BOUSFIELD LOCALIZATION FOR REPRESENTATION THEORISTS

JEREMY RICKARD

1. Introduction

Bousfield localization is a technique that has been used extensively in algebraic topology, specifically in stable homotopy theory, over the last quarter century. Its name derives from the fundamental work of Bousfield [3], although this is based on earlier work of Brown [4] and Adams [1]. In abstract terms, Bousfield localization deals with the inclusion of a thick subcategory (i.e., a triangulated subcategory closed under direct summands) into a triangulated category and the existence of adjoint functors to such an inclusion. In the original topological setting, the triangulated category was typically the stable homotopy category, and the thick subcategory was defined by the vanishing of some homology theory, but the techniques involved work much more generally. In particular, the triangulated category can be one of interest to representation theorists, such as a stable module category or the derived category of a module category: since the techniques rely heavily on limiting procedures, however, one is forced to work with infinite dimensional modules.

In this article I will look at the simplest form of Bousfield localization – so-called ‘finite localization’ – which is good enough for most of the applications to representation theory to date, and I will briefly describe a more general form.

For the most part, I will eschew applications, referring instead to the article by Dave Benson in these proceedings. However, in the final section I will describe a recent application of a technique very similar to Bousfield localization to the construction of equivalences of derived categories.

Except for the last section, almost nothing in this article is original. Much of what I know about Bousfield localization I learned from the papers of Neeman [2, 7, 8, 9, 10, 11], who has enthusiastically argued that these techniques are much underused outside algebraic topology and has shown that they can be used to simplify substantially the proofs of certain theorems of algebraic geometry and algebraic K-theory. This article owes a lot to this work of Neeman, though I mostly restrict my attention to the triangulated categories that are of interest in representation theory.

2. Homotopy colimits

Many of the constructions that we shall describe here depend on taking ‘direct limits’ of sequences of objects in some triangulated category. However, direct limits of such sequences, in the strict category-theoretic sense, rarely exist in a triangulated category, so we shall have to use a substitute.

To motivate this, let us first recall one method of constructing the direct limit of a sequence

X_{0} ⟶^{α} X_{1} ⟶^{α} X_{2} ⟶ …

of objects in an abelian category, such as a module category (for convenience of notation we shall use α to denote all the maps in this sequence – it will be clear from the context which of the maps we are referring to). This method is to express the direct limit as the cokernel of the injective endomorphism 1 − α of the direct sum of the objects X_{i}, so we have a short exact sequence

0 ⟶ ⨁_{i=0}^{∞} X_{i} ⟶^{1−α} ⨁_{i=0}^{∞} X_{i} ⟶ colim(X_{i}) ⟶ 0.

In a triangulated category we can mimic this, so long as the category has countable direct sums, by substituting a distinguished triangle for the short exact sequence. Thus we define the homotopy colimit of the sequence by the distinguished triangle

⨁_{i=0}^{∞} X_{i} ⟶^{1−α} ⨁_{i=0}^{∞} X_{i} ⟶ hocolim(X_{i}) ⟶ ⨁_{i=0}^{∞} X_{i}[1].

The homotopy colimit shares some, but not all, of the properties of a genuine colimit. Let Y be another object of the category. If we apply the functor Hom(?, Y) to the distinguished triangle above, then we get a long exact sequence including

Hom(hocolim(X_{i}), Y) ⟶ Hom(⨁_{i=0}^{∞} X_{i}, Y) ⟶^{1−α} Hom(⨁_{i=0}^{∞} X_{i}, Y).

If we choose a collection of maps

β_{i} : X_{i} ⟶ Y

with the compatibility conditions that αβ_{i} = β_{i−1}, then this represents an element of the kernel of the second map in the exact sequence. By exactness, this implies that there is a homomorphism

β : hocolim(X_{i}) ⟶ Y

such that each β_{i} is the composition of the natural map

X_{i} ⟶ hocolim(X_{i})

with β. However, for a genuine colimit we would need β to be unique, which is only the case if the first map

Hom(hocolim(X_{i}), Y) ⟶ Hom(⨁_{i=0}^{∞} X_{i}, Y)

in the exact sequence is injective, which will not usually be the case.

The paper [2] of Bökstedt and Neeman deals in much greater depth with the notion of homotopy colimits in triangulated categories.

3. Finite localization

Throughout this section, T will denote a triangulated category with arbitrary (i.e., possibly infinite) direct sums of sets of objects. To give some representation-theoretic examples, T could be the stable module category StMod(Λ) of (possibly infinite-dimensional) modules for a finite-dimensional self-injective algebra Λ, or the derived category D(Mod(Λ)) of (possibly unbounded) complexes of modules for a finte-dimensional algebra Λ. Note that the bounded derived category D^{b}(Mod(Λ)) will not do, since the direct sum of an infinite number of bounded complexes may not be bounded itself.

It is automatically true that the direct sum of a set of distinguished triangles is itself a distinguished triangle: this is certainly clear in the examples above.

We shall be interested in the following finiteness condition on objects of T .

Definition 3.1. An object C of T is compact if the functor Hom(C,?) (from T to abelian groups) preserves arbitrary direct sums.

Another way of saying this is that any map from C to an infinite direct sum must factor through a finite subsum.

In the example StMod(Λ), it is easy to see that the compact objects are precisely those isomorphic to finitely generated modules. In the example D(Mod(Λ)), the compact objects are those isomorphic to bounded complexes of finitely generated projective modules.

A related finiteness condition on the category T , which is easily seen to be satisfied by both of the examples above, is the following.

Definition 3.2. The category T is compactly generated if there is a set X of compact objects of T such that for each object T of T there is an object C of X such that Hom(C, T) ≠ 0.

The reason for choosing the set X of objects rather than using all compact objects is that the class of compact objects may not be a set.

The notion of a compact object is related to that of a homotopy colimit by the following important but easy result.

Proposition 3.3. If C is a compact object of T and

X_{0} ⟶^{α} X_{1} ⟶^{α} X_{2} ⟶ …

is a sequence of objects, then there is a natural isomorphism

Hom(C, hocolim(X_{i})) ≅ colim Hom(C, X_{i}).

Proof. Since C is compact, the map

Hom(C, ⨁_{i=0}^{∞} X_{i}[n]) ⟶^{1−α} Hom(C, ⨁_{i=0}^{∞} X_{i}[n])

is isomorphic to

⨁_{i=0}^{∞} Hom(C, X_{i}[n]) ⟶^{1−α} ⨁_{i=0}^{∞} Hom(C, X_{i}[n]),

it is injective for all n, and the long exact sequence obtained by applying the functor Hom(C,?) to the distinguished triangle that defines hocolim(X_{i}) splits into short exact sequences, one of which is

0 ⟶ ⨁_{i=0}^{∞} Hom(C, X_{i}) ⟶^{1−α} ⨁_{i=0}^{∞} Hom(C, X_{i}) ⟶ Hom(C, hocolim(X_{i})) ⟶ 0.

But since this is the short exact sequence that calculates colim Hom(C, X_{i}), we get the required isomorphism.

Definition 3.4. A localizing subcategory of T is a triangulated subcategory that is closed under taking direct sums of sets of objects.

The simplest version of Bousfield localization, so-called ‘finite localization’, involves a compactly generated localizing subcategory of T .

Theorem 3.5. Let C be a localizing subcategory of T and assume that C is compactly generated by a set X of objects that are compact in T (not just in C). Then the inclusion functor

C ⟶ T

has a right adjoint.

We shall describe the construction of the right adjoint and sketch the proof.

We may as well assume that the set X of compact objects is closed under the shift functors C ↦ C[n]. Let X be an object of T and set X_{0} = X. We shall construct, by induction on n, a sequence of objects X_{n}.

Given X_{k}, let S_{k} ⟶ X_{k} be a map from a (possibly infinite) direct sum of objects of X with the property that every map from an object of X to X_{k} factors through it. This can be achieved by considering all maps C ⟶ X_{k} from objects of X to X_{k} and setting S_{k} to be the direct sum of the objects C, one for each such map. Now form the distinguished triangle

S_{k} ⟶ X_{k} ⟶ X_{k+1} ⟶ S_{k}[1]

to define X_{k+1}. Note that, for any object C of X, the exact sequence

Hom(C, S_{k}) ⟶ Hom(C, X_{k}) ⟶ Hom(C, X_{k+1})

shows that any map C ⟶ X_{k} composes with X_{k} ⟶ X_{k+1} to give zero, since the first map in the exact sequence is surjective by construction, and hence the second map is zero.

If we set F X = hocolim(X_{i}), then by Proposition 3.3 we have that

Hom(C, F X) ≅ colim(Hom(C, X_{k})) = 0

for any C in X.

If we define an object EX by forming the distinguished triangle

EX ⟶ X ⟶ F X ⟶ EX[1],

then we shall show that EX is in C. Let us first give a proof of this making the extra assumption that the quotient triangulated category T /C exists. Then we shall describe how to sidestep this assumption.

If T /C does exist, then proving that EX is in C is equivalent to proving that the map X ⟶ F X is an isomorphism in T /C. But all the maps

X = X_{0} ⟶ X_{1} ⟶ … ⟶ X_{k} ⟶ X_{k+1} ⟶ …

are isomorphisms in T /C by construction. The natural functor

T ⟶ T /C

preserves (countable) direct sums because C is closed under (countable) direct sums. It follows that F X is the homotopy colimit of the sequence

X = X_{0} ⟶ X_{1} ⟶ … ⟶ X_{k} ⟶ X_{k+1} ⟶ …

in the quotient category, not just in the category T . Hence X ⟶ F X is an isomorphism in T /C, being the homotopy colimit of a sequence of isomorphisms.

The potential problem with the existence of the quotient T /C is a set-theoretic one. Given objects U and V of T , a map from U to V in T /C would be represented by a diagram U ⟵ W ⟶ V in T , where the map from W to U occurs in a distinguished triangle

W ⟶ U ⟶ C ⟶ W [1]

whose third object C is in C. Since the objects of C may not form a set, it is not clear that the equivalence classes of diagrams U ⟵ W ⟶ V form a set. Thus it is not clear that we can form the quotient category within our original set-theoretic universe.

However, for the purpose of this proof we can sidestep these difficulties. First, if one is willing to accept that the quotient category can be constructed in some larger set-theoretic universe (which may depend on accepting some large cardinal axioms), then this will be good enough. Even if one is not, then the following argument can be used.

Let C′ be the smallest triangulated subcategory of T that contains all the objects S_{k} used in the construction of F X and closed under countable direct sums. Then there is a set of isomorphism classes of objects of C′, and hence there is no set-theoretic problem involved in the construction of the quotient T /C′. Since C′ is closed under countable direct sums, the natural functor

T ⟶ T /C′

preserves homotopy colimits of sequences, and so X ⟶ F X is an isomorphism in T /C′. So EX is in C′, and hence in C.

Having shown that EX is in C, we next want to show that X ↦ EX is the right adjoint of the inclusion functor C ⟶ T .

The key step that remains to prove this is to show that Hom(C, F X) = 0 for every C in C, not just for C in X. It is easy to prove that Hom(C, F X) = 0 for every C in the localizing subcategory C′′ of T generated by X. Let us show that C′′ = C.

To do this, we just need to note that everything we have proved so far about EX and F X works equally well if we replace C by C′′. In particular, if X is in C, then EX is in C′′, but also F X ≅ 0 and so EX ≅ X. Hence X is in C′′.

Although he did not state it in terms of triangulated categories, this theorem essentially goes back to Brown [4].

4. Applications of finite localization in modular representation theory

As we mentioned, the stable module category StMod(Λ) of a self-injective algebra (e.g., a finite group algebra kG) is an example of a category T to which Theorem 3.5 applies. For a finite group algebra there are many natural choices for the localizing subcategory C: in particular subcategories defined in terms of ‘varieties for modules’. We shall not describe here the applications that have been found, but shall refer to Dave Benson’s article in these proceedings for more details.

5. More general Bousfield localization

In this section we shall assume that the triangulated category T has direct sums and is compactly generated, but we shall not make any assumption of compact generation for the localizing subcategory C.

There is a quite general theorem asserting the existence of a right adjoint for the inclusion functor C ⟶ T . Only a set-theoretic problem prevents a completely general theorem.

Theorem 5.1. Let T be a compactly generated triangulated category in which direct sums of sets of objects exist. Let C be a localizing subcategory of T . The inclusion functor C ⟶ T has a right adjoint if and only if the quotient category T /C (in the sense of triangulated categories) exists.

As we remarked when dealing with finite localization, the potential problem with the existence of the quotient category is that when localizing a category, one may be forced to construct a Hom-set that is too large to be a set. To the best of my knowledge there are no examples – with T compactly generated – where this is known to be a genuine problem, although there are cases that arise in stable homotopy theory where it may well depend on the model of set theory that one uses.

We shall very briefly sketch the proof. I would like to thank Amnon Neeman for describing this to me.

In his paper [10], Neeman gives a proof of a ‘Brown representability theorem’ in a general compactly generated triangulated category T with direct sums. This states that if H is a cohomological contravariant functor from T to abelian groups that takes direct sums to direct products, then H is representable: i.e., there is an object T_{H} of T such that H ≅ Hom(?, T_{H}). The proof of this is in many ways reminiscent of the proof of Theorem 3.5 that we sketched.

This can be applied to the construction of an adjoint as follows. Let X be an object of T and let H = Hom_{T /C}(?, X). This is a cohomological functor that takes direct sums to direct products, and so the representability theorem applies. Thus

Hom_{T /C}(?, X) ≅ Hom_{T} (?, F X)

for some object F X of T . There is a natural map X ⟶ F X, and we can form a distinguished triangle

EX ⟶ X ⟶ F X ⟶ EX[1].

Then EX is in C and X ↦ EX is the right adjoint of the inclusion functor.

6. Constructing equivalences of derived categories

In his article in these proceedings, Dave Benson briefly describes Broué’s conjecture on equivalences of derived categories between blocks of finite group algebras. A construction rather like that involved in finite Bousfield localization has recently proved useful in verifying the existence of such equivalences.

The key theorem is the following.

Theorem 6.1. Let B be a finite-dimensional symmetric algebra over a field k, and let X_{0}, …, X_{n} be objects of the derived category D^{b}(mod(B)) such that

(i) Hom(X_{i}, X_{j}) = 0 if i ≠ j and Hom(X_{i}, X_{i}) = k for all i,

(ii) Hom(X_{i}, X_{j}[m]) = 0 for all m < 0 and all i, j, and

(iii) X_{0}, …, X_{n} generate D^{b}(mod(B)) as a triangulated category.

Then there is another algebra C (also a finite-dimensional symmetric k-algebra) and an equivalence D^{b}(mod(B)) ≈ D^{b}(mod(C)) of triangulated categories that takes X_{0}, …, X_{n} to the simple C-modules.

Note that conditions (i) to (iii) are obviously satisfied by the simple modules: for example, condition (i) is just Schur’s Lemma.

Let us sketch the proof before describing why this theorem is useful.

The idea of the proof is to mimic a construction of the injective hull of a simple module. When applied to the objects X_{1}, …, X_{n}, this will produce objects T_{0}, …, T_{n} such that T = T_{0} ⊕⋯⊕T_{n} is a ‘tilting complex’ [13] for B. The algebra C will be the endomorphism algebra of T, and so, by the main theorem of [13], the derived categories D^{b}(mod(B)) and D^{b}(mod(C)) are equivalent.

One condition that would be satisfied if X_{0}, …, X_{n} were the simple B-modules and T_{0}, …, T_{n} were their injective hulls is that Hom(X_{i}, T_{j}[m]) = 0 for m ≠ 0. If we regard X_{j} as a ‘first approximation’ to T_{j}, then this Hom-space only vanishes for m < 0. By adjusting X_{j} to ‘kill’ all maps from X_{i}[−m] to X_{j}, we shall eventually produce T_{j}.

Let X be {X_{i}[t] : 0 ≤ i ≤ n, t < 0}. Note that this is only closed under negative shifts. Set Y_{0} = X_{j} for some j. We shall define a sequence Y_{0}, Y_{1}, …, Y_{k}, … of objects of

D(Mod(B)) by induction on k as follows.

Given Y_{k}, let S_{k} ⟶ Y_{k} be a map from a direct sum of objects of X to Y_{k}, such that every map from an object of X to Y_{k} factors through it. Also, choose S_{k} to be ‘minimal’: no proper direct summand of S_{k} has the property just described. Form a distinguished triangle

S_{k} ⟶ Y_{k} ⟶ Y_{k+1} ⟶ S_{k}[1]

to define Y_{k+1}. Finally, define T_{j} = hocolim(Y_{k}).

To show that Hom(X_{i}, T_{j}[m]) = 0 for m > 0, we want to know that

Hom(X_{i}[−m], hocolim(Y_{k})) = colim(Hom(X_{i}[−m], Y_{k}),

which is zero by construction (just as in the proof of Theorem 3.5), since X_{i}[−m] is in X. Unfortunately, we cannot deduce this from Proposition 3.3, since X_{i}[−m] is not usually compact. However, the complexes Y_{k} are uniformly bounded below, and the functor Hom(X_{i},?) does preserve direct sums for such families of objects, so all is well.

The fact that B is a symmetric algebra is used in the proof (which we shall omit) that T has all the other properties of a tilting complex. The details will appear in [14].

In particular, one must show that T is isomorphic to a bounded complex of finitely generated projective modules. For a general symmetric algebra B, the intermediate objects Y_{k} certainly need not be bounded: however, if B is the group algebra of a finite group, it follows from the finite generation of group cohomology that it is possible to take all the objects Y_{k} to be bounded complexes of finitely generated modules.

Finally, let us describe the reason that this theorem is useful. This is based on an idea of Okuyama [12]. In many of the simpler situations where Broué’s conjecture has yet to be verified, it is known that the blocks A and B involved are stably equivalent (i.e., stmod(A) and stmod(B) are equivalent as triangulated categories) even though it is not known that D^{b}(mod(A)) and D^{b}(mod(B)) are equivalent. Moreover, the stable equivalence is of ‘Morita type’: i.e., it is induced by an exact functor between the module categories (often this exact functor is just restriction or some mild generalization).

Usually one of the blocks involved (A, say) is much more complicated than the other one: in fact, its structure (in terms of a quiver with relations, say) is often not known. If one can construct a tilting complex T for the simpler block B, then one knows that the derived categories of B and of C = End(T) are equivalent, but it may be very hard to decide whether A and C are isomorphic. However, by composing the known stable equivalence stmod(A) ≈ stmod(B) with the equivalence stmod(B) ≈ stmod(C) induced by the equivalence of derived categories D^{b}(mod(B)) ≈ D^{b}(mod(C)), one obtains a stable equivalence stmod(A) ≈ stmod(C) of Morita type.

Okuyama’s idea was to use a theorem of Linckelmann [6] that states that any stable equivalence of Morita type (between self-injective algebras) that takes simple modules to simple modules is induced by an equivalence of module categories. If one can choose the tilting complex T so that the equivalence stmod(A) ≈ stmod(C) has this property, then mod(A) ≈ mod(C), and so one can prove that

D^{b}(mod(A)) ≈ D^{b}(mod(C)) ≈ D^{b}(mod(B))

without needing to know anything about the structure of A except what the images of its simple modules are under the stable equivalence to stmod(B).

Using this idea, Okuyama verified several particular cases of Broué’s conjecture.

The main problem, of course, is to construct an equivalence D^{b}(mod(B)) ≈ D^{b}(mod(C)) with the required properties. Theorem 6.1 shows that one need only produce objects X_{0}, …, X_{n} of D^{b}(mod(B)) that are stably isomorphic to the images of the simple Amodules (i.e., isomorphic in stmod(B), considered as a quotient category of D^{b}(mod(B))), and have properties (i) to (iii) of Theorem 6.1. This tends to be a lot easier than directly constructing the tilting complex T, since X_{0}, …, X_{n} usually have a much simpler structure than T does.

Let us give an example where the coefficient field k has characteristic 3. Let G be the alternating group A_{6} (which is isomorphic to P SL_{2}(9)), let P ≅ C_{3} × C_{3} be a Sylow 3-subgroup of G and let H = N_{G}(P), which is the semidirect product of P with a cyclic group of order 4. Let A and B be the principal block algebras of kG and kH respectively. It is well-known that restriction of modules induces a stable equivalence between A and B. Okuyama [12] proved, by different methods, that A and B have equivalent derived categories, but we shall describe how Theorem 6.1 gives a simple proof.

The algebra B has four simple modules, all one-dimensional, which we shall call k, 1, 2, 3, the number referring to how the cyclic quotient C_{4} of H acts.

Also, A has four simple modules, k = V_{0}, V_{1}, V_{2}, V_{3} with dimensions 1, 3, 4 and 3 respectively. Let Z_{i} be the restriction of V_{i} to B (for i = 0, 1, 2, 3). Then Z_{0} = k and the structure of Z_{1}, Z_{2}, Z_{3} is described by the following diagrams (the rows give the Loewy layers).

Z_{1} = \begin{matrix}1 \\ k \\ 3 \end{matrix} , Z_{2} = 2 1  3 2 , Z_{3} = \begin{matrix}3 \\ k \\ 1 \end{matrix}

Clearly the modules Z_{0}, …, Z_{3}, considered as objects of the derived category D^{b}(mod(B)), do not satisfy the hypotheses of Theorem 6.1: there are non-zero maps between Z_{1} and Z_{3}, and Z_{2} has a two-dimensional endomorphism algebra.

But let X_{0} = Z_{0} = k, X_{1} = Ω^{2}Z_{1}[2], X_{2} = ΩZ_{2}[1] and X_{3} = Ω^{2}Z_{3}[2]. These objects are stably isomorphic to Z_{0} …, Z_{3} since, for example, X_{1} is isomorphic in D^{b}(mod(B)) to the complex

… 0 ⟶ P_{1} ⟶ P_{0} ⟶ Z_{1} ⟶ 0 ⟶ …

with Z_{1} in degree zero, obtained by truncating a minimal projective resolution of Z_{1}. But this is clearly isomorphic to Z_{1} in stmod(B), considered as the quotient of D^{b}(mod(B)) by the subcategory of bounded complexes of finitely generated projective modules.

The structure of Ω^{2}Z_{1}, ΩZ_{2} and Ω^{2}Z_{3} is described by the following diagrams.

Ω^{2}Z_{1} = \begin{matrix}k \\ 3 \\ 2 \end{matrix} , ΩZ_{2} = \begin{aligned}k & & & & k \\ & 1 & & 3 \\ & & 2 \end{aligned} , Ω^{2}Z_{3} = \begin{matrix}k \\ 1 \\ 2 \end{matrix}

These modules, and hence the objects X_{0}, …, X_{3}, have one-dimensional endomorphism algebras, so that the second part of condition (i) of Theorem 6.1 is satisfied.

The rest of condition (i) and condition (ii) follow in most cases from the fact that Hom_{D^{b}(mod(B))}(U, V [m]) = 0 if U and V are modules and m < 0. The remaining cases are:

Hom(k, X_{j}[−2]) = Hom(k, Ω^{2}Z_{j}) = 0 (for j = 1, 3),
Hom(k, X_{j}[−1]) = Ext^{1}(k, Ω^{2}Z_{j}) = \underline{Hom}(k, ΩZ_{j}) = 0 (for j = 1, 3),
Hom(k, X_{j}) = Ext^{2}(k, Ω^{2}Z_{j}) = \underline{Hom}(k, Z_{j}) = 0 (for j = 1, 3),
Hom(k, X_{2}[−1]) = Hom(k, ΩZ_{2}) = 0,
Hom(k, X_{2}) = Ext^{1}(k, ΩZ_{2}) = \underline{Hom}(k, Z_{2}) = 0,
Hom(X_{2}, X_{j}[−1]) = Hom(ΩZ_{2}, Ω^{2}Z_{j}) = 0 (for j = 1, 3),
Hom(X_{2}, X_{j}) = Ext^{1}(ΩZ_{2}, Ω^{2}Z_{j}) = \underline{Hom}(Z_{2}, Z_{j}) = 0 (for j = 1, 3),

and

Hom(X_{i}, X_{j}) = Hom(Ω^{2}Z_{i}, Ω^{2}Z_{j}) = 0 (for {i, j} = {1, 3}).

Condition (iii) is quite easy to verify. We just need to show that all the simple Bmodules are in the triangulated subcategory C of D^{b}(mod(B)) generated by X_{0}, …, X_{3}. Since k, Ω^{2}Z_{1}, ΩZ_{2} and Ω^{2}Z_{3} are in C, so are the modules

{3 \atop 2} , 1 3 2 , {1 \atop 2}

that are kernels of surjections from Ω^{2}Z_{1}, ΩZ_{2} and Ω^{2}Z_{3} to direct sums of copies of k. Hence 1 and 3, which are quotients of the second module above by submodules isomorphic to the first and third, are also in C, and finally 2 is in C, since it is the kernel of a surjection from the first module to 3.

Theorem 6.1 has also been used to verify some new cases of Broué’s conjecture:

Theorem 6.2 (Chuang [5]). If G = SL_{2}(p^{2}), H is the normalizer of a Sylow p-subgroup of G, and k is algebraically closed of characteristic p, then the principal blocks of kG and kH have equivalent derived categories.

The case p = 5 of this theorem was independently proved by Holloway, who has also proved the corresponding theorem for G = SL_{2}(27) and p = 3.

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[14] J. Rickard, Constructing derived equivalences for symmetric algebras, in preparation.

University of Bristol, School of Mathematics, University Walk, Bristol BS8 1TW, ENG-LAND

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