Proceedings of the Summer School on “Representation theory of Algebras, Finite and Reductive Groups” “Babeş-Bolyai” University Cluj, 1998; pp. 67–100.

III. BLOCKS OF FINITE GROUPS AND

STABLE EQUIVALENCES

Markus Linckelmann

Modular representation theory studies the relations between representations of a finite group G in characteristic zero and in positive characteristic p via the study of the structure of the group algebra of G over a complete discrete valuation ring O having a residue field k of characteristic p and a quotient field K of characteristic zero. The link between representations of G over K and over k is then established by the functors “reduction modulo p” k ⊗_{O} − and “extension of scalars” K ⊗_{O} −.

One of the first observations is, that the group algebra OG need not be indecomposable as an algebra in general. We decompose OG into its indecomposable direct algebra summands and study the structure of each of these summands separately. Such a direct summand is called a block of OG. We introduce in section 1 some fundamental concepts from block theory, emphasizing that the typical objects that we have to deal with are pairs consisting of a finite group and a conjugacy class of primitive idempotents in some fixpoint algebra. For instance, a block can be viewed as a pair consisting of a finite group G and a centrally primitive idempotent of the group algebra OG. The local structure of such a block can then be expressed in terms of Brauer pairs or pointed groups. Also Vertices and sources of modules can be interpreted as such pairs. This way of looking at modular representation theory, beyond rephrasing “classical” ideas, gives rise to the notion of source algebras of blocks of finite groups.

Section 2 contains results which do not depend on the ground ring - such as Frobenius’ reciprocity, Mackey’s formula or Higman’s criterion on relative projectivity (and there is deliberately some overlap with chapter I, since we think it is useful to see different presentations of this material).

In section 3 we introduce then Puig’s concept of pointed groups which is a systematic approach to block theory unifying the various definitions given in section 1.

Both the Green correspondence and the Brauer correspondence, described in section 9 of Chapter I as correspondences between (isomorphism classes of) modules can be rephrased more generally as correspondences between pointed groups in a very simple way; this is the content of section 4.

It happens frequently that the module category of a block algebra can be determined only “up to projectives”; that is, up to stable equivalences. It turns out to be a difficult problem to reconstruct the algebra structure from its stable category. Section 5 contains a description of some methods trying to get around this problem.

In section 6 we bring the material of the previous sections together and describe the standard situation where block theory and stable equivalences meet.

Finally, we give in section 7 a short (and, of course, incomplete) account on the fundamental problem in block theory, namely the determination of the structure of the source algebras of blocks of finite groups with a given defect group.

1. Blocks, defect groups and vertices

The aim of this section is to show, that some of the fundamental definitions in block theory can be given with relatively little technical effort. We postpone a deeper understanding of the motivations leading to these definitions to later sections.

We fix a prime p, a complete discrete valuation ring O having a residue field k = O/J(O) of characteristic p, and denote by K the quotient field of O.

We adopt the convention, that all modules are finitely generated left unitary modules, unless explicitely stated otherwise, and that all O−algebras are finitely generated as O−modules.

Definition 1.1. A block of a finite group G is a primitive idempotent b in the center Z(OG) of the group algebra OG of G over O. The algebra OGb is called the block algebra of the block b.

A defect group of the block b is a minimal subgroup P of G such that the homomorphism OGb ⊗_{OP} OGb ⟶ OGb induced by multiplication in OGb (i.e. mapping a ⊗ a′ to aa′, where a, a′ ∈ OGb) splits as homomorphism of OGb − OGb−bimodules. A source idempotent of b is a primitive idempotent i ∈ (OGb)^{P} such that the induced map OGi ⊗_{OP} iOG ⟶ OGb is still split surjective as homomorphism of OGb − OGb−bimodules. The algebra iOGi, considered together with the group homomorphism P ⟶ (iOGi)^{×} mapping u ∈ P to ui, is called a source algebra of b.

A vertex of an indecomposable OG−module U is a minimal subgroup Q of G such that the homomorphism OG ⊗_{OQ} U ⟶ U mapping a ⊗ u to au, where a ∈ A, u ∈ U, splits as homomorphism of OG−modules. A source of U is an indecomposable direct summand V of Res_{Q}^{G}(U) such that U is isomorphic to a direct summand of Ind_{Q}^{G}(V).

Let us explain the vocabulary used in this definition.

For any finite group G, the algebra OG is the free O−module with basis G (cf. I, section 1); that is, we can write formally OG = ⊕_{x∈G}Ox, considered as algebra with the multiplication induced by the product in G. The center of OG is the commutative subalgebra

Z(OG) = {z ∈ OG|az = za for all a ∈ OG} .

An idempotent of an O−algebra A is a non-zero element i ∈ A satisfying i^{2} = i. Note that if i ≠ 1_{A} then 1_{A} − i is an idempotent, too. Two idempotents i, j in A are orthogonal if ij = 0 = ji; in that case i + j is again an idempotent. Note again that if i is an idempotent different from 1_{A} then i and 1_{A} −i are in fact orthogonal idempotents. An idempotent in A is called primitive if it cannot be written as sum of two orthogonal idempotents.

A primitive decomposition of an idempotent e ∈ A is a finite set I of pairwise orthogonal primitive idempotents in A such that e = ∑_{i∈I}i. Note that then ei = i = ie for

any i ∈ I and that the left A−module Ae can be writtent as a direct sum Ae = ⊕_{i∈I}Ai (and similarly for the right A−module eA). From this follows easily that - using our convention that A is finitely generated as O−module - every idempotent in A has a primitive decomposition.

We recall the following fundamental properties of idempotents (see I, section 4 for proofs)

- We have a “Krull-Schmidt-theorem” (I, 4.2): any two primitive decompositions I, I′ of an idempotent e in an O−algebra A are conjugate by an element of the group A^{×} of invertible elements in A; that is, there is u ∈ A^{×} such that uIu^{−1} = I′.

- For any two idempotents i, i′ in A we have an isomorphism Ai ≅ Ai′ of left A−modules if and only if i, i′ are conjugate.

- If i is a primitive idempotent in A, the A−module Ai has a unique maximal submodule, namely J(A)i, and then the quotient module S_{i} = Ai/J(A)i is a simple A−module. The correspondences

i ↦ Ai ↦ S_{i}

induce bijections between the sets of conjugacy classes of primitive idempotents, isomorphism classes of projective indecomposable A−modules and isomorphism classes of simple A−modules.

Let still G be a finite group. Any two blocks b, b′ of G are either equal or orthogonal (easy exercise). Thus the set B of blocks of G is a primitive decomposition of 1 in Z(OG); that is, 1 = ∑_{b∈B}b, and whence, we have a decomposition

OG = ⊕_{b∈B}OGb

as direct sum of two-sided ideals, each of which is itself an indecomposable algebra.

The module category Mod(OG) of finitely generated left OG−modules decomposes into a direct sum of the module categories Mod(OGb), with b running over the set B of blocks of G. More explicitely, if U is an OG−module, we have a direct decomposition

U = ⊕_{b∈B}bU

as direct sum of OG−modules. Thus if U is indecomposable, there is a unique block b of G such that U = bU, and then necessarily b′U = {0} for any block b′ of G which is different from b. In that situation we say that U belongs to the block b, and we may view U as OGb−module, since the block idempotent b acts as identity on U.

Recall, with the notation of 1.1, that the homomorphism OGb ⊗_{OP} OGb ⟶^{μ} OGb is split if there is a homomorphism ι : OGb ⟶ OGb ⊗_{OP} OGb such that μ ∘ ι = Id_{OGb}. This means in particular, that OGb is isomorphic to a direct summand of OGb ⊗_{OP} OGb as OGb − OGb−bimodule. Moreover, for some indecomposable direct summand X of OGb ⊗_{OP} OGb the restriction of μ to X yields still a split surjection X ⟶ OGb; this holds because otherwise any composition OGb ⟶ X ⟶ OGb would be an endomorphism of OGb as OGb − OGb−bimodule which is not an automorphism, thus induced by multiplication with an element of J(Z(OGb)), and taking the sum over a direct decomposition of OGb ⊗_{OP} OGb would imply that μ ∘ ι is induced by multiplication with an element of J(Z(OGb)), contradicting the fact that μ ∘ ι = Id_{OGb}. This implies that there is some primitive idempotent i ∈ (OGb)^{P} such that the map

OGi ⊗_{OP} iOG ⟶ OGb

induced by multiplication in OGb is still split surjective; in other words, source idempotents do exist. We will see in section 3 how to get all source idempotents and discuss up to what source idempotents are unique.

Finally, for any OGb−module U, tensoring μ and ι by − ⊗_{OGb} U together with the canonical isomorphism U ≅ OGb ⊗_{OGb} U shows that the natural map

OGb ⊗_{OP} U ⟶ U

splits as homomorphism of OGb−modules. In particular, for every indecomposable OGb−module U and any defect group P of b there is a vertex Q of U contained in P.

We will see later that one can be much more precise about defect groups and vertices (section 3 below).

The next definition is a fundamental technical tool establishing a link between global and local structure of blocks. It goes back to Brauer [5] and has been subsequently generalized to modules in Feit [14] and G−algebras in Broué-Puig [8].

Definition 1.2. Let G be a finite group, P a p−subgroup of G and set (OG)^{P} = {a ∈ OG|uau^{−1} = a for all u ∈ P }. The Brauer homomorphism associated with P is the O−linear map

Br_{P}^{G} : (OG)^{P} ⟶ kC_{G}(P)

sending ∑_{x∈G}λ_{x}x ∈ (OG)^{P} to ∑_{x∈C_{G}(P)}\overline{λ}_{x}x, where λ_{x} ∈ O and \overline{λ}_{x} is the image of λ_{x} in

k, for any x ∈ G.

It is easy to describe the subalgebra (OG)^{P} of P −stable elements in OG as follows:

we have ∑_{x∈G}λ_{x}x ∈ (OG)^{P} if and only if ∑_{x∈G}λ_{x}uxu^{−1} = ∑_{x∈G}λ_{x}x for any u ∈ P, or

equivalently, if and only if λ_{x} = λ_{u^{−1}xu} for any x ∈ G and any u ∈ P. Thus (OG)^{P} is O−free with the set of P −conjugacy class sums of elements of G as an O−basis. That is, if x ∈ G and u, v ∈ P we have ^{u}x = ^{v}x if and only if v^{−1}u ∈ C_{P} (x). Therefore the set of the different P −conjugates of x is {^{u}x}_{u∈[P/C_{P} (x)]} , where [P/C_{P} (x)] is a set of representatives of the cosets uC_{P} (x) in P. The corresponding conjugacy class sum is \underline{x} = ∑_{u∈[P/C_{P} (x)]}^{u}x. Such a P −conjugacy class consists of a single element x if and only

if x ∈ C_{G}(P), or equivalently, C_{P} (x) = P. Thus the linear map Br_{P}^{G} has as kernel the O−submodule generated by all non trivial conjugacy class sums and J(O)(OG)^{P} .

One might wonder, why we chose to define the Brauer homomorphism in such a way that its image is kC_{G}(P) and not OC_{G}(P). The reason is the following theorem.

Theorem 1.3. Let G be a finite group and P a p−subgroup of G. The Brauer homomorphism

Br_{P}^{G} : (OG)^{P} ⟶ kC_{G}(P)

is a surjective algebra homomorphism.

Proof. The surjectivity of Br_{P}^{G} is trivial, because OC_{G}(P) ⊂ (OG)^{P} maps onto kC_{G}(P) under the Brauer homomorphism. We have to show that Br_{P}^{G} is multiplicative. Let x, y ∈ G and consider their P −conjugacy class sums \underline{x} = ∑_{[P/Q]}^{u}x, \underline{y} = ∑_{[P/R]}^{u}y, where

Q = C_{P} (x), R = C_{P} (y). We have to show that Br_{P}^{G}(\underline{x}\underline{y}) = Br_{P}^{G}(\underline{x})Br_{P}^{G}(\underline{y}). If P = Q = R then \underline{x} = x, \underline{y} = y, we have x, y ∈ C_{G}(P), whence xy ∈ C_{G}(P), and we are done. If P ≠ Q then Br_{P}^{G}(\underline{x}) = 0, thus Br_{P}^{G}(\underline{x})Br_{P}^{G}(\underline{y}) = 0. We have to show that Br_{P}^{G}(\underline{x}\underline{y}) = 0.

Since \underline{y} is P −stable, we can write \underline{x}\underline{y} = ∑_{u∈[P/Q]}^{u}(x\underline{y}) = ∑_{v∈[P/R]} ∑_{u∈[P/Q]}^{u}(xvyv^{−1}).

If in this sum we have a summand xvyv^{−1} which is not contained in C_{G}(P), then for any u ∈ P, the summand ^{u}(xvyv^{−1}) is not contained in C_{G}(P), so all these terms lie in ker(Br_{P}^{G}). If however xvyv^{−1} ∈ C_{G}(P), then for any u ∈ P we have ^{u}(xvyv^{−1}) = xvyv^{−1} , so this summand occurs [P : Q] times. As P ≠ Q and k has characteristic p, it follows that the total sum is zero.

Recall from I,4.3 the nice behaviour of idempotents with respect to surjective algebra homomorphisms: if A, B are O−algebras and f : A ⟶ B is a surjective algebra homomorphism, then f(J(A)) = J(B), f(A^{×}) = B^{×}, and the following hold:

- for any primitive idempotent i in A either f(i) = 0 or f(i) is a primitive idempotent in B;

- for any primitive idempotent j in B there is a primitive idempotent i in A such that f(i) = j;

- any two primitive idempotents i, i′ in A not contained in ker(f) are conjugate by an element of A^{×} if and only if their images f(i), f(i′) are conjugate by an element of B^{×}.

The systematical interplay between idempotents and Brauer homomorphisms will give rise to the definition of the local structure of a block in terms of its Brauer pairs, introduced in [2], analogously to the p−local structure of a finite group. One of the fundamental definitions in this context is the following (an amalgam of definitions occurring in [2], [8], [31]):

Definition 1.4. Let G be a finite group and b a block of G. 1.4.1. A Brauer-pair of OG is a pair (P, e) consisting of a p−subgroup P of G and a block e of kC_{G}(P). 1.4.2. A b−Brauer pair is a Brauer pair (P, e) consisting of a p−subgroup P of G and a block e of kC_{G}(P) such that Br_{P}^{G}(b)e = e. 1.4.3. For two Brauer pairs (P, e), (Q, f) we write (Q, f) ⊂ (P, e) and say that (Q, f) is contained in (P, e) if there is a primitive idempotent i in (OG)^{P} such that Br_{P} (i)e ≠ 0 and Br_{Q}(i)f ≠ 0. 1.4.4. For any two Brauer pairs (P, e), (Q, f) of OG we denote by E_{G}((Q, f), (P, e)) the set of all classes of group homomorphisms φ : Q ⟶ P modulo inner automorphisms of P for which there is an element x ∈ G satisfying φ(u) = xux^{−1} for all u ∈ G and x(Q, f) ⊂ (P, e), where ^{x}(Q, f) = (^{x}Q,^{x}f); the elements of E_{G}((Q, f), (P, e)) are called G−fusions from (Q, f) to (P, e).

The previous definition makes sense: clearly Br_{P}^{G} maps Z(OG) to Z(kC_{G}(P)), but not surjectively, in general. Thus either Br_{P}^{G}(b) = 0 or Br_{P}^{G}(b) is an idempotent in Z(kC_{G}(P)), whence a sum of blocks of kC_{G}(P). If i is a primitive idempotent in (OG)^{P} then either Br_{P} (i) = 0 or Br_{P} (i) is a primitive idempotent in kC_{G}(P) since Br_{P} is surjective. Thus, in the latter case, there is a unique block e of kC_{G}(P) such that Br_{P} (i)e ≠ 0, which forces in fact Br_{P} (i)e = Br_{P} (i) by the primitivity of Br_{P} (i). If however Q is a proper subgroup of P, the idempotent Br_{Q}(i) need no longer be primitive in kC_{G}(Q), and so, at a first glance, one might expect that there could be several blocks f of kC_{G}(Q) satisfying Br_{Q}(i)f ≠ 0, or equivalently, satisfying (Q, f) ⊂ (P, e). It is a remarkable fact of Brauer pairs, that this does not happen:

Theorem 1.5. (Broué-Puig [8, 1.8]) Let G be a finite group, (P, e) a Brauer pair of OG and let Q be a subgroup of P.

There is a unique block f of kC_{G}(Q) such that (Q, f) ⊂ (P, e). In other words, f is a (necessarily unique) block of kC_{G}(Q) with the property that for any primitive idempotent i ∈ (OG)^{P} satisfying Br_{P} (i)e = Br_{P} (i) ≠ 0 we have Br_{Q}(i)f = Br_{Q}(i) ≠ 0.

It follows that the “inclusion relation” on Brauer pairs is transitive (this is not obvious from the definition). As a consequence of this theorem, the next definition makes sense:

Definition 1.6. Let G be a finite group, b a block of OG and The Brauer category of the block b is the category whose objects are the b−Brauer pairs and whose sets of morphisms are the sets E_{G}((Q, f), (P, e)) of G−fusions between the b−Brauer pairs (Q, f) and (P, e).

The Brauer category expresses “how” the defect groups are embedded in the block algebra in terms of the action of G by conjugation on Brauer pairs. This is a way of describing the p−local structure of a block. There is a significant refinement of this philosophy, using pointed groups, introduced by Puig in [30], instead of Brauer pairs, that we are going to study in section 3.

Remarks 1.7. Let G be a finite group. 1.7.1. There is a unique surjective algebra homomorphism

OG ⟶ O

mapping x ∈ G to 1, called augmentation homomorphism of OG.

Since 1 is the unique idempotent of O, the augmentation homomorphism maps exactly one block b_{0} of OG to 1 and any other block to zero. The block b_{0} is called the principal block of OG. Equivalently, b_{0} is the unique block of OG to which the trivial OG−module O belongs.

The defect groups of the principal block b_{0} are precisely the Sylow-p−subgroups of G (because one easily checks that the Sylow- p−subgroups are the vertices of the trivial OG−module O).

1.7.2. Suppose that p = 2. Then OA_{4} has a unique block (namely 1), and OA_{5} decomposes into two blocks, OA_{5} = OA_{5}b_{0}⊕ OA_{5}b_{1} .

1.7.3. If P is a finite p−group, OP has 1_{OP} as unique block. More generally, if E is a p′−group acting on P such that the semi-direct product P ⋊ E is a Frobenius group (i.e. any non trivial element of E acts without fixpoints on P − {1}), then O(P ⋊ E) has a unique block.

1.7.4. If b_{0} is the principal block of OG, for any p−subgroup P of G the pair (P, e_{0}) is a b_{0}−Brauer pair, where e_{0} is the principal block of kC_{G}(P), and all b_{0}−Brauer pairs are of this form. It follows that the Brauer category of b_{0} is equivalent to what is sometimes called the Frobenius category, namely the category whose objects are the p−subgroups of G and whose morphisms are the classes E_{G}(Q, P) of group homomorphisms φ : Q ⟶ P modulo inner automorphisms of P for which there is an element x ∈ G satisfying φ(u) = xux^{−1} for any u ∈ Q.

1.7.5. We will see in section 3 that the maximal b−Brauer pairs (with respect to the inclusion relation), where b is a block of OG, are all G−conjugate, and that if (P, e) is a maximal b−Brauer pair, then P is a defect group of b; thus in particular, the defect groups of b are all G−conjugate.

2. Group algebras and groups acting on algebras

We collect in this section basic material on algebras endowed with an action of a finite group which does not depend on specific properties of the ground ring over which we work.

Let R be a commutative ring and G a finite group.

Recall from I, section 1, that the group algebra RG of G over R is the free R−module having the set of elements of G as R−basis, endowed with the R−bilinear multiplication induced by multiplication in G. We denote as usual by Mod(RG) the category of finitely generated left RG−modules.

For any subgroup H of G we denote by Res_{H}^{G} : Mod(RG) ⟶ Mod(RH) the restriction functor sending an RG−module to its restriction to RH. This functor is naturally isomorphic to the functor _{H}(RG) ⊗_{RH} − , where _{H}(RG) is the restriction to RH on the left of the regular RG − RG−bimodule RG.

We denote by Ind_{H}^{G} : Mod(RH) ⟶ Mod(RG) the induction functor Ind_{H}^{G}− = RG ⊗_{RH} −, mapping an RH−module V to the RG−module RG ⊗_{RH} V .

Recall that an RG−module U is called relatively RH−projective, or simply relatively H−projective, if U is isomorphic to a direct summand of Ind_{H}^{G}(V) for some RH−module V . See I,8.3 for a characterization of relatively H−projective modules in the case where R is a field (some of these characterizations hold over arbitrary R, see 2.6 below).

If x, y ∈ G then any two right H−cosets xH, yH are either equal or disjoint, and hence the right H−cosets in G form a partition of G. We denote by [G/H] a subset of G such that G = ∪_{x∈[G/H]}xH is a disjoint union; the set [G/H] is called a set of representatives of the set of right H−cosets in G. Similarly, we denote by [H\G] a set of representatives of the left H−cosets Hx in G, where x ∈ G. If L is another subgroup of G, again any two double cosets HxL, HyL are either equal or disjoint, and we denote by [H\G/L] a system of representatives in G of the set of H − L−double cosets in G.

A G−set is a set M endowed with an action of G on M; this is a map G×M ⟶ M, written (x, m) ↦ xm, where x ∈ G, m ∈ M, such that 1_{G}m = m and x(ym) = (xy)m for all x, y ∈ G and m ∈ M. In particular, x^{−1}(xm) = m for all x ∈ G and m ∈ M, and so the map ρ(x) : M ⟶ M defined by ρ(x)(m) = xm is a permuation of M with inverse permutation ρ(x^{−1}). The condition x(ym) = (xy)m adds that the map

ρ : G ⟶ S_{M}

obtained this way is a group homomorphism, where S_{M} is the group of all permutations of M.

The stabilizer of m ∈ M in G is the group G_{m} = {x ∈ G|xm = m}. In a similar way we define right G−sets and G − G′−bisets, if G′ is another finite group. Note that we can consider any left G−set M as right G−set (and vice versa) by defining the right action of x ∈ G on m ∈ M through m.x = x^{−1}m; that is, through the original action of x^{−1} on the left.

By taking a G−set M as basis of a free R−module, we obtain an RG−module R[M]: the elements of R[M] are again formal sums ∑_{m∈M}r_{m}m, where r_{m} ∈ R for any m ∈ M,

and this is an RG−module by extending R−bilinearly the action of G on M. Of corse, if M is a right G−set, then R[M] is a right RG−module, and if M is a G − G′−biset, then R[M] is an RG − RG′−bimodule. In particular, for any two subgroups H, L of G and x ∈ G, the left coset Hx gives rise to the left RH−module R[Hx], the right coset xH gives rise to a right RH−module, and from the double coset HxL we obtain the RH − RL−bimodule R[HxL].

Right multiplication by x and x^{−1} on RH and R[Hx], respectively, induce mutually inverse isomorphisms

RH ≅ R[Hx]

of left RH−modules, and thus we have isomorphisms of left RH−modules

RG = ⊕_{x∈[H\G]}R[Hx] ≅ (RH)^{[G:H]} ,

which show that RG is free of rank [G : H] as left RH−module. Similarly, RG is free of rank [G : H] as right RH−module.

The RH −RL−bimodule R[HxL] is free as left RH−module because HxL is a union of certain left H−cosets, and, for similar reasons, it is also free as right RL−module.

The algebra RG is isomorphic to its opposite algebra (RG)^{0} as defined in 1.1 by means of the anti-automorphism of RG mapping x ∈ G to x^{−1} . This allows us to consider any right RG−module V as left RG−module (and vice versa) through x.v = vx^{−1} , where x ∈ G, v ∈ V . In particular, if U is a left RG−module, its R−dual U^{∗} = Hom_{R}(U, R) is a right RG−module through u^{∗}.x(u) = u^{∗}(xu) for all x ∈ G, u ∈ U, u^{∗} ∈ U^{∗}, and hence U^{∗} becomes again a left RG−module through x.u^{∗}(u) = u^{∗}(x^{−1}u).

More generally, an RG − RH−bimodule M can be considered as R(G × H)−module (and vice versa) through (x, y).m = xmy^{−1} , where x ∈ G, y ∈ H and m ∈ M. Remember also that M^{∗} = Hom_{R}(M, R) becomes an RH − RG−bimodule through y.m^{∗}.x(m) = m^{∗}(xmy), where x ∈ G, y ∈ H, m ∈ M and m^{∗} ∈ M^{∗}.

Theorem 2.1. Let G be a finite group, H a subgroup of G, U an RG−module and V an RH−module.

(i) The map sending χ ∈ (RG)^{∗} to ∑_{x∈G}χ(x^{−1})x ∈ RG is an isomorphism of RG −

RG−bimodules; in particular, the R−algebra RG is symmetric.

(ii) The covariant functors Ind_{H}^{G} and Res_{H}^{G} are exact.

(iii) We have a natural isomorphism

\{ \begin{matrix}Hom_{RG}(Ind_{H}^{G}(V), U) & ≅ Hom_{RH}(V, Res_{H}^{G}(U)) \\ φ & ↦ (v ↦ φ(1_{G} ⊗ v)) ; \end{matrix}

in particular, Ind_{H}^{G} has Res_{H}^{G} as right adjoint.

(iv) We have a natural R−linear isomorphism

\{ \begin{matrix}Hom_{RH}(Res_{H}^{G}(U), V) & ≅ Hom_{RG}(U, Ind_{H}^{G}(V)) \\ ψ & ↦ (u ↦ ∑_{x∈[G/H]}x ⊗ ψ(u)) ; \end{matrix}

in particular, Ind_{H}^{G} has Res_{H}^{G} as right adjoint.

Proof. All statements can be proved by a straightforward verification. In (i) one constructs an explicit inverse sending x ∈ G to the unique form on RG mapping x^{−1} to 1_{R} and all other elements of G to 0_{R}. Since RG is free as left or right RH−module and RG−module, the exactness of induction and restriction follows from the fact that tensoring by a finitely generated projective module is always an exact functor.

The adjunction of Ind_{H}^{G} and Res_{H}^{G} is frequently called Frobenius reciprocity; it is a particular case of a more general adjunction property of functors induced by bimodules between so-called symmetric algebras.

Definition 2.2. Let G be a finite group. A G−algebra over R is an R−algebra A endowed with an action G × A ⟶ A of G on A, written (x, a) ↦ ^{x}a for any a ∈ A and x ∈ G, such that the map sending a ∈ A to ^{x}a is an R−algebra automorphism of A for all x ∈ G. We denote then for any subgroup H of G by A^{H} the subalgebra of A of H−fixpoints in A; that is, A^{H} = {a ∈ A|^{y}a = a for all y ∈ H}.

An interior G−algebra over R is an R−algebra A endowed with a group homomorphism σ : G ⟶ A^{×}, called then the structural homomorphism of A.

Every interior G−algebra A with structural homomorphism σ : G ⟶ A is in particular a G−algebra, with the action of x ∈ G on a ∈ A defined by ^{x}a = σ(x)aσ(x^{−1}); that is, by “conjugation with σ(x) ”.

The main examples of (interior) G−algebras that will be relevant for us are the following: the group algebra RG is an interior G−algebra with structural homomorphism G ⟶ (RG)^{×} mapping x ∈ G to its natural image in RG; if U is an RG−module, then the R−algebra End_{R}(U) becomes an interior G−algebra with structural homomorphism

G ⟶ End_{R}(U)^{×} = Gl_{R}(U)

mapping x ∈ G to the R−linear automorphism of U sending u ∈ U to xu. As mentioned before, End_{R}(U) is therefore in particular a G−algebra, and the action of x ∈ G on φ ∈ End_{R}(U) is given by

(^{x}φ)(u) = xφ(x^{−1}u) ,

where u ∈ U. This formula shows that for any subgroup H of G we have (End_{R}(U))^{H} = End_{RH}(U), the endomorphism algebra of U considered as RH−module.

There are two elementary construction principles for (interior) G−algebras, that we frequently employ without further comment: If A is an interior G−algebra with structural homomorphism σ : G ⟶ A^{×}, then for any subgroup H of G and any idempotent i ∈ A^{H}, the algebra iAi becomes an interior H−algebra with structural homomorphism H ⟶ (iAi)^{×} mapping x ∈ H to σ(x)i = iσ(x). Note that the source algebra defined in 1.1 is a particular case of this concept. If A′ is an interior G′−algebra for another finite group G′, with structural homomorphism σ′ : G′ ⟶ (A′)^{×}, then the algebra A ⊗_{R} A′ becomes an interior G × G′−algebra with structural homomorphism mapping (x, x′) ∈ G × G′ to σ(x) ⊗ σ(x′). If moreover G = G′, we can consider A ⊗_{R} A′ again as interior G−algebra through the diagonal embedding Δ_{G} : G ⟶ G × G mapping x ∈ G to (x, x). Similar constructions work for G−algebras.

For a G−algebra A and any subgroup H of G we have A^{G} ⊂ A^{H}, and this inclusion is in particular an R−linear map. We define now an R−linear map A^{H} ⟶ A^{G}. If x, y ∈ G such that xH = yH, there is h ∈ H such that y = xh. Then for any a ∈ A^{H} we have ^{y}a = ^{xh}a = ^{x}a. This implies that the sum ∑_{x∈[G/H]}^{x}a is G−stable and does not depend on the choice of the system of representatives [G/H] of the set of right H−cosets in G. Therefore the following definition makes sense:

Definition 2.3. (Relative traces, J. A. Green [17]) Let G be a finite group, A a G−algebra and H a subgroup of G. We define the R−linear map

T r_{H}^{G} : A^{H} ⟶ A^{G}

by T r_{H}^{G}(a) = ∑_{x∈[G/H]}^{x}a for any a ∈ A^{H}, and call T r_{H}^{G} the relative trace map from H

to G on A. Moreover, we set A_{H}^{G} = Im(T r_{H}^{G}).

We collect from [17] a few elementary properties of relative trace maps.

Proposition 2.4. Let G be a finite group, A a G−algebra and H, L subgroups of G.

(i) For any a ∈ A^{H} and b ∈ A^{G} we have T r_{H}^{G}(a)b = T r_{H}^{G}(ab) and bT r_{H}^{G}(a) = T r_{H}^{G}(ba); in particular, A_{H}^{G} is a two-sided ideal in A^{G}.

(ii) If L ⊂ H we have T r_{H}^{G}T r_{L}^{H} = T r_{L}^{G}.

(iii) If L ⊂ H and x ∈ G, we have ^{x}(T r_{L}^{H}(a)) = T r_{^{x}L}^{^{x}H}(^{x}a) for any a ∈ A^{L}.

(iv) For any a ∈ A^{L} we have T r_{L}^{G}(a) = ∑_{x∈[H\G/L]}T r_{H∩^{x}L}^{H}(^{x}a).

Proof. The statements (i), (ii), (iii) are trivial verifications. For (iv) one only needs to observe, that in the disjoint union G = ∪_{x∈[H\G/L]}HxL any double coset HxL is again a disjoint union HxL = ∪_{y∈[H/H∩^{x}L]}yxL. In other words, we can take for [G/L] the set of all yx, with x running over [H\G/L] and, for any such x, with y running over [H/H ∩^{x} L], which implies the formula.

The formula in 2.4(iv) is called Mackey formula; it is basically a reinterpretation of the decomposition of G as disoint union of double cosets. A formula of this type has first been proved by Mackey in the context of induction and restriction; see I, 7.6. In order to state this properly, observe that for any subgroup H of G and any x ∈ G, the right RH−module R[xH] is also a left R^{x}H−module; indeed, xH is a left xH−set since (^{x}h)(xh′) = x(hh′) ∈ xH for any h, h′ ∈ H. Thus R[xH] is in fact an R^{x}H − RH−bimodule, and for any RH−module V we have an R^{x}H−module xV = R[xH] ⊗_{RH} V , obtained from V by “transport de structure” through the group isomorphism ^{x}H ≅ H mapping ^{x}h to h for any h ∈ H.

Theorem 2.5. (Mackey) Let G be a finite group, H, L subgroups of G.

(i) We have RG = ⊕_{x∈[H\G/L]}R[HxL] as RH − RL−bimodules.

(ii) For any x ∈ G there is a unique isomorphism of RH − RL−bimodules

R[HxL] ≅ RH ⊗_{R(H∩^{x}L)} R[xL]

mapping yxz to y ⊗ xz, where y ∈ H and z ∈ L.

(iii) For any RL−module W there is a natural isomorphism of RH−modules

Res_{H}^{G}Ind_{L}^{G}(W) ≅ ⊕_{x∈[H\G/L]}Ind_{H∩^{x}L}^{H}Res_{H∩^{x}L}^{^{x}L}(^{x}W) .

Proof. Statement (i) is trivial, (ii) is easily checked, and (iii) follows by tensoring the formulae in (i), (ii) with − ⊗_{RH} W.

Relative traces allow the following elegant reformulation of relative projectivity:

Theorem 2.6. (Higman’s criterion) Let G be a finite group, H a subgroup of G and U an RG−module. The following statements are equivalent.

(i) The module U is relatively H−projective.

(ii) We have Id_{U} ∈ (End_{R}(U))_{H}^{G} .

Proof. Suppose that Id_{U} = T r_{H}^{G}(φ) for some φ ∈ (End_{R}(U))^{H} = End_{RH}(U). This means that ∑_{y∈[G/H]}yφ(y^{−1}u) = u for all u ∈ U. Thus the natural surjection RG ⊗_{RH}

U ⟶ U mapping x ⊗ u to xu splits with section U ⟶ RG ⊗_{RH} U mapping u to ∑_{y∈[G/H]}y ⊗ φ(y^{−1}u), where u ∈ U and x ∈ G. In particular, U is isomorphic to a direct

summand of RG ⊗_{RH} U = Ind_{H}^{G}Res_{H}^{G}(U) and hence relatively H−projective.

In order to prove the converse, let V be an RH−module, set W = RG ⊗_{RH} V and define τ ∈ End_{RH}(W) by τ(x ⊗ v) = x ⊗ v if x ∈ H and τ(x ⊗ v) = 0 if x ∈ G − H, where v runs over V . Then T r_{H}^{G}(τ)(x ⊗ v) = ∑_{y∈[G/H]}yτ(y^{−1}x ⊗ v). In this sum there is

exactly one term for which y^{−1}x ∈ H, and then this term is equal to y(y^{−1}x⊗v) = x⊗v while all other terms vanish by the definition of τ. This means that T r_{H}^{G}(τ) = Id_{W} .

Now if U is isomorphic to a dirct summand of W, there are maps ι : U ⟶ W and π : W ⟶ U such that πι = Id_{U}. Then Id_{U} = π(Id_{W})ι = πT r_{H}^{G}(τ)ι = T r_{H}^{G}(πτι), where the last equality comes from the fact that π, ι commute to the action of G.

We refer to I, 7.3 to a longer list of equivalent statements to those in Higman’s criterion (formulated there over a field).

3. Pointed groups and the Brauer construction

This section contains some fundamental technical concepts from block theory. The notions of pointed groups and source algebras are due to L. Puig [30], who developed an approach to this theory which makes a systematic use of idempotents in algebras endowed with a group action.

As mentioned at the beginning of this chapter, the typical objects that we have to deal with in block theory are pairs consisting of a finite group and an idempotent in some fixpoint algebra (or, more precisely, a conjugacy class of idempotents, as we will see later). For instance, a block is a pair consisting of a finite group G and a primitive idempotent of (OG)^{G}, and an indecomposable direct summand M′ of an OG−module M gives rise to an idempotent in End_{OG}(M), namely the projection of M onto M′. Other fundamental examples are defect groups coming along with their source idempotents (cf. 1.1) and Brauer pairs (cf. 1.4).

As before, p is a prime, O a complete discrete valuation ring having a residue field k = O/J(O) of characteristic p and G a finite group. Note that if A is an O−algebra, by our conventions the algebra A/J(O)A is a finite-dimensional k−algebra.

Definition 3.1. (i) Let A be an O−algebra. A point of A is an A^{×}−conjugacy class α of primitive idempotents in A. The multiplicity of the point α is the cardinal m_{α} of the finite set α ∩ I, where I is a primitive decomposition of 1_{A} in A.

(ii) Let A be a G−algebra and H a subgroup of G. A point of H on A is a point β of A^{H}, and the pair H_{β} is called a pointed group on A.

Remarks 3.2. 3.2.1. With the vocabulary introduced in 3.1, the lifting properties of idempotents I, 4.1 can be rephrased as follows: if f : A ⟶ B is a surjective homomorphism of O−algebras, for any point α on A either α ⊂ ker(f) or f(α) is a point of B; moreover, for any point β of B there is a point α of A such that f(α) = β, and in that situation we have m_{α} = m_{β}. Note also, that the multiplicity m_{α} is independent of the chosen decomposition I, since all primitive decompositions of 1_{A} in A are conjugate by some elements of A^{×}. The reason, why a surjective algebra homomorphism preserves multiplicities comes from the fact that orthogonal idempotents go again to orthogonal idempotents (or zero). 3.2.2. Remember from I, 4.4 that for any O−algebra A, a projective indecomposable A−module is isomorphic to Ai for some primitive idempotent i in A, and that for any two primitive idempotents i, j in A we have Ai ≅ Aj if and only if i, j are conjugate by an element of A^{×}, which in turn holds if and only if the simple quotients Ai/J(A)i and Aj/J(A)j are isomorphic. Thus we have bijections between the sets

\{ \begin{matrix}points of A ↔ proj. indec. A − modules ↔ simple A − modules \\ i ∈ α  ↦  Ai  ↦  S_{i} = Ai/J(A)i \end{matrix}

3.2.3. Let U be an OG−module and H a subgroup of G. If ι is an idempotent in End_{OH}(U), then ι(U) is a direct summand of Res_{H}^{G}(U), and ι is primitive in End_{OH}(U) if and only if this direct summand is indecomposable as OH−module. Since End_{OH}(U) = End_{O}(U)^{H}, the map sending ι to ι(U) induces a bijection between the set of points of H on End_{O}(U) and the set of isomorphism classes of indecomposable direct summands of Res_{H}^{G}(U). 3.2.4. A point of G on OG (with G acting by conjugation) is of the form {b}, where b is a block of OG, since (OG)^{G} = Z(OG). 3.2.5. A matrix algebra M_{n}(O) has a unique point α; we have as primitive decomposition of 1_{M_{n}(O)} the set {e_{i}}_{1≤i≤n}, where e_{i} is the idempotent matrix with entry 1 in (i, i) and all other entries zero, and one easily checks that the e_{i} are all conjugate. This holds, of course, also for k instead of O, and more generally for any division algebra D. 3.2.6. The canonical surjection s : A ⟶ A/J(A) induces a bijection between the sets of points of A and of A/J(A), since ker(s) = J(A) contains no idempotent. By Wedderburn’s theorem, the semi-simple algebra A/J(A) is a finite direct sum of matrix algebras over division algebras, one for each point; thus

A/J(A) = ⊕_{α}M_{m_{α}}(D_{α}) ,

where α runs over the set of points of A and D_{α} is the division algebra D_{α} = End_{A}(Ai/J(A)i) for some i ∈ α. This shows that for any point α of A there is a unique maximal ideal M_{α} in A not containing α, and then A/M_{α} ≅ M_{m_{α}}(D_{α}). This explains also the terminology: the points of an affine variety correspond bijectively to the maximal ideals in the associated affine algebra. 3.2.7. For any G−algebra A, the group G acts on the set of pointed groups on A; indeed, clearly if H_{β} is a pointed group on A, the set ^{x}β is a point of ^{x}H on A, and thus x(H_{β}) = ^{x}H_{^{x}β} is a pointed group on A.

We define now an inclusion on the set of pointed groups on a G−algebra A (cf. [30]). Observe that if H, L are subgroups of G, we have A^{H} ⊂ A^{L}, and thus, if i is a primitive idempotent in A^{H}, this idempotent need no longer be primitive in A^{L}.

Definition 3.3. Let A be a G−algebra, H_{β}, L_{γ} pointed groups on A. We say that L_{γ} is contained in H_{β} and write L_{γ} ⊂ H_{β}, if L ⊂ H and if for some i ∈ β there is j ∈ γ such that ij = j = ji (and then such a j can be found for any i ∈ β, since (A^{H})^{×} ⊂ (A^{L})^{×}).

Observe that the inclusion L_{γ} ⊂ H_{β} is equivalent to L ⊂ H and j ∈ A^{L}iA^{L} for some (and whence any) i ∈ β, j ∈ γ; indeed, if ji = j = ij then j = ij ∈ iA^{L} ⊂ A^{L}iA^{L}, and conversely, if j ∈ A^{L}iA^{L} we have j ∈ jA^{L}iA^{L}j, which means that iA^{L}j is not contained in iJ(A^{L})j, or equivalently, there is an A^{L}−homomorphism A^{L}i ⟶ A^{L}j whose image is not in J(A^{L})j, thus which is surjective because the projective indecomposable A^{L}−module A^{L}j has J(A^{L})j as unique maximal submodule, and therefore A^{L}i has a direct summand isomorphic to A^{L}j, which in turn means that some conjugate of j appears in a primitive decomposition of i in A^{L}.

Remark 3.4. Let G be a finite group, U an OG−module, H_{β}, L_{γ} pointed groups on the interior G−algebra A = End_{O}(U), and let i ∈ β, j ∈ γ. We have L_{γ} ⊂ H_{β} if and only if L ⊂ H and j(U) is isomorphic to a direct summand of Res_{L}^{H}(i(U)).

We generalize now the Brauer construction in 1.3 to arbitrary G−algebras (see [8]).

Definition 3.5. Let G be a finite group, A a G−algebra and P a subgroup of G. We set

A(P) = A^{P} /( ∑_{Q<P}A_{Q}^{P} + J(O)A^{P})

and denote by Br_{P}^{A} : A^{P} ⟶ A(P) the canonical surjection.

For any Sylow-p−subgroup Q of P the index [P : Q] is invertible in O and whence a = T r_{Q}^{P}(\frac{1}{[P:Q]}a) for any a ∈ A^{P} , which shows that A(P) is zero unless P is a p−group. In the above definition, Br_{P}^{A} is a surjective algebra homomorphism; for x ∈ N_{G}(P) we have ^{x}(A^{P}) = A^{P} and ^{x}(A_{Q}^{P}) = A_{^{x}Q}^{P}, thus the action of N_{G}(P) on A induces an action of N_{G}(P) on A(P), making Br_{P}^{A} into a homomorphism of N_{G}(P)−algebras. If A = OG and P is a p−subgroup of G we have a natural isomorphism (OG)(P) ≅ kC_{G}(P), which shows that the definition 3.5 generalizes 1.3.

Definition 3.6. Let G be a finite group, A a G−algebra and P a subgroup of G. A point γ of P on A is called a local point of P on A if Br_{P}^{A}(γ) ≠ 0, and P_{γ} is then called a local pointed group on A.

Note that if P_{γ} is a local pointed group on A, the image Br_{P}^{A}(γ) is then a point of kC_{G}(P) because Br_{P}^{A} is a surjective algebra homomorphism. If Br_{P}^{A}(γ) = 0 we have γ ∈ A_{Q}^{P} for some proper subgroup Q of P by Rosenberg’s lemma I,4.6 and the fact that J(O)A^{P} ⊂ J(A^{P}) contains no idempotent.

In view of Higman’s criterion, we investigate now what happens if a point of a finite group G on a G−algebra A lies in the image of the relative trace map A_{H}^{G} for some subgroup H of G. The following results are due to Puig [30].

Theorem 3.7. Let G be a finite group, A a G−algebra, α a point of G on A and H a subgroup of G such that α ⊂ A_{H}^{G}.

(i) There is a point β of H on A such that α ⊂ T r_{H}^{G}(A^{H}βA^{H}), and H_{β} ⊂ G_{α}. (ii) For any subgroup P in G such that Br_{P}^{A}(α) ≠ 0 there is x ∈ G such that

xP ⊂ H.

(iii) For any local pointed group P_{γ} on A such that P_{γ} ⊂ G_{α} there is x ∈ G such that ^{x}(P_{γ}) ⊂ H_{β}.

Proof. Let e ∈ α and I a primitive decomposition of e ∈ A^{H}; that is, e = ∑_{i∈I}i.

Let a ∈ A^{H} such that T r_{H}^{G}(a) = e. Then

e = e^{2} = T r_{H}^{G}(a)e = T r_{H}^{G}(ae) = ∑_{i∈I}T r_{H}^{G}(ai)

which is an element of the sum of ideals ∑_{i∈I}T r_{H}^{G}(A^{H}iA^{H}) in A^{G}. Thus, by Rosen-

berg’s lemma I, 4.6, there is i ∈ I such that e ∈ T r_{H}^{G}(A^{H}iA^{H}). Take for β the (A^{H})^{×}−conjugacy class of i. This shows (i). Moreover, by Mackey’s formula 2.4(iv) we have

e = T r_{H}^{G}(a) = ∑_{y∈[P \G/H]}T r_{P ∩^{y}H}^{P}(^{y}a).

If P ∩ ^{y}H ≠ P, the corresponding summand lies in ker(Br_{P}^{A}). Thus, if Br_{P}^{A}(e) ≠ 0, there is y ∈ G such that P ∩ ^{y}H = P, or equivalently, such that ^{x}P ⊂ H, where x = y^{−1} . Statement (ii) follows.

For (iii) we just refine the above argument: by (i) we have e ∈ T r_{H}^{G}(A^{H}iA^{H}) ⊂

∑_{y∈[P \G/H]}T r_{P ∩^{y}H}^{P}(A^{P ∩^{y}H}(^{y}i)A^{P ∩^{y}H}), thus for a suitable j ∈ γ we have j = je ∈

T r_{P ∩^{y}H}^{P}(A^{P ∩^{y}H}(^{y}i)A^{P ∩^{y}H}) for some y ∈ G. But Br_{P}^{A}(j) ≠ 0, so P = P ∩^{y}H, whence j ∈ A^{P} (^{y}i)A^{P} , and therefore P_{γ} ∈ ^{y}(H_{β}) by the remark following 3.3, or again setting x = y^{−1} , we have ^{x}(P_{γ}) ⊂ H_{β}.

What one should remember from the above proof is that it just combines Rosenberg’s lemma and the Mackey formula, the rest is then routine. The reader should translate the previous theorem to a module theoretic statement involving relative projectivity by applying it to the particular case A = End_{O}(M), where M is an OG−module.

We define now defect (pointed) groups for arbitrary pointed groups, and show then that this definition generalizes indeed our initial definition of defect groups of blocks and of vertices of modules.

Definition 3.8. Let G be a finite group, A a G−algebra and α a point of G on A. (i) A subgroup P of G is called a defect group of α if P is minimal such that α ⊂ A_{P}^{G}. (ii) A pointed group P_{γ} on A is called a defect pointed group of G_{α} if P_{γ} is maximal

local such that P_{γ} ⊂ G_{α}.

(iii) If P_{γ} is a defect pointed group of G_{α} and i ∈ γ, the interior P −algebra iAi, endowed with the group homomorphism P ⟶ (iAi)^{×} mapping u ∈ P to ui, is a source algebra of G_{α}.

The next theorem is basically a consequence of 3.7:

Theorem 3.9. Let G be a finite group, A a G−algebra, α a point of G on A and P a subgroup of G. The following are equivalent:

(i) P is a defect group of α.

(ii) P is a maximal subgroup of G such that Br_{P}^{A}(α) ≠ 0.

(iii) There is a local point γ of P on A such that P_{γ} ⊂ G_{α} and α ⊂ T r_{P}^{G}(A^{P} γA^{P}). (iv) For every subgroup Q of P such that Br_{Q}^{A}(α) ≠ 0 there is x ∈ G such that

xQ ⊂ P.

In particular, all defect groups of α are G−conjugate.

Proof. If (i) holds so does (iv) by 3.7. Also by 3.7, there is a point γ of P on A such that α ⊂ T r_{P}^{G}(A^{P} γA^{P}). Then γ is not contained in A_{Q}^{P} for any proper subgroup Q of P by the minimality of P with α ⊂ A_{P}^{G}. Thus γ is local, so (iii) holds. Yet another immediate consequence of 3.7 is that (iii) implies (ii). If (ii) holds, then applying 3.7 to a defect group R of α yields x ∈ G such that ^{x}P ⊂ R, whence ^{x}P = R. If (iv) holds clearly P is maximal such that Br_{P}^{A}(α) ≠ 0, so (ii) holds.

Combining 3.7 and 3.9 yields then the following result (whose proof is left to the reader):

Theorem 3.10. Let G be a finite group, A a G−algebra, α a point of G on A and P_{γ} a pointed group on A. The following are equivalent:

(i) P_{γ} is a defect pointed group of G_{α}.

(ii) P_{γ} is a local pointed group on A such that P_{γ} ⊂ G_{α} and α ⊂ T r_{P}^{G}(A^{P} γA^{P}). (iii) P is a defect group of α and γ is a local point of P on A such that P_{γ} ⊂ G_{α}. (iv) For any local pointed group Q_{δ} on A such that Q_{δ} ⊂ G_{α} there is x ∈ G such

that ^{x}(Q_{δ}) ⊂ P_{γ}.

In particular, all defect pointed groups of G_{α} are G−conjugate.

Remarks 3.11. Let G be a finite group and b a block of OG.

3.11.1. The set α = {b} is a point of G on OG since (OG)^{G} = Z(OG). In order to establish a link between 1.1 and the content of this section we observe that for any subgroup P of G, the bimodule homomorphism

OGb ⊗_{OP} OGb ⟶ OGb

splits if and only if there is c ∈ (OGb)^{P} such that b = T r_{P}^{G}(c).

Indeed, if b = T r_{P}^{G}(c) we can give explicitely a section OGb ⟶ OGb ⊗_{OP} OGb, namely the map sending a ∈ OGb to ∑_{[G/P]}axc ⊗ x^{−1} ; one only has to check that this

map is independent of the choice of [G/P], from which follows that this map is indeed a homomorphism of OGb − OGb−bimodules. Conversely, if there is a section ι : OGb ⟶ OGb ⊗_{OP} OGb, the image ι(b) of b in OGb ⊗_{OP} OGb is G−stable (since b is so and ι is a homomorphism of OGb − OGb−bimodules). Now (OGb ⊗_{OP} OGb)^{G} ⊂ (OG ⊗_{OP} OG)^{G}. As [G/P] is a basis of OG viewed as right OP −module, any element of OG ⊗_{OP} OG can be written uniquely in the form ∑_{x∈[G/P]}x ⊗ c_{x}x^{−1} for some c_{x} ∈ OG. If this element is

G−stable, the uniqueness of the c_{x} implies that c = c_{x} does not depend on x. Thus ι maps b to an element of the form ∑_{x∈[G/P]}x ⊗ cx^{−1} for some c ∈ OGb, and then

b = ∑_{x∈[G/P]}xcx^{−1} = T r_{P}^{G}(c) because ι is a section of the homomorphism induced by

multiplication in OGb.

3.11.2. Let U be an indecomposable OG−module. Then {Id_{U}} is the unique point of G on A = End_{O}(U). Let P_{γ} be a defect pointed group of G_{{Id_{U} }}. By Higman’s criterion 2.6, P is a vertex of U, and the indecomposable OP −module i(U) is a source of U (cf. 1.1).

3.11.3. Let P be a defect group of the block b of OG. Let i ∈ (OGb)^{P} be a primitive idempotent such that the bimodule homomorphism

OGi ⊗_{OP} iOG ⟶ OGb

induced by multiplication in OGb is still split. By the minimality of P the idempotent i is not a relative trace from a proper subgroup Q of P, thus i belongs to a local point γ of P on OGb. As all defect pointed groups of G_{{b}} are conjugate, this shows that the source idempotents defined in 1.1 are exactly the primitive idempotents belonging to a local point of a defect group P on OGb. Note that at the level of source algebras, multiplying the above homomorphism by i on both sides implies that the iOGi − iOGi−bimodule homomorphism

iOGi ⊗_{OP} iOGi ⟶ iOGi

induced by multiplication in iOGi is still split surjective. Furthermore this shows that the map

OGi ⊗_{iOGi} iOG ⟶ OGb

induced by multiplication is surjective, and as the obvious map

iOG ⊗_{OGb} OGi ⟶ iOGi

is always an isomorphism, it follows that the OGb − iOGi−bimodule OGi and its dual iOG induce a Morita equivalence between OGb and iOGi.

4. Correspondences

Let p, O and k be as before. In I, section 9 the Green correspondence between modules and the Brauer correspondence between blocks are described in module theoretic terms. Using Higman’s criterion and the material developped in section 3, both correspondences can be interpreted as correspondences between (conjugacy classes of) primitive idempotents induced by the Brauer homomorphism. Technically, the key observation is the following elementary lemma:

Lemma 4.1. Let G be a finite group, A a G−algebra and P a subgroup of G. For any a ∈ A^{P} we have

Br_{P}^{A}(T r_{P}^{G}(a)) = T r_{P}^{N_{G}(P)}(Br_{P}^{A}(a)) ;

in particular, the Brauer homomorphism Br_{P}^{A} : A^{P} ⟶ A(P) maps A_{P}^{G} onto A(P)_{P}^{N_{G}(P)} and induces whence a bijection between the set of points of G on A having P as a defect group and the set of points of N_{G}(P) on A(P) having P as defect group.

Proof. By Mackey’s formula 2.4(iv) we have

T r_{P}^{G}(a) = ∑_{x∈[P \G/P]}T r_{P ∩^{x}P}^{P} (^{x}a) ;

thus all terms in this sum but those for which x lies in N_{G}(P) are contained in ker(Br_{P}^{A}). Therefore, applying Br_{P}^{A} to the above equation yields

Br_{P}^{A}(T r_{P}^{G}(a)) = ∑_{x∈[N_{G}(P)/P]}^{x}Br_{P}^{A}(a) = T r_{P}^{N_{G}(P)}(Br_{P}^{A}(a)) .

The rest follows then from the standard lifting theorems for idempotents (I, section 4).

Theorem 4.2. (Brauer correspondence) Let G be a finite group, P a p−subgroup of G and H a subgroup of G containing N_{G}(P).

For any block b of OG having P as defect group there is a unique block c of OH having P as defect group such that Br_{P} (b) = Br_{P} (c).

Moreover, the correspondence b ↦ c is a bijection between the sets of blocks of OG and of OH having P as defect group.

Proof. The blocks of OG with defect group P are exactly the primitive idempotents of (OG)_{P}^{G} not contained in ker(Br_{P}), thus correspond bijectively to the primitive idempotents of

Br_{P} ((OG)_{P}^{G}) = kC_{G}(P)_{P}^{N_{G}(P)} = kC_{H}(P)_{P}^{N_{H}(P)} = Br_{P} ((OH)_{P}^{H}) ,

where the first and last equality comes from 4.1 while the second equality follows from the hypothesis N_{G}(P) ⊂ H. Again, the primitive idempotents on the right side correspond exactly to the blocks of OH having P as defect group.

Theorem 4.3. (Green correspondence) Let G be a finite group, P a p−subgroup of G and H a subgroup of G containing N_{G}(P).

(i) If U is an indecomposable OG−module with vertex P, then Res_{H}^{G}(U) has up to isomorphism, a unique indecomposable direct summand FU with vertex P.

(ii) If V is an indecomposable OH−module with vertex P, then Ind_{H}^{G}(V) has, up to isomorphism, a unique indecomposable direct summand GV with vertex P such that V is isomorphic to a direct summand of Res_{H}^{G}(GV).

Moreover, with this notation, we have GFU ≅ U and FGV ≅ V .

Proof. (i) We apply 4.1 to the interior G−algebra A = End_{O}(U). Since {Id_{U}} is the unique point of G on A and P a defect group of this point (Higman’s criterion 2.6) there is a unique point β of H on A such that Br_{P}^{A}(β) = Br_{P}^{A}({Id_{U}}) = {1_{A(P)}}, because Br_{P}^{A} maps A_{P}^{G} and A_{P}^{H} onto A(P)_{P}^{N_{G}(P)} = A(P)_{P}^{N_{H}(P)} , where the last equality holds by the assumption N_{G}(P) ⊂ H. Take j ∈ β and set FU = j(U). Since the indecomposable direct summands of Res_{H}^{G}(U) with vertex P correspond by Higman’s criterion 2.6 exactly to the points of H on A having P as defect group, (i) follows.

(ii) We apply 4.1 this time to the interior G−algebra A = End_{O}(Ind_{H}^{G}(V)). Since V has vertex P, Higman’s criterion 2.6 implies that A^{G} = A_{H}^{G} = A_{P}^{G}. Note that V is naturally isomorphic to a direct summand of Res_{H}^{G}(Ind_{H}^{G}(V)), thus determines a unique point β of H on A with defect group P. Again, since by 4.1 the Brauer homomorphism Br_{P}^{A} maps A_{P}^{G} and A_{P}^{H} to A(P)_{P}^{N_{G}(P)} = A(P)_{P}^{N_{H}(P)} there is a unique point α of G on A having P as defect group such that Br_{P}^{A}(α) = Br_{P}^{A}(β). Take i ∈ α and set GV = i(Ind_{H}^{G}(V)). The theorem follows.

Note that the statements in I, section 9 are slightly more precise, since they give information on the remaining direct summands of Res_{H}^{G}(U) and Ind_{H}^{G}(V) with vertex not conjugate to P. We leave it to the reader to check, that this could also be done in the present context of correspondences between sets of points. Observe that the Brauer correspondence is a particular case of the Green correspondence (see I, 9.3).

5. Stable categories

Stable equivalences are one of the main tools for trying to understand the relation between the structure of a block algebra OGb and its local structure. In general, stable equivalences remain a mysterious phenomeneon; the stable equivalences occurring in block theory are at least always induced by bimodules, and in this context it is possible to be a little bit more precise.

The stable category \overline{C} of an additive category C is the additive category having the same objects as C and having as morphisms the equivalence classes of morphisms f : U ⟶ V with respect to the equivalence relation declaring two homomorphisms f, f′ : U ⟶ V to be equivalent, if their difference f − f′ factors through a projective object in C; composition of morphisms in \overline{C} is induced by composition in C. In other words, \overline{C} is obtained from C by “identifying projective objects in C to zero”.

Let A be an O−algebra. Remind from I, 5.2 that an A−module U is relatively O−projective if U is isomorphic to a direct summand of A ⊗_{O} V for some O−module V . If A is O−free of finite rank over O one can show that this is equivalent to requiring that the natural map A ⊗_{O} U ⟶ U mapping A ⊗ u to au, where a ∈ A, u ∈ U is split surjective.

In the case O = k the relatively k−projective A−modules are precisely the projective A−modules, because any k−module V is then free (a k−vectorspace), whence A ⊗_{k} V is a free A−module, and so any direct summand of A ⊗_{k} V is projective.

Definition 5.1. Let A be an O−algebra. The O−stable category \overline{Mod}(A) of Mod(A) is the O−linear category whose objects are the objects of Mod(A) and whose morphism space, for any two A−modules U, V , is the quotient space

\overline{Hom}_{A}(U, V) = Hom_{A}(U, V)/Hom_{A}^{1}(U, V)

where Hom_{A}^{1}(U, V) is the subspace of Hom_{A}(U, V) of all A−homomorphisms from U to V which factor through a relatively O−projective A−module, and where the composition of morphisms is induced by the composition of A−homomorphisms.

That is, \overline{Mod}(A) has to be understood as the “quotient” of Mod(A) obtained by “identifying all relatively O−projective A−modules to zero”. If O = k then \overline{Mod}(A) is just the usual stable category of Mod(A) by the previous remark. We need criteria, when a homomorphism factors through a projective module. Remind that for any O−algebra A and any A−module U (which is finitely generated by our conventions) we have J(A)U = rad(U), the intersection of all maximal submodules of U. Furthermore, remind that a k−algebra A is self-injective, if A is injective as left A−module. One easily shows that then A is also injective as right A−module. Moreover, a k−algebra A is self-injective if and only if the classes of projective and injective A−module conincide. Indeed, if A is self-injective, every projective A−module is isomorphic to a direct summand of a free A−module, and every free A−module is injective since A is injective; moreover, the number of isomorphism classes of projective indecomposable and of injective indecomposable A−modules coincide (they are both equal to the number of isomorphism classes of simple A−modules), thus every injective A−module is also projective. Conversely, if every projective A−module is injective, in particular A is injective.

Lemma 5.2. Let A be an O-algebra, let U, V be A-modules and let φ ∈ Hom_{A}(U, V). The following are equivalent:

(i) φ factors through a projective A−module.

(ii) φ factors through any surjective A−homomorphism W ⟶ V .

(iii) φ factors through a minimal projective cover P_{V} ⟶^{π_{V}} V of V .

If moreover A is a self-injective k−algebra, then any of the above statements is equivalent to any of the following statements:

(iv) φ factors through an injective A−module.

(v) φ factors through any injective A−homomorphism U ⟶ W.

(vi) φ factors through a minimal injective envelope U ⟶^{ι_{U}} I_{U} of U.

Proof. Suppose that (i) holds. Then there is a projective A−module P and homomorphisms α : U ⟶ P, β : P ⟶ V , such that φ = β ∘ α. Since P is projective, β lifts through any surjective homomorphism W ⟶ V , thus (ii) holds. The implications (ii) ⇒ (iii) ⇒ (i) are trivial.

If A is a self-injective k−algebra, the classes of projective and of injective A−modules coincide. Thus (i) is equivalent to (iv), and the equivalence with the remaining statements (v), (vi) is then shown by dualizing the above proof.

Proposition 5.3. Let A be an O−algebra, U, V non zero A−modules such that U has no non zero projective direct summand and let φ ∈ Hom_{A}(U, V). If φ factors through a projective A−module then Im(φ) ⊂ rad(V); in particular, φ is not surjective. If V is simple we have \overline{Hom}_{A}(U, V) ≅ Hom_{A}(U, V).

Proof. If Im(φ) is not contained in rad(V) there is a maximal submodule M in V such that Im(φ) is not contained in M. Thus the composition U ⟶^{φ} V ⟶ V/M is non zero. If φ factors through a projective A−module, this composition factors through a minimal projective cover P_{V/M} ⟶^{π} V/M. As M is maximal in V , the module V/M is simple and whence P_{V/M} is projective indecomposable; in particular, rad(P_{V/M}) is the unique maximal submodule of P_{V/M}. Since U has no non zero projective direct summand, every homomorphism from U to P_{V/M} has therefore an image contained in rad(P_{V/M}), and whence its composition with π is zero. Therefore no non zero homomorphism from U to V/M factors through P_{V/M}. This contradiction shows the first statement, and the rest follows immediately.

For a self-injective k−algebra we can dualize 5.3, and obtain:

Proposition 5.4. Let A be a self-injective k−algebra, let non zero U, V be A−modules having no non zero projective direct summands and let φ ∈ Hom_{A}(U, V). If φ factors through projective A−module then Im(φ) ⊂ rad(V) and soc(U) ⊂ ker(φ); in particular, φ is neither surjective nor injective. If one of U, V is simple, we have \overline{Hom}_{A}(U, V) ≅ Hom_{A}(U, V).

Proof. If φ factors through a projective A−module we have Im(φ) ⊂ rad(V) by 5.3. But then the k−dual φ^{∗} : V^{∗} ⟶ U^{∗} factors through a projective (right) module, too, since the k−dual of a projective module is an injective right module, and thus projective as A is self-injective. Thus, again by 5.3, we have Im(φ^{∗}) ⊂ rad(U^{∗}). Applying k−duality again shows that this means soc(U) ⊂ ker(φ). Alternatively, if soc(U) is not contained in ker(φ), there is a simple submodule S of U such that the composition S ⟶ U ⟶^{φ} V is non zero and factors through a minimal injective envelope S ⟶ I_{S} of S by 5.2 But then V has a direct summand isomorphic to I_{S}, contradicting the hypothesis on V .

Let A be an O−algebra. The Heller operator Ω_{A} on Mod(A) sends any A−module U to the A−module Ω_{A}(U) = ker(π_{U}), where π_{U} : P_{U} ⟶ U is a minimal projective cover of U. By the uniqueness of minimal projective covers, this module is up to isomorphism independent of the choice of π_{U}.

If A is a k−algebra, we define analogously the operator Ω_{A}^{−1} by setting Ω_{A}^{−1}(U) = coker(ι_{U}), where ι_{U} : U ⟶ I_{U} is a minimal injective envelope of the A−module U. This notation is not a coincidence:

Theorem 5.5. Let A be a self-injective k−algebra. The operators Ω_{A} and Ω_{A}^{−1} extend to mutually inverse equivalences on the stable category \overline{Mod}(A).

Proof. For simplicity we write Ω instead of Ω_{A}. Let U, V be A−modules. Consider a diagram of A−modules of the following form:

0 ⟶ ΩU ⟶ P_{U} ⟶^{π_{U}} U ⟶ 0
γ↓  ^{β}↓  ↓^{α}
0 ⟶ ΩV ⟶ P_{V} ⟶_{π_{V}} V ⟶ 0

Given α, there is β making the right square commutative, as π_{V} is surjective and P_{U} is projective. Then β maps ker(π_{U}) to ker(π_{V}) and whence induces a homomorphism γ which makes the diagram commutative.

The other way round, given the homomorphism γ, there is a homomorphism β making the left square commutative, as ι_{U} is injective and the module P_{V} is injective, too (this is where we use that A is self-injective). Again, β induces a homomorphism α making the diagram commutative. However, in general α and γ do not determine each othere uniquely. But we show that they determine uniquely their classes in \overline{Hom}_{A}(U, V) and \overline{Hom}_{A}(ΩU, ΩV). For this we have to show that α factors through a projective module if and only if γ does so.

Suppose that α factors through a projective A−module. Then α factors through π_{V} by 5.2. Let φ : U ⟶ P_{V} be a homomorphism such that α = π_{V} ∘ φ. Then π_{V} ∘ (β − φ ∘ π_{U}) = α ∘ π_{U} − α ∘ π_{U} = 0, which means that β − φ ∘ π_{U} maps P_{U} to ker(π_{V}) = ΩV ; in other words, there is ψ : P_{U} ⟶ ΩV such that ψ = β − φ ∘ π_{U}.

Then γ = β|_{ΩU} = (β − φ ∘ π_{U})|_{ΩU} = ψ|_{ΩU}, so γ factors through P_{U}. A dual argument shows that conversely, if γ factors through a projective A−module, so does α.

Together this means that the correspondence α ↦ γ induce a k−isomorphism

\overline{Hom}_{A}(U, V) ≅ \overline{Hom}_{A}(ΩU, ΩV) .

This shows that Ω induces a self equivalence of \overline{Mod}(A). Since P_{U} is also injective, the inclusion ΩU ⟶ P_{U} is an injective envelope of ΩU. This implies that Ω and Ω^{−1} induce mutually inverse equivalences on \overline{Mod}(A).

In block theory all stable equivalences that usually occur are in fact induced by tensoring with a suitable bimodule. This led M. Broué to the following definition:

Definition 5.6. (Broué [6]) Let A, B be O−algebras, M an A − B−bimodule, N a B −A−bimodule. We say that M, N induce a stable equivalence of Morita type between A, B, if M, N are projective as left modules and as right modules, and if

M ⊗_{B} N ≅ A ⊕ X for some projective A − A − bimodule X ,
N ⊗_{A} M ≅ B ⊕ Y for some projective B − B − bimodule Y .

The bimodule X is projective if it is isomorphic to a direct summand of a direct sum of (finitely many) copies of A ⊗_{O} A. Thus if U is an A module, we have (A ⊗_{O} A) ⊗_{A} U ≅ A ⊗_{O} U, which implies that X ⊗_{A} U is a relatively O−projective A−module. In other

words, the functor X ⊗_{A} − induces on \overline{Mod}(A) the zero functor. Therefore, if M, N induce a stable equivalence of Morita type, the functors M ⊗_{B} − and N ⊗_{A} − induce

mutually inverse O−linear equivalences between the O−stable categories \overline{Mod}(A) and \overline{Mod}(B). Note that if X and Y are zero in the above definition, the bimodules M, N induce a Morita equivalence between A and B, which explains the terminology.

We show now that if A is self-injective, the stable equivalence given by the Heller operator on Mod(A) (cf. 5.5) is in fact of Morita type if A:

Proposition 5.7. Let A be a self-injective k−algebra. Then A ⊗_{k} A^{0} is self-injective, and the A − A−bimodules Ω_{A⊗A^{0}}(A) and Ω_{A⊗A^{0}}^{−1}(A) induce a stable equivalence of Morita type on A. Moreover, the functors on \overline{Mod}(A) induced by Ω_{A⊗A^{0}}(A) ⊗_{A} − and Ω_{A} are naturally isomorphic.

Proof. Multiplication in A induces a surjective homomorphism of A − A−bimodules A ⊗_{k} A ⟶^{μ} A sending a ⊗ a′ to aa′, where a, a′ ∈ A. This is a projective cover of A as A − A−bimodule (or equivalently, as A ⊗_{k} A^{0}−module). Thus, by taking a suitable

direct summand P_{A} of A ⊗_{k} A we get a minimal projective cover P_{A} ⟶^{π} A of A, where π is the restriction of μ to P_{A}. Thus we have an exact sequence of A − A−bimodules

0 ⟶ Ω_{A⊗A^{0}}(A) ⟶ P_{A} ⟶^{π} A ⟶ 0 .

If U is an A−module then A⊗_{k}A⊗_{A}U ≅ A⊗_{k}U is a projective A−module, thus P_{A} ⊗_{A}U is

a projective A−module; therefore the map P_{A} ⊗_{A}U ⟶^{π⊗Id_{U}} A⊗_{A}U ≅ U is a projective cover of U (but not necessarily a minimal one). This means that ker(π ⊗ Id_{U}) ≅ Ω_{A}(U) ⊕ W for some projective A−module W. Tensoring the above exact sequence by − ⊗_{A} U yields an exact sequence

0 ⟶ Ω_{A⊗A^{0}}(A) ⊗_{A} U ⟶ P_{A} ⊗_{A} U ⟶^{π⊗Id_{U}} A ⊗_{A} U ⟶ 0

and therefore Ω_{A⊗A^{0}}(A) ⊗_{A} U ≅ Ω_{A}(U) ⊕ W. This shows that the functors on \overline{Mod}(A) induced by Ω_{A⊗A^{0}}(A) ⊗_{A} − and by Ω_{A} are isomorphic. Moreover, applying the above argument to the A − A−bimodule Ω_{A⊗A^{0}}^{−1}(A) instead of U shows that Ω_{A⊗A^{0}}(A) ⊗_{A} Ω_{A⊗A^{0}}^{−1}(A) ≅ A ⊕ X for some projective A − A−bimodule X. The proposition follows.

We quote (without proof) the following important relationship between derived and stable equivalences for symmetric algebras (see also Chapter II, section 6):

Proposition 5.8. (Rickard [35]) Let A, B be symmetric O−algebras. If A, B are derived equivalent, there is a stable equivalence of Morita type between A and B.

The converse of 5.8 is not true in general. The next theorem is a list of basic properties of stable equivalences of Morita type.

Theorem 5.9. ([24, 2.1], [25, 3.4]) Let A, B be O−algebras such that k ⊗_{O} A, k ⊗_{O} B are self-injective indecomposable non simple k−algebras whose semi-simple quotients are separable, M a B − A−bimodule and N an A − B−bimodule which induce a stable equivalence of Morita type between A and B.

(i) Up to isomorphism, M and N have unique indecomposable non-projective direct summands as bimodules.

(ii) If M is indecomposable as B − A−bimodule, for any simple A−module S the B−module M ⊗_{A} S is indecomposable and non projective as k ⊗_{O} B−module.

(iii)If for any simple A−module S the B−module M ⊗_{A} S is simple then the functor M ⊗_{A} − is a Morita equivalence Mod(A) ≅ Mod(B).

Proof. (i) (Rouquier [37]) Let M′, M′′ be subbimodules of M such that M = M′ ⊕M′′. Then (N ⊗_{B} M′) ⊕ (N ⊗_{B} M′′) = N ⊗_{B} M ≅ A ⊕ X for some projective A − A−bimodule X. Since A is indecomposable as A − A−bimodule we may assume that A is a direct summand of N ⊗_{B} M′. Then N ⊗_{B} M′′ is projective as A − A−bimodule. Tensoring by M ⊗_{A} − implies that M′′ is projective as B − A−bimodule.

(ii) We may assume O = k. Note that soc(B⊗_{O}A^{0}) = soc(B)⊗soc(A^{0}), as A, B have separable semi-simple quotients. If M is indecomposable, M has no projective direct summand and so soc(B)Msoc(A) = 0. Thus M ⊗_{A} soc(A) ≅ Msoc(A) is annihilated by soc(B), and therefore has no non zero projective direct summand as left B−module, as B is self-injective. Since A is self-injective, any simple A−module occurs as direct summand of soc(A) and it follows that M⊗_{A}S has no non zero projective direct summand, thus is indecomposable non projective, as the functor M ⊗_{A}− induces a stable equivalence.

(iii) By (i) we may assume that N is indecomposable. If M ⊗_{A} S is simple, where S is a simple A−module, then by (ii) the A−module N ⊗_{B} M ⊗_{A} S is indecomposable, thus isomorphic to S, as M and N induce mutually inverse stable equivalences. Since N ⊗_{B}M ≅ A⊕X for some projective A−A−bimodule X it follows that (A⊗_{A}S)⊕(X⊗_{A}S) ≅ S, thus X = 0. Therefore M, N induce mutually inverse equivalences between Mod(A) and Mod(B).

Using 5.9 it is not difficult to prove the following generalization of 5.7:

Proposition 5.10. Let A, B be O−algebras such that k⊗_{O}A, k⊗_{O}B are indecomposable non simple self-injective k−algebras, M a B −A−bimodule and N an A−B−bimodule such that M, N induce a stable equivalence of Morita type between A and B. Suppose that M, N are indecomposable. Then for any integer n, the B − A−bimodule Ω_{B⊗A^{0}}^{n}(M) is, up to isomorphism, the unique indecomposable non projective direct summand of both Ω_{B⊗B^{0}}^{n}(B)⊗_{B}M and M ⊗_{A}Ω_{A⊗A^{0}}^{n}(A); moreover, the bimodules Ω_{B⊗A^{0}}^{n}(M), Ω_{A⊗B^{0}}^{−n}(N) again induce a stable equivalence of Morita type between A and B.

One of the most prominent and still open conjectures on stable equivalences is

5.11 Auslander’s conjecture: Let A, B be two (finite-dimensional) algebras over a field k such that \overline{Mod}(A) and \overline{Mod(B)} are equivalent. Then A, B have the same number of isomorphism classes of non projective simple modules.

6. The trivial intersection case

The “trivial intersection case” is the typical situation where we have a stable equivalence of Morita type, and this is a good illustration for the interplay between block theory as developped in the sections 1 through 4 above and the material of section 5.

Theorem 6.1. Let G be a finite group, b a block of OG having a non trivial defect group P, let H be a subgroup of G satisfying P ∩^{x}P = 1 for any x ∈ G − H and let c be the block of OH corresponding to b through the Brauer correspondence.

The OGb − OHc−bimodule bOGc and the OHc − OGb−bimodule cOGb induce a stable equivalence of Morita type between OGb and OHc.

Note that in the situation of 6.1 the subgroup H of G contains the normalizer of every non trivial subgroup of P; in particular, H contains N_{G}(P), and so the Brauer correspondence 4.2 applies. The standard example for this situation is the case where P is cyclic non trivial and where H is the normalizer in G of the unique subgroup of order p of P; indeed, if x ∈ G such that P ∩^{x}P ≠ 1 then P ∩^{x}P contains the unique subgroup of order p of P and therefore x ∈ H. Another standard example is the case where p = 2 and P is genarized quaternionian, since then P has a unique subgroup Z of order 2, and whence H = N_{G}(Z) = C_{G}(Z) fulfills the trivial intersection hypothesis in 6.1.

For the proof of 6.1 we need the following technical result (cf. [25],6.2):

Lemma 6.2. With the notation of 6.1, for any non trivial subgroup Q of P we have Br_{Q}(b) = Br_{Q}(c); in particular, c − bc ∈ (OG)_{1}^{H}.

Proof. Let Q be a non trivial subgroup of P and a ∈ (OG)^{P} . We have T r_{P}^{G}(a) = ∑_{x}T r_{P^{x}∩Q}^{Q}(a^{x}) , where x runs over a set of representatives of the double cosets P \G/Q

in G. As P^{x} ∩Q = 1 if x ∈ G−H, applying Br_{Q} yields Br_{Q}(T r_{Q}^{G}(a)) = Br_{Q}(T r_{P}^{H}(a)), thus Br_{Q}((OG)_{P}^{G}) = Br_{Q}((OG)_{P}^{H}) = Br_{Q}((OH)_{P}^{H}), where the last equality holds since C_{G}(P) ⊂ H. Consequently, Br_{Q}(b) and Br_{Q}(c) are primitive idempotents in the ideal Br_{Q}((OG)_{P}^{G}) of the commutative subalgebra Br_{Q}((OG)^{G}) of kC_{G}(Q) and therefore are either equal or orthogonal. Since Br_{P} (Br_{Q}(b)) = Br_{P} (b) = Br_{P} (c) = Br_{P} (Br_{Q}(c)) they cannot be orthogonal. As c − bc ∈ (OG)_{P}^{H} and c − bc ∈ ker(Br_{Q}) for any non trivial subgroup Q of P, the last statement follows easily.

Proof of 6.1. Clearly bOGc and cOGb are projective as left and right modules because they are direct summands of OG.

We show first, that the OHc−OHc−bimodule cOGb ⊗_{OGb}bOGc ≅ cOGbc has OHc as unique non projective indecomposable direct summand, up to isomorphism. Since OH is a direct summand of OG as OH − OH−bimodule, clearly OHc is a direct summand of cOGc as OHc − OHc−bimodule.

If x ∈ G − H we have P ∩ P^{x} = 1; thus it follows easily that the P × P −permutation module O[HxH] is projective as P × P −module and therefore cO[HxH]c is projective as OHc − OHc−bimodule, since it is relatively P × P −projective and projective as P ×P −module. Consequently, cOGc has OHc as unique non projective indecomposable direct summand, up to isomorphism. Now cOGc = cOGbc ⊕ cOG(c − bc), and we only have to observe that OHc is not a direct summand of cOG(c−bc). This however follows easily, using Higman’s criterion, as c − bc ∈ (OG)_{1}^{H} by 6.2, and therefore cOG(c − cb) is projective as OHc − OHc−bimodule.

It remains to show that the OGb − OGb−bimodule bOGc ⊗_{OHc}cOGb has OGb as unique indecomposable non projective direct summand, up to isomorphism. Since Br_{P} (b) = Br_{P} (c), in a primitive decomposition of cb in (OG)^{P} occurs an idempotent belonging to a local point of P on OGb. This implies, by [30],3.5, that multiplication by c induces a Morita equivalence between OGb and cOGbc. It suffices thus to show that cOGbc ⊗_{OHc}cOGbc has cOGbc as unique non projective indecomposable direct summand, up to isomorphism, as cOGbc − cOGbc−bimodule. Now multiplication induces a surjective bimodule homomorphism cOGbc ⊗_{OHc}cOGbc ⟶ cOGbc, which has a section as OHc − cOGbc−bimodule homomorphism, namely the map sending a ∈ cOGbc to bc ⊗ a. Thus it has a section as a cOGbc − cOGbc−bimodule homomorphism, since any cOGbc − cOGbc−bimodule is relatively OHc − cOGbc−projective; in particular, cOGbc is a direct summand of cOGbc ⊗_{OHc}cOGbc. There is no other non projective direct summand, since even as OHc − OHc−bimodule, OHc is, up to isomorphism, the unique non projective direct summand of cOGbc, thus of cOGbc ⊗_{OHc}cOGbc, too.

7. Perspectives

The fundamental problem of block theory can be phrased as follows: given any finite p−group P, dress a list of all interior P −algebras which occur, up to isomorphism, as source algebra of a block of a finite group having P as a defect group.

Puig has conjectured that for a given finite p−group P, there are only finitely many isomorphism classes of interior P −algebras which occur as source algebras of some block of a finiite group G with defect group P.

Puig’s conjecture would imply Donavan’s conjecture, claiming that given a finite p−group P, there are only finitely many Morita equivalence classes of block algebras of blocks over k with P as defect group.

Puig’s conjecture is proved in full generality only in one case, namely if P is cyclic [25]. It is also proved for certain classes of finite groups, such as symmetric groups by Puig [34], generalizing prior work on Donovan’s conjecture by Scopes [38], and double covers of symmetric and alternating groups (Kessar [20]).

Donovan’s conjecture is proved of course in all of the above cases and for 2−groups of tame representation type by Erdmann [13] except for certain blocks of quaternion type.

Another case where the source algebras are determined “up to a power of the Heller operator” is the case p = 2 and P a Klein four group [23]. However, in this case the results are just not precise enough to prove Puig’s conjecture. At least, Donovan’s conjecture holds both over k (Erdmann [12]) and over O as a consequence of the structure of the source algebras [23].

We describe now (informally) the classifications of the source algebras in [25], [23], as far as they are complete in the two cases mentioned above, that is, if P is cyclic or a Klein four group. We keep the notation for p, O, k. In the cyclic case this is based on prior work of many authors such as [4], [9], [16], [18], for instance; in the Klein four group case the main reference is Erdmann [12].

7.1. Let G be a finite group and b a block of OG having a non trivial cyclic defect group. Let Z be the unique subgroup of order p of P.

Set H = N_{G}(Z) and C = C_{G}(Z). Denote by c the block of OH corresponding to b through the Brauer correspondence; this makes sense as H contains N_{G}(P). Let e be a block of OC such that Br_{P} (be) ≠ 0, or equivalently, such that (Z, e) is a b−Brauer pair. Note that then (Z, e) is also a c−Brauer pair since C ⊂ H.

Our first step is a description of the source algebras of OCe.

7.1.1. As a block of OC, e has P as defect group, and the source algebras of OCe are isomorphic to

S ⊗_{O} OP ,

where S = End_{O}(N) for some indecomposable OP −module N with vertex P such that N^{∗} ⊗_{O} N is a permutation OP −module.

In particular, the block algebra OCe is Morita equivalent to OP.

The interior P −algebra structure of S ⊗_{O} OP is given by the map sending u ∈ P to σ(u)⊗u, where σ(u) is the O−linear automorphism of N given by left multiplication by u on N. An OP −module N with the property that N^{∗} ⊗_{O}N is a permutation module is called an endo-permutation OP −module [10]. For abelian p−groups, endo-permutation modules are classified in Dade [11]. Note that S ≅ N^{∗}⊗_{O}N as OP −module, where P acts on S “by conjugation” and “diagonally” on N^{∗} ⊗_{O} N. Thus N is an endo-permutation module if and only if S has a P −stable O−basis.

The next step is the description of the structure of OHc.

7.1.2. The source algebras of OHc are isomorphic to

S ⊗_{O} O(P ⋊ E) ,

where E is the p′−subgroup of Aut(P) isomorphic to H/C. In particular, OHc is Morita equivalent to O(P ⋊ E).

The proof of 7.1.2 requires p′−extensions of nilpotent blocks, for instance studied in [28]. The structure of k(P ⋊ E) is very well understood: the algebra k(P ⋊ E) is symmetric and serial; that is, every indecomposable k(P ⋊ E)− module U has a unique composition series. From this one can easily describe all indecomposable modules (there are only finitely many isomorphism classes) and their homomorphism spaces.

7.1.3. There is a stable equivalence of Morita type between the block algebras OGb and OHc given by the bimodules cOGb and bOGc.

This is a particular case of 6.1. Self-injective algebras over a field stably equivalent to a self-injective serial algebra have been classified in [16]; it turns out, that the Morita equivalence class of any such algebra can be read off a tree, called the Brauer tree, and any such algebra is therefore called a Brauer tree algebra (see Chapter II, section 5 or [1]) . We can use this to obtain finally the structure of the source algebras of OGb:

7.1.4. The source algebras of OGb are determined by the isomorphism class of the endopermutation module N and the Brauer tree of OGb (see [25] for a precise statement).

At this point the classification of possible source algebras is almost complete - but we still do not know which trees do actually occur as Brauer tree of a block. Using the classification of finite simple groups, W. Feit showed that most trees do not occur as Brauer tree of a block. In any case, since the Brauer tree of a block with cyclic defect P has at most p − 1 edges, the number of isomorphism classes of Brauer trees for a given p is bounded, and since there are only finitely many isomorphism classes of indecomposable endo-permutation OP −modules with vertex P, it follows that Puig’s conjecture holds.

Another consequence of the precise knowledge on the structure of blocks with cyclic defect groups is that we can show that Broué’s abelian defect conjecture holds:

7.1.5. The algebras OGb and O(P ⋊ E) are derived equivalent.

It is possible to make 7.1.4 more precise, using splendid derived equivalences, a notion of derived equivalences which takes into account the local structure of the considered blocks (see [36], [26], [27]).

The second case where we have an “almost complete” description of source algebras is the Klein four defect case.

7.2. Let G be a finite group and b a block of OG having a Klein four defect group P; that is, p = 2 and P ≅ ℤ/2ℤ × ℤ/2ℤ. Denote by c the block of ON_{G}(P) with P as defect group corresponding to b through the Brauer correspondence.

Note that P is isomorphic to a Sylow-2−subgroup of A_{4} and of A_{5} . Thus P can be identified to a defect group of the unique block 1 of OA_{4} and of the principal block b_{0} of OA_{5} . Moreover, the block algebras OA_{4} and OA_{5}b_{0} , when viewed as interior P −algebras through any injective group homomorphisms from P to A_{4} and A_{5} , are isomorphic to their source algebras, because it is not hard to show that 1_{A_{4}} and b_{0} are already primitive in (OA_{4})^{P} and (OA_{5}b_{0})^{P} , respectively.

7.2.1. (Erdmann[12]) The block algebra kG\overline{b} is Morita equivalent to one of kP, kA_{4} , kA_{5}\overline{b_{0}} , where \overline{b} (resp. \overline{b_{0}} ) is the obvious image in kG (resp. kA_{5} ) of the block b of OG (resp. principal block b_{0} of OA_{5} ).

Using 7.2.1 it is possible to be more precise about source algebras:

7.2.2. ([23, 1.1]) The source algebras of OGb are isomorphic to one of the interior P −algebras

S ⊗_{O} OP , S ⊗_{O} OA_{4} , iS ⊗_{O} OA_{5}i ,

where S = End_{O}(Ω_{P}^{n} (O)) for some integer n and where i is a primitive idempotent in (S ⊗_{O} OA_{5}b_{0})^{P} not contained in ker(Br_{P}).

The modules Ω_{P}^{n} (O) are particular cases of endo-permutation modules; they are in fact endo-trivial modules; that is, Ω_{P}^{n} (O)⊗Ω_{P}^{n} (O)^{∗} is isomorphic to the direct sum of the trivial OP −module O and a projective OP −module. The problem in the statement of 7.2.2 is that we do not know, which integers n actually occur from source algebras. We do not even know, whether only finitely many integers occur - in other words, Puig’s conjecture is still open in this case.

For the interior P −algebra structure of the algebras occurring in 7.2.2 it is not relevant which embedding of P into A_{4} and A_{5} we choose, because any group automorphism of a Sylow-2−subgroup of A_{4} and A_{5} extends to a group automorphism of A_{4} and A_{5} , respectively. We note two consequences of 7.2.2: first, since OGb is Morita equivalent to any of its source algebras (cf. [30, 3.5]), Erdmann’s classification 7.2.1 “lifts” over O:

7.2.3. The block algebra OGb is Morita equivalent to one of the algebras OP, OA_{4} , OA_{5}b_{0} .

Second, by a result of J. Rickard, the algebras OA_{4} and OA_{5}b_{0} are splendidly derived equivalent; we deduce from this again Broué’s conjecture (in its strongest version):

7.2.4. The block algebras OGb and ON_{G}(P)c are splendidly derived equivalent.

7.3. Instead of fixing the defect group P one might try to put hypotheses on the local structure of the given block b of OG. This strategy has been successfully used for blocks with “trivial” local structure; that is, for blocks where the defect group P “controls G−fusion” or equivalently, where all G−fusions between Brauer pairs are induced by elements in P. Such a block is called nilpotent. The definition goes back to Broué-Puig [7] where is also determined the character theory of nilpotent blocks; their source algebras are determined by Puig in [32]. In fact, the knowledge of the structure of nilpotent blocks is used in both 7.1 and 7.2 above. General extensions of nilpotent blocks are treated in [21]. Puig determined also the source algebras for all blocks of p−solvable groups. As a consequence of this and some work of Rickard, Broué’s conjecture holds again in its strong form for blocks of p−solvable groups with abelian defect groups [19].

7.4. For further reading we recommend the books [1], [3], [14], [39] that we quote. Alperin’s book [1] is a module theoretic approach to block theory in general and blocks with cyclic defect groups in particular; Benson’s volumes [3] emphasize aspects including cohomology and algebraic topology. Thévenaz’ book [39] comes closest to our presentation of block theory using Puig’s concepts of pointed groups and source algebras, and contains detailed proofs of much of the material on block theory we present here.

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Markus Linckelmann, CNRS, Université Paris 7, UFR Mathématiques, 2, place Jussieu, 75251 Paris Cedex 05, FRANCE

E-mail address: linckelm@math.jussieu.fr