matrix}

B.1.2 G = J_{2}

M_{1} = 0,  M_{2} = Mst_{2},  M_{3} = Mst_{3} ⊕ P_{3} ⊕ P_{6},
M_{4} = P_{3} ⊕ P_{6}  M_{5} = 2P_{4} ⊕ 2P_{5},  M_{6} = Mst_{6} ⊕ 2P_{4} ⊕ 2P_{5} ⊕ 4P_{3} ⊕ 4P_{6}
Mst_{3} = \begin{matrix}3 & 6 \\ k & 5 \\ 3 & 6 \\ k & 5 \\ 3 & 6 \end{matrix} , Mst_{2} ≅ Mst_{6} = \begin{matrix}4 & 5 \\ 3 & 2 & 6 & 6 \\ k & 4 & 4 & 5 & 5 & 5 \\ 2 & 2 & 3 & 6 & 6 & 6 \\ k & 4 & 4 & 5 & 5 \\ 3 & 2 & 6 & 6 \\ 4 & 5 \end{matrix}

B.1.3 G = Sp_{4}(4)

As there are two conjugacy classes in H of cyclic 5-subgroups, Q_{1}, Q_{2} with different centralizers, let M_{i}^{r} (resp. Mst_{i}^{r}) denote the M_{i} (resp. Mst_{i}) associated with Q_{r}.

M_{1}^{1} = 0, M_{2}^{1} = Mst_{2}^{1} ⊕ P_{2} ⊕ P_{3}, M_{3}^{1} = Mst_{3}^{1},
M_{4}^{1} = 2P_{2} ⊕ 2P_{3},  M_{5}^{1} = Mst_{5}^{1} ⊕ 4P_{5},
Mst_{2}^{1} = 2  3 5 3  2 5 3  2 ,  Mst_{3}^{1} ≅ Mst_{5}^{1} = \begin{aligned}& 5 \\ k & & 4 \\ & 5 \\ k & & 4 \\ & 5 \end{aligned}
M_{1}^{2} = 0,  M_{2}^{2} = Mst_{2}^{2} ⊕ P_{3} ⊕ P_{4},  M_{3}^{2} = 2P_{3} ⊕ 2P_{4},
M_{4}^{2} = Mst_{4}^{2},  M_{5}^{2} = Mst_{5}^{2} ⊕ 4P_{5},
Mst_{2}^{2} = 4  3 5 4  3 5 4  3 ,  Mst_{4}^{2} ≅ Mst_{5}^{2} = \begin{aligned}& 5 \\ k & & 2 \\ & 5 \\ k & & 2 \\ & 5 \end{aligned}

B.2 Non-principal blocks

B.2.1 G = 2.J_{2}

Only Mst_{i} were calculated. The non-zero ones were:

Mst_{2} = 6 2 4 5 1 3 6 6 2 4 5 5 5 1 3 6 6 6 2 4 5 5 1 3 6 5 ,  Mst_{4} = 2 5 1 3 6 6 2 2 4 5 5 5 1 1 3 6 6 6 2 2 4 5 5 5 1 3 6 6 2 5

82

Appendix C

Images of simple modules under derived equivalence

As described in Section 6.5 the information contained here and in Appendix D is sufficient to determine the {X_{i}}, up to isomorphism in K^{b}(fkH).

The degree zero term of the following cochain complexes is the first non-zero module reading from the right. P_{i} denotes the projective cover of the simple fkH-module indexed by i.

C.1 Principal blocks

C.1.1 G = U_{3}(4)

X_{1} =  ⋯ → 0 ⟶ k ⟶ 0 → ⋯
X_{2} =  ⋯ → 0 ⟶ P_{3} ⟶ Mst_{2} ⟶ RG_{2} ⟶ 0 → ⋯
X_{3} =  ⋯ → 0 ⟶ P_{2} ⟶ 2P_{2} ⟶ Mst_{3} ⟶ RG_{3} ⟶ 0 → ⋯

C.1.2 G = J_{2}

X_{1} =  ⋯ → 0 ⟶ k ⟶ 0 → ⋯
X_{2} =  ⋯ → 0 ⟶ P_{3} ⟶ P_{3} ⊕ P_{4} ⟶ Mst_{2} ⟶ RG_{2} ⟶ 0 → ⋯
X_{3} =  ⋯ → 0 ⟶ P_{3} ⟶ P_{3} ⊕ Mst_{3} ⟶ RG_{3} ⟶ 0 → ⋯
X_{4} =  ⋯ → 0 ⟶ P_{5} ⟶ P_{6} ⟶ P_{3} ⟶ RG_{4} ⟶ 0 → ⋯
X_{5} =  ⋯ → 0 ⟶ P_{5} ⟶ P_{6} ⟶ P_{2} ⊕ P_{3} ⟶ P_{5} ⟶ RG_{5} ⟶ 0 → ⋯
X_{6} =  ⋯ → 0 ⟶ P_{3} ⟶ P_{3} ⊕ P_{5} ⟶ 2P_{5} ⟶ Mst_{6} ⊕ P_{2} ⊕ P_{3} ⟶ RG_{6} ⟶ 0 → ⋯

C.1.3 G = Sp_{4}(4)

X_{1} =  ⋯ → 0 ⟶ k ⟶ 0 → ⋯
X_{2} =  ⋯ → 0 ⟶ P_{5} ⟶ P_{2} ⊕ P_{3} ⊕ P_{4} ⟶ P_{3} ⊕ Mst_{2}^{1} ⊕ Mst_{2}^{2} ⟶ RG_{2} ⟶ 0 → ⋯
X_{3} =  ⋯ → 0 ⟶ P_{2} ⟶ P_{5} ⟶ P_{4} ⊕ Mst_{3}^{1} ⟶ RG_{3} ⟶ 0 → ⋯
X_{4} =  ⋯ → 0 ⟶ P_{4} ⟶ P_{5} ⟶ P_{2} ⊕ Mst_{4}^{2} ⟶ RG_{4} ⟶ 0 → ⋯
X_{5} = ⋯ → 0 ⟶ P_{3} ⟶ 2P_{3} ⟶ P_{3} ⊕ P_{5} ⟶ P_{5} ⊕ Mst_{5}^{1} ⊕ Mst_{5}^{2} ⟶ RG_{5} ⟶ 0 → ⋯

C.2 Non-principal blocks

C.2.1 G = 2.J_{2}

X_{1} =  ⋯ → 0 ⟶ P_{1} ⟶ P_{1} ⟶ RG_{1} ⟶ 0 → ⋯
X_{2} =  ⋯ → 0 ⟶ P_{2} ⟶ P_{6} ⟶ P_{4} ⊕ Mst_{2} ⟶ RG_{2} ⟶ 0 → ⋯
X_{3} = ⋯ → 0 ⟶ P_{1} ⊕ P_{3} ⟶ P_{1} ⊕ P_{3} ⊕ P_{5} ⟶ 2P_{5} ⟶ P_{1} ⊕ P_{2} ⊕ P_{5} ⟶ RG_{3} ⟶ 0 → ⋯
X_{4} =  ⋯ → 0 ⟶ P_{1} ⟶ P_{1} ⊕ P_{2} ⟶ P_{6} ⊕ Mst_{4} ⟶ RG_{4} ⟶ 0 → ⋯
X_{5} =  ⋯ → 0 ⟶ P_{1} ⊕ P_{3} ⟶ P_{5} ⟶ P_{5} ⟶ P_{1} ⊕ P_{3} ⟶ P_{6} ⟶ RG_{5} ⟶ 0 → ⋯
X_{6} =  ⋯ → 0 ⟶ P_{1} ⊕ P_{2} ⊕ P_{3} ⟶ P_{3} ⊕ P_{5} ⊕ P_{6} ⟶ RG_{6} ⟶ 0 → ⋯

Appendix D

Loewy layers of cohomology groups of complexes in Appendix C

D.1 Principal blocks

D.1.1 G = U_{3}(4)

H^{0}(X_{1}) = k,  H^{−1}(X_{2}) = k,  H^{−2}(X_{2}) = \begin{matrix}k \\ 3 \end{matrix} ,
H^{−3}(X_{3}) = \begin{matrix}k \\ 3 \\ 3 \\ 2 \end{matrix}

D.1.2 G = J_{2}

H^{0}(X_{1}) = k,  H^{−3}(X_{2}) = \begin{matrix}k \\ 6 \\ 5 \\ 3 \end{matrix} ,  H^{−1}(X_{3}) = k,

H^{−2}(X_{3}) = \begin{aligned}4 & & k \\ & 6 \\ & 5 \\ & 3 \end{aligned} , H^{−2}(X_{4}) = 2,  H^{−3}(X_{4}) ≅ H^{−4}(X_{5}) = \begin{aligned}k & & 4 \\ & 6 \\ & 5 \\ 2 & & 3 \\ & 5 \end{aligned} ,

H^{−2}(X_{6}) = k,  H^{−3}(X_{6}) = \begin{aligned}k & & 4 \\ & 6 \\ & 5 \\ 2 & & 3 \end{aligned} ,  H^{−4}(X_{6}) = 3

We have the following short exact sequences:

0 ⟶ H^{−3}(X_{2}) ⟶ H^{−2}(X_{3}) ⟶ 4 ⟶ 0  (D.1)
0 ⟶ {2 \atop 5} ⟶ H^{−3}(X_{4}) ⟶ H^{−2}(X_{3}) ⟶ 0  (D.2)
0 ⟶ 5 ⟶ H^{−3}(X_{4}) ⟶ H^{−3}(X_{6}) ⟶ 0  (D.3)

D.1.3 G = Sp_{4}(4)

H^{0}(X_{1}) = k,  H^{−2}(X_{2}) = k,  H^{−3}(X_{2}) = \begin{matrix}k \\ 5 \end{matrix} ,
H^{−1}(X_{3}) = k,  H^{−3}(X_{3}) = 2,  H^{−1}(X_{4}) = k,

H^{−3}(X_{4}) = 4,  H^{−4}(X_{5}) = \begin{aligned}& k \\ & 5 \\ 4 & & 2 \\ & 5 \\ & 3 \end{aligned}

D.2 Non-principal blocks

D.2.1 G = 2.J_{2}

H^{−1}(X_{1}) = {4 \atop 6} ,  H^{−3}(X_{2}) = 2,  H^{−2}(X_{1}) ≅ H^{−3}(X_{4}) = 6 2  4 6 5 1 ,

H^{−2}(X_{3}) = 4 6 5 3 ,  H^{−3}(X_{3}) ≅ H^{−4}(X_{3}) ≅ H^{−5}(X_{5}) = 6 2  4 6 5 1  3 ,  H^{−2}(X_{4}) = {6 \atop 2} ,

H^{−1}(X_{6}) = 6 2  4 6 5 3 ,  H^{−2}(X_{6}) = 6  2 2  4 6 5 1  3

We have the following short exact sequences:

0 ⟶ H^{−5}(X_{6}) ⟶ H^{−2}(X_{6}) ⟶ 2 ⟶ 0  (D.4)
0 ⟶ 1 ⟶ H^{−5}(X_{5}) ⟶ H^{−1}(X_{6}) ⟶ 0  (D.5)
0 ⟶ H^{−2}(X_{3}) ⟶ H^{−1}(X_{6}) ⟶ H^{−2}(X_{4}) ⟶ 0  (D.6)

88

Appendix E

MAGMA code computer programs

The following programs were written in the MAGMA code for implementing on the computer algebra package MAGMA [23].

The main function is homotopy which takes as inputs two complexes X, Y of modules for a group G and returns an array of pairs (m, dim Hom_{K^{b}(kG)}(X, Y [m])) such that the second entry ranges over all non-zero values.

The function homotopy calls in turn the subroutines doublecomplex and totalcomplex which calculate Hom_{kG}(X^{•}, Y^{•}) and Hom_{kG}^{•}(X, Y) respectively.

/*****************************************************************************
* A function, called homotopy,to take two bounded complexes, say X,Y of      *
* modules for a group, or at least whose terms are of Type ModMatGrpElt or   *
* ModMatFldElt, and return a matrix with only two columns,  call this return *
* matrix hom. The first column of hom records the integers r for which the   *
* vectorspace Hom_{K^b(mod-kG)}(X,T^r(Y)) is non-zero. Here Hom is           *
* calculated in the homotopy category and T is the Translation functor.      *
* If hom[i] = [n,d] say, then not only is the above vector space (with r=n)  *
* non-zero, but it has dimension = d.                                        *
*       The inputs to the function homotopy are to be:                       *
* 1. Two lists: array1,array2, that contain the maps of the two complexes.   *
*                                                                            *
* 2. Two integers p,q that describe the degree zero terms of the complexes   *
* in arry1 and array2 respectively.                                          *
*       If array1 is the n long list [* f1,f2,...,fn *], then  the           *
* complex is:                                                                *
*         ..-->0-->X0--f1-->X1--f2-->....--fn-->Xn-->0-->..                  *
* Viewing this as a chain complex with the X0 term in degree 0,              *
* the integer p then associated with array1 is the left shift by p places    *

* to obtain the complex desired as  input, so the term Xp will be in         *
* degree 0. Note p is allowed to be negative or greater than n, similarly    *
* for q and array2. The order of the input parameters is:                    *
* >homotopy(array1,p,array2,q);                                              *
*****************************************************************************/

/*****************************************************************************
**            Now define the various subroutines we'll need                 **
*****************************************************************************/
map2mat1:=function(X,Y,f)
/** Returns matrix of Hom(-,f):X-->Y **/
dx:=Dimension(X);;
dy:=Dimension(Y);;
BF:=BaseRing(X);
if (dx eq 0) then
return Random(KMatrixSpace(BF,dx,dy));
elif (dy eq 0) then
return Random(KMatrixSpace(BF,dx,dy));
else
return Matrix([Coordinates(Y,f*x):x in Basis(X)]);
end if;
end function;
map2mat2:=function(X,Y,f)
/** Returns matrix of Hom(f,-):X-->Y **/
dx:=Dimension(X);;
dy:=Dimension(Y);;
BF:=BaseRing(X);
if (dx eq 0) then
return Random(KMatrixSpace(BF,dx,dy));
elif (dy eq 0) then
return Random(KMatrixSpace(BF,dx,dy));
else
return Matrix([Coordinates(Y,x*f):x in Basis(X)]);
end if;
end function;

doublecomplex:=function(array1,n1,array2,n2)
/*****************************************************************************
* array1 is an array of maps: f1,f2,...,fn                                   *
* array2 is an array of maps: g1,g2,...,gm.                                  *
* These define complexes:                                                    *
* (1)  0-->X0--f1-->X1--f2-->X2-->......-->X(n-1)--fn-->Xn-->0-->0           *
* (2)  0-->Y0--g1-->Y1--g2-->Y2-->......-->Y(m-1)--gm-->Ym-->0-->0           *
*                                                                            *
* This program calculates the double complex                                 *
* GHom(Xi,Yj),returning 3 arrays C,diffV,diffH.                              *
* Where C[i+1][j+1]=GHom(X(n-i),Y(m-j)) as a vector space,                   *
* with i in [1..n],j in [1..m]. The double complex C looks like:             *
*                                                                            *
* C[1][m+1] ---> C[2][m+1] -> ..->C[n][m+1] ---> C[n+1][m+1]  --->0          *
*     |dv[1][m]      |                |                |dv[n][m]             *
*     |              |                |                |                     *
* C[1][m-1] ---> C[2][m-1] -> ..->C[n][m-1] ---> C[n+1][m]    --->0          *
*     |              |                |                |                     *
*     :              :                :                :                     *
*     |              |                |                |                     *
* C[1][2]   ---> C[2][2]   -> ..->C[n][2]    ---> C[n+1][2]  --->0           *
*     |              |                |                |                     *
*     |   dh[1][1]   |                |   dh[n][1]     |                     *
* C[1][1] -----> C[2][1]   -> ..->C[n][1] ------> C[n+1][1]  --->0           *
*                                                                            *
* Maps (already with correct signs!), run from top left to bottom right:     *
* dv[i][j]:= C[i][j+1] --->C[i][j]    for j in range 1..m                    *
* dh[i][j]:= C[i][j]   --->C[i+1][j]  for i in range 1..n                    *
*****************************************************************************/
C:=[[]];              /* Store the double complex as AHom's */
CV:=[* [* *] *];      /* Store the double complex as Vector spaces */
diffV:=[* [* *] *];   /* Store the vertical differentials */
diffH:=[* [* *] *];   /* Store the horizontal differentials */
/************************************************
*Error detection in inputs                      *
*Expect elements in array1,array2 to be of type *
*ModMatGrpElt, or                               *
*ModMatFldElt- if not signal error!             *
************************************************/
err:=0;

for i in [1..n1] do
if not ((Type(array1[i]) eq ModMatGrpElt) or (Type(array1[i]) eq ModMatFldElt) ) then
err:=1;
end if;
end for;
for i in [1..n2] do
if not((Type(array2[i]) eq ModMatGrpElt)or (Type(array2[i]) eq ModMatFldElt) )  then
err:=1;
end if;
end for;
if (err eq 1) then print
"Error in input arrays - expect arrays to contain ModMatGrpElt or MoadMatFldElt";
return "FAIL";
end if;
/************************************************
* End of Error detection in input               *
************************************************/
FF:=BaseRing(Domain(array1[1]));
for i in [1..(n1+1)] do
C[i]:=[];
CV[i]:=[* *];
diffV[i]:=[* *];
end for;
for i in [1..n1] do
diffH[i]:=[* *];
end for;
for i in [1..n1] do
for j in [1..n2] do
C[i][j]:=AHom(Codomain(array1[n1+1-i]),Codomain(array2[n2+1-j]));
end for;
C[i][n2+1]:=AHom(Codomain(array1[n1+1-i]),Domain(array2[1]));
end for;
for i in [1..n2] do
C[n1+1][i]:=AHom(Domain(array1[1]),Codomain(array2[n2+1-i]));
end for;
C[n1+1][n2+1]:=AHom(Domain(array1[1]),Domain(array2[1]));
for i in [1..(n1+1)] do
if (i mod 2 eq 0) then s:=-1;
else s:=1;
end if;
for j in [1..n2] do
diffV[i][j]:=map2mat2(C[i][j+1],C[i][j],s*array2[n2+1-j]);

end for;
end for;
for i in [1..n1] do
for j in [1..(n2+1)] do
diffH[i][j]:=map2mat1(C[i][j],C[i+1][j],array1[n1+1-i]);
end for;
end for;
for i in [1..(n1+1)] do
for j in [1..(n2+1)] do
CV[i][j]:=VectorSpace(FF,Dimension(C[i][j]));
end for;
end for;
return CV,diffV,diffH;
end function;
join2mat:=function(mat1,mat2,flag)
/*****************************************************
* A function to take two matrices mat1 and mat2      *
* and if flag = "v" return matrix provided           *
* col's(mat1) > col's (mat2) vertical join of mat2   *
* with zeros appended to mat1 and if flag ="h"       *
* (a horizontal joint) return provided               *
* row's(mat1)> row's(mat2) horizontal join of mat1   *
* with mat2 appended by zero's on top of it          *
*****************************************************/
BF:=BaseRing(mat1);
if (flag eq "v") /* a vertical join */ then
c1:=NumberOfColumns(mat1);
c2:=NumberOfColumns(mat2);
if (c1 lt c2) then
return "error incorrect sizes of matrices";
else
r2:=NumberOfRows(mat2);
temp:=RMatrixSpace(BF,r2,(c1-c2))!0;
temp1:=HorizontalJoin(temp,mat2);
return VerticalJoin(mat1,temp1);
end if;
elif (flag eq "h") /* a horizontal join */ then
r1:=NumberOfRows(mat1);

r2:=NumberOfRows(mat2);
if (r1 lt r2) then
return "error incorrect sizes of matrices";
else
c2:=NumberOfColumns(mat2);
temp:=RMatrixSpace(BF,(r1-r2),c2)!0;
temp1:=VerticalJoin(temp,mat2);
return HorizontalJoin(mat1,temp1);
end if;
else
return "error incorrect flag settings:use \"v\" or \"h\" ";
end if;/*if(flag eq "v") */
end function;

totalcomplex:=function(C,n,m,dv,dh)
/*****************************************************************************
* C to be a list of a list of Vector spaces representing the double complex  *
* So the picture is of a double complex:                                     *
*                                                                            *
*C[1][m]   ---> C[2][m]    -> ..->C[n-1][m]   ---> C[n][m]  --->0            *
*    |dv[1][m-1]    |                 |                |dv[n][m-1]           *
*    |              |                 |                |                     *
*C[1][m-1] ---> C[2][m-1]  -> ..->C[n-1][m-1] ---> C[n][m-1]--->0            *
*    |              |                 |                |                     *
*    :              :                 :                :                     *
*    |              |                 |                |                     *
*C[1][2]   ---> C[2][2]   -> ..->C[n-1][2]    ---> C[n][2]  --->0            *
*    |              |                 |                |                     *
*    |   dh[1][1]   |                 |    dh[n-1][1]  |                     *
*C[1][1] -----> C[2][1]   -> ..->C[n-1][1] ------> C[n][1]  --->0            *
*                                                                            *
* Maps (already with correct signs!), run from top left to bottom right:     *
* dv[i][j]:= C[i][j+1] --->C[i][j]    for j in range 1..(m-1)                *
* dh[i][j]:= C[i][j]   --->C[i+1][j]  for i in range 1..(n-1)                *
*                                                                            *
* Total complex will be:                                                     *
*                                                                            *
*          [dv[1][m-1]]                                                      *
*          [dh[1][m]  ]                              [dh[n-1][1],dv[n][1]]   *

*  C[1][m]  ------> C[1][m-1]+C[2][m] -> ..->C[n-1][1]+C[n][2]--->C[n][2]    *
*                                                                            *
* from degree -m+1 through to degree +n-1, so it has length n+m-1            *
* NOTE the return lists T and D will de indexed by 1,..,n+m+1, so the degree *
* i term of the Total complex will be T[i+m]                                 *
*****************************************************************************/
BF:=BaseField(C[1][1]);
T:=[* *];    /* Store entries for total complex,as a list of vector spaces  */
D:=[* *];    /* Store diffirentials of total complex,as a list of matrices  */
/*****************************************************************************
* First calculate the terms of total complex from degree -m+1 through to     *
* degree -1 together with the differentials                                  *
*****************************************************************************/
for deg:=(-m+1) to -1 by 1 do   /*big outer loop*/
dim:=0;
for i in [1..n] do
if ((i-deg) le m) then
dim := dim + Dimension(C[i][i-deg]);
end if;
end for;
T[deg+m]:=VectorSpace(BF,dim);
/*
Now define the differential from degree deg term to degree deg+1 term
as a matrix
*/

d:=dv[1][-deg];
i:=1;
while (((i-deg) le m) and (i le (n-1))) do
d:=join2mat(d,dh[i][i-deg],"h");
if ((i-deg) le (m-1)) then  /* Can still join on a vertical differential */
d:=join2mat(d,dv[i+1][i-deg],"v");
end if;
i:=i+1;
end while;
D[deg+m]:=d; /*The differential from degree deg term to deg+1 term */

end for; /*for deg:=(-m+1) to -1 by 1 do */

/*
The last differential was from degree -1 term to degree 0 term
Still need to define this degree 0 term in T
*/
/*****************************************************************************
* Next calculate the terms of total complex from degree 0 through to degree  *
* n-1 together with the differentials                                        *
*****************************************************************************/

for deg:=0 to n-1 by 1 do  /*big outer loop*/
dim:=0;
for i in [1..m] do
if ((deg+i) le n) then
dim := dim + Dimension(C[deg+i][i]);
end if;
end for;
T[deg+m]:=VectorSpace(BF,dim);
/*
Now define the differential from degree deg term to degree deg+1 term
as a matrix
*/
if(deg le (n-2)) /* Still have a differential to define */
then
d:=dh[deg+1][1];
i:=2;
while(((deg+i) le n) and (i le m)) do
d:=join2mat(d,dv[deg+i][i-1],"v");
if ((deg+i) le (n-1)) then
d:=join2mat(d,dh[deg+i][i],"h");
end if;
i:=i+1;
end while;

D[deg+m]:=d;  /* The differential from degree deg term to deg+1 term */
end if;  /*if(deg le (n-2))*/
end for; /*big outer loop*/

return T,D;
end function;

/***********************************************************
**           End of subroutine declarations               **
***********************************************************/

homotopy:=function(array1,p,array2,q)
/***********************************************************
* Could do some error checking here e.g. are they          *
* complexes                                                *
************************************************************/
n:=#array1;
m:=#array2;
mat:=[[]];
hom:=[];
for i in [1..(n-1)] do
if not (Dimension(Image(array1[i]*array1[i+1])) eq 0) then
print "Error in first array - not a complex";
return -1;
end if;
end for;
for i in [1..(m-1)] do
if not (Dimension(Image(array2[i]*array2[i+1])) eq 0) then
print "Error in second array - not a complex";
return -1;
end if;
end for;
/* Now call doublecomplex */
C,dv,dh:=doublecomplex(array1,n,array2,m);
/* Now call totalcomplex */
T,d:=totalcomplex(C,n+1,m+1,dv,dh);

/*check on length of T */
t:=#T;
if not ((t eq (n+m+1)) or (#d eq (t-1))) then
print "Error, something gone wrong incorrect length of total complex ";
return -1;
end if;
/********************************
* Now calculate the homology of *
* total complex T               *
********************************/
hom[1]:=Dimension(Kernel(d[1]));
for i in [1..(t-2)] do
hom[i+1]:=Dimension(quo<Kernel(d[i+1])|Image(d[i])>);
end for;
hom[t]:=Dimension(quo<T[t]|Image(d[t-1])>);
/********************************
* Now calculate the shift r     *
* needed for the original       *
* complexes to yield            *
* the non-zero degrees          *
*********************************/
j:=1;
for i in [1..t] do
if not (hom[i] eq 0) then
r:=i-m-1+p-q;r1:=p-q-n+i-1;
mat[j]:=[r1,hom[i]];
j:=j+1;
end if;
end for;
return mat;
end function;

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