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Brauer-friendly modules and slash functors

This paper introduces the notion of Brauer-friendly modules, a generalisation of endo-p-permutation modules. A module over a block algebra OGe is said to be Brauer-friendly if it is a direct sum of indecomposable modules with compatible fusion-stable endopermutation sources. We obtain, for these modules, a functorial version of Dade's slash construction, also known as deflation-restriction. We prove that our slash functors, defined over Brauer-friendly categories, share most of the very useful properties that are satisfied by the Brauer functor over the category of p-permutation OGe-modules. In particular, we give a parametrisation of indecomposable Brauer-friendly modules, which opens the way to a complete classification whenever the fusion-stable sources are classified. Those tools have been used to prove the existence of a stable equivalence between non-principal blocks in the context of a minimal counter-example to the odd Z*p-theorem.


Introduction
The p-permutation modules, or trivial-source modules, are an essential tool of the local representation theory of finite groups. What makes them so useful is the existence of a localising tool, the Brauer functor, which satisfies nice properties such as transitivity, and provides a simple classification of indecomposable p-permutation modules, as proven by Broué in [4,Theorem 3.2 (3)]. In particular, the properties of the Brauer functor have enabled Rickard, in [13], to prove that a derived equivalence defined by a so-called splendid complex of bimodules induces a collection of local derived equivalences.
A possible generalisation of p-permutation modules is the notion of endo-p-permutation modules, as described by Urfer in [18]. Let O be a discrete valuation ring with positive residual characteristic p. An indecomposable OG-module M is an endo-p-permutation module if its source V with respect to a given vertex P is an endopermutation OP -module that is fusionstable with respect to the fusion system F G (P ) defined by the group G on the p-subgroup P . This characterisation suggests that endo-p-permutation modules are not the most natural notion when one wants to take blocks into account. Indeed, a non-principal block e of the group G usually defines a fusion system on a p-subgroup P that is finer than the fusion system F G (P ) defined by the group G. This is why we introduce a more general notion. We say that an OGe-module M is Brauer-friendly if it is a direct sum of indecomposable modules with endopermutation sources that are fusion-stable with respect to the block e, as defined in [8], and moreover compatible with one another.
Although they do not seem to have been explicitly described before, these modules are ubiquitous in modular representation theory. Indeed, let M be an indecomposable bimodule that induces a Morita or stable equivalence between two block algebras. If M admits a diagonal vertex, it follows from [11, §7] that a source of M must be a fusion-stable endopermutation module, i.e., M must be a Brauer-friendly module. Puig usually studies Brauer-friendly modules by turning them into endopermutation modules over source algebras. However, it seems reasonable to study Brauer-friendly modules directly at the level of block algebras, a more common framework in modular representation theory. Such is the aim of the present paper.
In Sections 1 and 2, we review the theory of subpairs that has been defined in [1]. Then we refine the definition of the Brauer functor, as well as Green's theory of vertices and sources (following [15]) to take subpairs into consideration. In Section 3, we define the notion of a Brauer-friendly module. We prove that a module M over a block algebra OGe with defect group D is Brauer-friendly if, and only if, the corresponding module over a source algebra A of OGe is an endopermutation OD-module.
In Sections 4 and 5, we prove that Dade's localising tool, known as the slash construction or deflation-restriction, can be applied to Brauer-friendly modules. We also show that this construction can be turned into a functor, provided that one focuses on what we call a Brauerfriendly category, i.e., a category of compatible Brauer-friendly modules. This functoriality may have important consequences in local representation theory. For instance, by allowing one to localise a complex of compatible Brauer-friendly modules, it would make it easier to deal with the local properties of a derived equivalence between block algebras, even though this equivalence is not defined by a splendid complex.
In Section 6, we use the properties of slash functors to give a practical parametrisation of indecomposable Brauer-friendly modules. As mentioned above, a Morita or stable equivalence between block algebras is often defined by such a module, so our parametrisation can be an important tool when it comes to finding a bimodule that induces such an equivalence.
Let us give a more precise example of application. The search for a modular proof of the odd Z * p -theorem is an important open question of the representation theory of finite groups. In the context of a (putative) minimal counter-example to that theorem, thanks to the above tools, we have been able to consider a family of local Morita equivalences and glue them together to obtain a stable equivalence between non-principal block algebras, as appears in [3,Chapter 4].

Subpairs and the Brauer functor
We first give a few notations that we use throughout this article. We let O be a complete discrete valuation ring with algebraically closed residue field k of characteristic p. This includes the case O = k, so that every result that is proven over the ring O remains true over the field k.
For any finite group G, we denote by ∆G = {(g, g); g ∈ G} the diagonal subgroup of the direct product G × G. We denote by OG Mod the category of OG-modules and morphisms of OG-modules, and by OG Perm the full subcategory of p-permutation OG-modules, i.e., direct summands of OG-modules that are O-free and admit a G-stable basis. For an element g ∈ G and an object X, the notation g X stands for the object gXg −1 whenever this makes sense. For any element x of the group algebra OG, we denote byx its image by the natural projection map OG kG. For any two groups G and H, an (OG, OH)-bimodule M will be considered as an O(G × H)-module by setting We recall the definition of subpairs from [1]. Let G be a finite group, and e be a block of the algebra OG. A subpair of the group G is a pair (P, e P ), where P is a p-subgroup of G and e P is a block of the group algebra OC G (P ). The subpair (P, e P ) is an e-subpair ifē P br P (e) = 0, where br P : (OG) P → kC G (P ) denotes the Brauer morphism. The idempotent e P is a block of the algebra OH whenever H is a local subgroup of G with respect to the subpair (P, e P ), i.e., a subgroup such that C G (P ) H N G (P, e P ).
The group G acts by conjugation on the poset of e-subpairs of G. This action can be described by the Brauer category Br(G, e), defined as follows. An object is an e-subpair (P, e P ). An arrow φ : (P, e P ) → (Q, e Q ) is a group morphism φ : P → Q such that there exists an element g ∈ G that satisfies g (P, e P ) (Q, e Q ) and φ(x) = g x for any x ∈ P . The composition of arrows is the usual composition of group morphisms. The Brauer category Br(G, e 0 ) of the principal block e 0 actually describes the action of the group G on the poset of its p-subgroups, since an e 0 -subpair (P, e P ) is completely determined by the p-subgroup P . This is called the Frobenius category of the group G, and denoted by Fr(G).
Let (D, e D ) be a maximal e-subpair, i.e., an e-subpair such that the p-subgroup D of G is a defect group of the block e. Let F (G,e) (D, e D ) be the full subcategory of Br(G, e) of which the objects are the e-subpairs contained in (D, e D ). This subcategory is called the fusion system of the block e with respect to the subpair (D, e D ); it is equivalent to the Brauer category Br(G, e). A source idempotent of the block e with respect to the maximal subpair (D, e D ) is a primitive idempotent i of the algebra (OGe) D such thatē D br D (i) = 0. The D-interior algebra A = iOGi is called a source algebra of the block e. The (A, OGe)-bimodule iOG induces a Morita equivalence A ∼ OGe. Moreover, the fusion system F (G,e) (D, e D ) can be read in the O(D × D)-module A. More details on this approach may be found in [7].
We now recall and slightly generalise the classical definition of the Brauer functor. Let G be a finite group and P be a p-subgroup of G. We writeN G (P ) = N G (P )/P . For any OG-module M , we denote by Br P (M ) the Brauer quotient of M , which some authors denote by M (P ), i.e., the kN G (P )-module where M P is the submodule of P -fixed points in M , Tr P Q : M Q → M P is the relative trace map and m is the maximal ideal of the local ring O. We denote by br M P : M P → Br P (M ) the projection map. Any morphism of OP -modules u : L → M induces a k-linear map Br P (u) : Br P (L) → Br P (M ). If u is a morphism of OG-modules, then Br P (u) is a morphism of kN G (P )modules. This defines a functor Notice that we write the Brauer functor Br P with a capital B, and the Brauer map br P with a lowercase b. This tool can be adapted to take subpairs into consideration. Let e be a block of the algebra OG, (P, e P ) be an e-subpair of the group G, and M be an OGe-module. We define the Brauer quotient of M with respect to the subpair (P, e P ) by setting Br (P,e P ) (M ) = Br P (e P M ).
This generalised Brauer quotient has a natural structure of kN G (P, e P )ē P -module, where we writeN G (P, e P ) = N G (P, e P )/P . Notice that we slightly abuse notations by identifying the blockē P ∈ kN G (P, e P ) to a block of the quotient algebra kN G (P, e P ).
Let L and M be two OGe-modules, and let u : L → M be a morphism of OP -modules. Then the map u induces a morphism of OP -modules e P ue P : e P L → e P M . We set Br (P,e P ) (u) = Br P (e P ue P ), a k-linear map from Br (P,e P ) (L) to Br (P,e P ) (M ). If u : L → M is a morphism of OGe-modules, then Br (P,e P ) (u) is a morphism of kN G (P, e P )ē P -modules. This defines a functor Br (P,e P ) : OGe Mod → kN G (P,e P )ē P Mod.

Vertex subpairs and the Green correspondence
In this section, we review the notion of a vertex subpair of an indecomposable module, which has been defined in [15]. Let G be a finite group, and e be a block of the algebra OG. The following lemma extends Green's theory of vertices and sources, as well as the Green correspondence, to take e-subpairs into consideration.
Lemma 1. Let M be an indecomposable OGe-module and (P, e P ) be an e-subpair of the group G. The following conditions are equivalent.
(i) The p-subgroup P is contained in a vertex of M , and M is isomorphic to a direct summand of the OGe-module eOGe P ⊗ OP V for some indecomposable OP -module V .
(ii) The OG-module M is relatively P -projective, and the OP -module e P M admits an indecomposable direct summand V with vertex P .
(iii) The ON G (P, e P )-module e P M admits an indecomposable direct summand L with vertex P such that M is isomorphic to a direct summand of the induced module Ind G N G (P,e P ) L (iv) The p-group P is a vertex of M , and the Green correspondent of M with respect to this vertex is an ON G (P )-module M that belongs to the block e P = Tr N G (P,e P ) e P of the algebra ON G (P ).
If these conditions are satisfied, then the ON G (P, e P )-module L of (iii) and the ON G (P )-module M of (iv) satisfy the relations M Ind N G (P,e P ) L and L e P M .
Proof. With the notations of Condition (iv), we know from [6, Theorem 1.6] that the restriction/induction functors e P Res N G (P,e P ) and Ind N G (P,e P ) define a Morita equivalence between the block algebras ON G (P, e P )e P and ON G (P )e P . The equivalence (iii)⇔(iv), as well as the last statement of the Lemma, immediately follow from this Morita equivalence and the classical definition (and uniqueness) of the Green correspondent of M .
We now suppose that (iii) is satisfied. Then the restriction Res N G (P,e P ) P L admits a direct summand V with vertex P , so the OP -module e P M also admits V as a direct summand. Moreover, the induced module Ind G N G (P,e P ) L is relatively P -projective, so the OG-module M is relatively P -projective. Since V is also a direct summand of M and admits P as a vertex, it follows that the p-group P is a vertex of M . Moreover, the OP -module V is a source of L. Thus L is isomorphic to a direct summand of the ON G (P, e P )e P -module ON G (P, e P )e P ⊗ OP V , and M to a direct summand of the OGe-module We have proven that the condition in (iii) implies (i) and (ii). Next, we suppose that (i) is satisfied. The OG-module M is relatively P -projective and the p-group P is contained in a vertex of M , so P itself is a vertex of M . Moreover, M is isomorphic to a direct summand of the OGe-module The ON G (P )-module M is therefore a Green correspondent of M with respect to the vertex P , so that (i) implies (iv).
Finally, we suppose that (ii) is satisfied. Then the restriction Res G P M admits an indecomposable direct summand with vertex P , and the p-group P contains a vertex of M , so P itself is a vertex of M . Let M be a Green correspondent of M with respect to this vertex, and let f be the block of the algebra ON G (P ) such that f M = M . Then the uniqueness of the Green correspondent implies that no indecomposable direct summand of the OP -module (1 − f )M admits P as a vertex. Since e P M = f e P M ⊕ (1 − f )e P M , it follows that f e P = 0, so that f = Tr N G (P,e P ) e P . Thus (ii) implies (iv), and the four conditions are equivalent. We derive from the above lemma a few definitions and easy consequences that extend some of Green's classical definitions.
Let M be an indecomposable OGe-module. It follows from Nagao's theorem (see, e.g., [2, Theorem 6.3.1]) that there exists an e-subpair (P, e P ) of the group G that satisfies the statement in Lemma 1 (iv), hence the statements in (i), (ii) and (iii). Such an e-subpair is called a vertex subpair of the indecomposable module M (this is consistent with [15, Definition 2.6]). A source of M with respect to the vertex subpair (P, e P ) is an OP -module V that satisfies any one of the equivalent conditions (i) and (ii). A source triple of M is a triple (P, e P , V ), where V is a source of M with respect to the vertex subpair (P, e P ). The source triples of M form an orbit under the action of the group G by conjugation.
A Green correspondent of M with respect to the vertex subpair (P, e P ) is an ON G (P, e P )module L that satisfies the condition in (iii). The mapping M → L induces a one-to-one correspondence between the isomorphism classes of indecomposable OGe-modules with vertex subpair (P, e P ), and the isomorphism classes of indecomposable ON G (P, e P )e P -modules with vertex P .
The following properties of vertex subpairs and sources are closer to the approach of [15]. (i) There exists a primitive idempotent i of the algebra (OG) P such thatē P br P (i) = 0 and that M is isomorphic to a direct summand of the OGe-module OGi ⊗ OP V .
(ii) There exists a defect group D of the block e such that P D, there exists a primitive idempotent j of the algebra (OG) D such that br D (j) = 0 andē P br P (j) = 0, and that M is isomorphic to a direct summand of the OGe-module OGj ⊗ OD Ind D P V .
Proof. By assumption, M is a direct summand of the OGe-module eOGe P ⊗ OP V . Consider a decomposition ee P = i 1 + . . . + i n of the idempotent ee P into mutually orthogonal primitive idempotents in the algebra (OG) P . This brings Since the OGe-module M is indecomposable, by the Krull-Schmidt theorem, it must be isomorphic to a direct summand of the OGe-module OGi l ⊗ OP V for some l ∈ {1, . . . , n}. If e P br P (i l ) = 0, then the idempotent i l lies in Tr P Q ((OG) Q ) for some proper subgroup Q of P , so M is relatively Q-projective and cannot admit P as a vertex. This contradiction proves (i).
Let α be the point of the algebra (OGe) P that contains the idempotent i = i l . Since br P (i) = 0, the pointed group P α is local. Let D β be a defect pointed group of the block algebra OGe such that P α D β (see [16, §18]). Then the p-group D is a defect group of the block e and there exists an idempotent j ∈ β such that br D (j) = 0 and ij = ji = i. Then e P br P (j) = 0, and the OGe-module OGi ⊗ OP V is a direct summand of The following result deals with the behaviour of vertex subpairs and sources with respect to restriction. It is similar to [15,Corollary 2.7], with an additionnal statement about sources.
is an e-subpair of the group G, and there exists a source triple (R, e R , W ) of an indecomposable direct summand of the OGe-module M such that (Q, e Q ) (R, e R ) and that V is a direct summand of the OQ-module Res R Q W . Proof. We may assume that the OGe-module M is indecomposable. Let (P, e P , U ) be a source triple of M . By Lemma 2, there is a primitive idempotent i ∈ (OGe) P such thatē P br P (i) = 0 and X is (isomorphic to) an indecomposable direct summand of the OH-module OGi ⊗ OP U .
is also a block of the algebra OQC G (Q). The OQC G (Q)-module e Q X admits an indecomposable direct summand X with vertex Q and source V . Up to replacing X by X , we may now assume that H = QC G (Q), and that X is an indecomposable direct summand of the OH-module L = e Q OGi ⊗ OP U .
We consider the OH- We then consider the decomposition L 0 = g∈R L g , where R is a set of representatives for the double class set H \G/P , and L g = OHgP ⊗ OP U for any g ∈ R. This can be refined into a Krull-Schmidt decomposition L 0 = Y 0 ⊕ · · · ⊕ Y n of the OH-module L 0 . From the proof of the Krull-Schmidt theorem in [2], we know that there exists an l ∈ {1, . . . , n} such that the projection map from Y l onto X 0 along X 1 ⊕ · · · ⊕ X n is an isomorphism. We may assume l = 0. Then f (Y 0 ) is a complement of the direct summand X 1 ⊕ · · · ⊕ X m in the OH-module L, so the OH-modules f (Y 0 ) and X 0 are isomorphic. We may assume X 0 = f (Y 0 ).
By construction, there exists a g ∈ R such that the OH-module Y 0 is a direct summand of L g . The isomorphism L g Ind H H∩ g P Res The e-subpair (Q, e Q ) is normalised by the group H, so we have Q hg P but (Q, e Q ) hg (P, e P ). By definition, i is a primitive idempotent of (OGe) P such thatē P br P (i) = 0. Thus hg i is a primitive idempotent in (OGe) Q , and it follows from [16,Lemma 40 This contradiction proves that (Q, e Q ) g (P, e P ). Then the OH-module X = X 0 is isomorphic to a direct summand of the induced module L g Ind H H∩ g P Res g P H∩ g P gU. As a consequence, the source V of X is isomorphic to a direct summand of the OQ-module Res hg P Q hgU for some h ∈ H. Then we set (R, e R , W ) = hg (P, e P , U ), and the proof is complete.
The application of Theorem 3 that we have in mind is as follows. Let M be an indecomposable OGe-module, and (P, e P ) be an e-subpair of the group G. Let H be a subgroup of G such that P C G (P ) H N G (P, e P ). Let (Q, e Q , V ) be a source triple of an indecomposable direct summand of the OH-module e P M . If the vertex Q contains the p-group P , then we have C G (Q) C G (P ) H. Thus the theorem applies: there exists a source triple (R, e R , W ) of M such that (Q, e Q ) (R, e R ) and that V is a direct summand of the OQ-module Res R Q W . Notice that this statement is unlikely to be true without the assumption P Q.

Brauer-friendly modules
Let P be a p-group. An endopermutation OP -module, as defined in [5], is an OP -module V such that the endomorphism algebra End O (V ) is O-free and admits a ∆P -stable basis. Two endopermutation OP -modules V and W are said to be compatible if the direct sum V ⊕ W is an endopermutation OP -module.
In this section, we fix a finite group G and a block e of the algebra OG. We aim at studying certain OGe-modules with endopermutation sources. The following definition is derived from [8]; notice however that we work over the local ring O rather than its residue field k, and that we replace the language of fusion systems with that of Brauer categories. Definition 4. Let (P, e P ) be an e-subpair of the group G, and let V be an endopermutation OP -module. We say that V is fusion-stable in the group G with respect to the subpair (P, e P ) if the endopermutation OQ-modules Res φ1 V and Res φ2 V are compatible for any e-subpair (Q, e Q ) and any two arrows φ 1 , φ 2 : (Q, e Q ) → (P, e P ) in the Brauer category Br(G, e).
In practice, let (P, e P ) is an e-subpair and V an endopermutation OP -module. If we wish to prove that V is fusion-stable with respect to the subpair (P, e P ), it is enough to check that the endopermutation OQ-modules Res P Q V and Res g P Q gV are compatible, for any e-subpair (Q, e Q ) contained in (P, e P ) and any element g ∈ G such that (Q, e Q ) g (P, e P ).
To make it shorter, we say that a triple (P, e P , V ) is a fusion-stable endopermutation source triple in the group G if V is an endopermutation OP -module that is indecomposable, capped (i.e., with vertex P ), and fusion-stable with respect to the subpair (P, e P ) of the group G.
(i) If (P 1 , e 1 , V 1 ) is a fusion-stable endopermutation source triple, then any G-conjugate of (P 1 , e 1 , V 1 ) is a fusion-stable endopermutation source triple. If moreover (P 2 , e 2 , V 2 ) is a fusion-stable endopermutation source triple that is compatible with (P 1 , e 1 , V 1 ), then it is also compatible with any G-conjugate of (P 1 , e 1 , V 1 ).
are compatible fusion-stable endopermutation source triples.
(iii) Let (R, e R ) be an e-subpair and ψ 1 : (P 1 , e 1 ) → (R, e R ), ψ 2 : (P 2 , e 2 ) → (R, e R ) be arrows in the Brauer category Br(G, e). Then (P 1 , e 1 , V 1 ) and (P 2 , e 2 , V 2 ) are compatible fusionstable endopermutation source triples if, and only if, the direct sum Ind ψ1 V 1 ⊕ Ind ψ2 V 2 is an endopermutation OR-module that is fusion-stable in the group G with respect to the subpair (R, e R ).
Proof. The statements in (i) and (ii) are straightforward consequences of the definition, as well as the "if" part of (iii). The "only if" part of (iii) is more subtle.
With the notations of (iii), we suppose that (P 1 , e 1 , V 1 ) and (P 2 , e 2 , V 2 ) are compatible fusion-stable endopermutation source triples. Up to replacing these triples by conjugates, we may suppose that the e-subpairs (P 1 , e 1 ) and (P 2 , e 2 ) are contained in (R, e R ), and that the arrows ψ 1 and ψ 2 are the inclusion maps. We follow the lines of the proof of [5, Lemma 6.8]. The Mackey formula gives an isomorphism of O∆R-modules Then the inclusions (P 2 , e 2 ) (R, e R ) and Q x P 2 imply (Q x , e Qx ) (P 2 , e 2 ). Since x lies in R, we also have x −1 (Q x , e Qx ) (R, e R ) so the inclusions (P 1 , e 1 ) (R, e R ) and x −1 Q x P 2 imply x −1 (Q x , e Qx ) (P 2 , e 2 ). As a consequence, φ x 1 : (Q x , e Qx ) → (P 1 , e 1 ) and φ x 2 : (Q x , e Qx ) → (P 2 , e 2 ) are arrows in the Brauer category Br(G, e). Since the triples (P 1 , e 1 , V 1 ) and (P 2 , e 2 , V 2 ) are compatible, we deduce that are permutation O∆R-module. Thus Ind R P1 V 1 and Ind R P2 V 2 are compatible endopermutation OR-modules. More generally, let (Q, e Q ) be an e-subpair that is contained in (R, e R ) and let g ∈ G be an element such that (Q, e Q ) g (R, e R ). A similar use of the Mackey formula proves that is a permutation O∆Q-module. It follows that the restrictions is an endopermutation OR-module that is fusion-stable with respect to the subpair (R, e R ).
We know from Lemma 6 (i) that the notion of compatible fusion-stable endopermutation source triples is invariant by conjugation in the group G. The source triples of an indecomposable OGe-module are defined up to conjugation, so the following definition is unambiguous. Next, let M be an indecomposable endo-p-permutation OGe-module, and (P, e P , V ) be a source triple of M . By [18,Theorem 1.5], the source V is an endopermutation OP -module that is fusion-stable in the Frobenius category Fr(G) with respect to the p-subgroup P . A fortiori, V is fusion-stable in the Brauer-category Br(G, e) with respect to the e-subpair (P, e P ), so M is a Brauer-friendly OGe-module. If M and N are compatible endo-p-permutation OGe-modules (i.e., the direct sum M ⊕ N is an endo-p-permutation module), then they are compatible as Brauer-friendly modules.
Conversely, let M be a Brauer-friendly OGe-module. If e = e 0 is the principal block of the group G, then an e-subpair (P, e P ) is uniquely determined by the p-subgroup P of G. Thus it follows from [18, Theorem 1.5] that M is an endo-p-permutation OGe-module. But the fusion system of an arbitrary block is usually "finer" than the fusion system of the principal block, so M may not be an endo-p-permutation module in general.
The following lemma, of which a proof is straightforward from Lemma 6 and the proof of Theorem 3, gives a more general example of Brauer-friendly module. Lemma 8. Let (P, e P ) be an e-subpair, and let V be an endopermutation OP -module that is fusion-stable with respect to the subpair (P, e P ). Let i ∈ (OGe) P be an idempotent such that e P br P (i) = 0. Assume that, for any subgroup Q of P , the idempotent br Q (i) lies in a single block of the algebra kC G (Q). Then the OGe-module L = OGi ⊗ OP V is Brauer-friendly.
Notice that the compatibility of endopermutation modules is preserved by the reduction from the local ring O to the residue field k. For any Brauer-friendly OGe-module M , it follows that the reduction k ⊗ O M is a Brauer-friendly kGe-module. The notion is also partially compatible with the restriction to a local subgroup, as appears in the following lemma.
Lemma 9. Let M be a Brauer-friendly OGe-module. Let (P, e P ) be an e-subpair of the group G, and H be a subgroup of G such that P C G (P ) H N G (P, e P ).
(i) The OHe P -module e P M admits the decomposition e P M = L ⊕ L , where L is a Brauerfriendly OHe P -module and L is a direct sum of indecomposable OHe P -modules with vertices that do not contain the p-subgroup P .
(ii) The restriction Res H P L is an endopermutation OP -module. Proof. The OH-module e P M certainly admits the decomposition e P M = L ⊕ L , where L is a direct sum of indecomposable OH-modules with vertices that contain P and L is a direct sum of indecomposable OH-modules with vertices that do not contain P . Let X, X be indecomposable direct summands of the OHe P -module L, and let (Q, f, W ), ( We conclude this section with a lemma that connects our notion of Brauer-friendly modules with Linckelmann's notion of modules with fusion-stable endopermutation sources over a source algebra, which appears in [7, §3]. It should be mentioned here that it was Linckelmann who initially gave us the idea to study such modules. Lemma 10. Let G be a finite group and e be a block of the group G. Let (D, e D ) be a maximal e-subpair and i be a primitive idempotent of the algebra (OGe) D such thatē D br D (i) = 0, i.e., a source idempotent. Let A = iOGi be the corresponding source algebra. An OGe-module M is Brauer-friendly if, and only if, the A-module iM is an endopermutation OD-module.
Proof. Let M be an OGe-module, and let M = M 1 ⊕· · ·⊕M n be a Krull-Schmidt decomposition. For each integer l ∈ {1, . . . , n}, let (P l , e l , V l ) be a source triple of the indecomposable OGemodule M l . By Lemma 2 (ii), there exists a defect group D l and a source idempotent j l of the block e with respect to the defect group D l such that P l D l andē l br P l (j l ) = 0, and such that M l is isomorphic to a direct summand of the OGe-module OGj l ⊗ OD l Ind D l P l V l . Up to replacing the defect group D l and the idempotent j l by conjugates, we may assume that D l = D and j l = i. We set W = Ind D P1 V 1 ⊕ · · · ⊕ Ind D Pn V n . Then M is isomorphic to a direct summand of the OG-module L = OGi ⊗ OD W , and iM is isomorphic to a direct summand of the A-module iL = A ⊗ OD W .
Suppose that the OGe-module M is Brauer-friendly. Then, by Lemma 6 (iii), W is an endopermutation OD-module that is fusion-stable with respect to the subpair (D, e D ). By [7, Proposition 3.2 (i)], this implies that the A-module iM is an endopermutation OD-module.
Conversely, suppose that iM is an endopermutation OD-module. Let F be the fusion system of the block e with respect to the maximal subpair (D, e D ). We know from [16, §47] or [7, §2] that the fusion system F can be read in the (OD, OD)-bimodule structure of the source A = iOGi. Since iM is an A-module, it follows that the endopermutation OD-module iM is fusion-stable with respect to the subpair (D, e D ). For any l ∈ {1, . . . , n}, the source V l is isomorphic to a direct summand of the restriction Res D P l iM . Thus, by Lemma 6 (ii), the triples (P l , e l , V l ), 1 l n, are compatible fusion-stable endopermutation source triples. So M is a Brauer-friendly OGe-module.

Slash functors
In this section, we extend Dade's slash construction, which has been defined in [5,Theorem 4.15], to take subpairs and Brauer-friendly modules into account. The slash construction is often called deflation-restriction, e.g. in [7] or [17]. In our context, this would become deflationtruncation-restriction. Let us briefly recall Dade's original construction.
Let R be a p-group and P be a subgroup of R. Let V be an endopermutation OR-module. The Brauer quotient Br ∆P (End O (V )) has a natural structure of N R (P )/P -algebra over k. Moreover, there exists an endopermutation kN R (P )/P -module V [P ] and an isomorphism of End k (V [P ]). The kN R (P )/P -module V [P ], which is unique up to (non-unique) isomorphism, is called a P -slashed module relative to the OR-module V . In particular, if P = R and V is a capped indecomposable endopermutation OP -module, then the group N P (P )/P is trivial, and a slashed module V [P ] is just the k-vector space k.
If V is a permutation OR-module, then V is a fortiori an endopermutation OR-module, and the Brauer quotient Br P (V ) is a P -slashed module relative to the OR-module V . However, in general, the slash construction appears to be functorial in End O (V ), but not in V . In a first step, we prove that this construction can actually be turned into a functor in V , provided that it is restricted to a suitable category of compatible endopermutation modules. We It follows that Sl P is a P -slash functor.
We now consider another P -slash functor Sl : OP M → k Mod. The OP -module X is an object of the category OP M, and there is an idempotent i ∈ End OP (X) such that V = iX. We set j = Sl(i) ∈ End k (Sl(X)), and W = j Sl(X). The map Sl X : End O (X) → End k (Sl(X)) induces an isomorphism Br ∆P (End O (X))→ End k (Sl(X)) that sends the idempotent br P (i) to j, hence an isomorphism Br ∆P (End O (V ))→ End k (W ). It follows that W is a 1-dimensional k-vector space. The choice of an isomorphism ζ : k→ W brings a natural isomorphism So we obtain a natural isomorphism ξ = φ • ψ : Sl P→ Sl. Moreover the correspondence ζ ↔ ξ is one-to-one, so ξ is unique up to scalar multiplication.
The second step is to extend the notion of a slash functor to take subpairs into account. In such a situation, what we call a slash functor is actually a little more than a functor. Definition 12. Let G be a finite group, e be a block of the group G, and OGe M be a subcategory of the category OGe Mod of OGe-modules. Let (P, e P ) be an e-subpair of the group G, and H be a subgroup of G such that P C G (P ) H N G (P, e P ). WriteH = H/P . A (P, e P )-slash functor Sl : OGe M → kHē P Mod is defined by the following data: • We can easily prove the existence and uniqueness of slash functors over such categories in the case H = P C G (P ). We writeC G (P ) = P C G (P )/P .

Lemma 14.
Let G be a finite group, e be a block of the group G, and OGe M be a Brauer-friendly category of OGe-modules. Let (P, e P ) be an e-subpair of the group G.
(ii) If Sl : OGe M → kC G (P )ē P Mod is another (P, e P )-slash functor, then there exists an isomorphism of slash functors Sl (P,e P )→ Sl, which is unique up to scalar multiplication.
Proof. Let OP M be the full subcategory of OP Mod over the essential image of the truncationrestriction functor e P Res G P . By Lemma 9, the category OP M satisfies the assumptions of Lemma 11. Thus there exists a P -slash functor Sl P : OP M → k Mod.
For any object M in OGe M, we have a morphism of algebras Sl e P M P : End OP (e P M ) → End k (Sl P (e P M )), and a natural group morphism ι : C G (P ) → End OP (e P M ) × . The composition Sl e P M P : C G (P ) → End k (Sl P (e P M )) × makes the endomorphism ring End k (Sl P (e P M )) a C G (P )-interior algebra. As a consequence, it makes the slashed module Sl P (e P M ) a kC G (P )ē Pmodule, which we denote by Sl (P,e P ) (M ).
For any two objects L, M in OGe M and any morphism of OP -module u : L → M , we denote by e P ue P : e P L → e P M the morphism of OP -modules induced by u. We consider Sl P (e P ue P ) as a k-linear map Sl (P,e P ) (L) → Sl (P,e P ) (M ), and we denote it by Sl (P,e P ) (u). If u is a morphism of OGe-modules, then it follows from the functoriality of Sl P and from the definition of theC G (P )-interior structures on End k (Sl (P,e P ) (L)) and End k (Sl (P,e P ) (M )) that Sl (P,e P ) (u) is a morphism of kC G (P )-modules. This defines a (P, e P )-slash functor Sl (P,e P ) : Thus ξ : Sl (P,e P ) → Sl is what we would like to call an isomorphism of (P, e P )-slash functors. The uniqueness of ξ up to scalar multiplication follows from the similar uniqueness of ξ .
The third and last step is to consider a (P, e P )-slash functor with a codomain such as the category kHē P Mod, where H is any subgroup of G with P C G (P ) H N G (P, e P ). The existence of such a slash functor follows from a deep result proven by Puig in [9]. Unfortunately, we lose the uniqueness of a slash functor up to isomorphism. We need to explain how a (P, e P )slash functor Sl : OGe M → kHē P Mod may be twisted by a linear character χ :H/C G (P ) → k × .
If M is an object of the category OGe M, then Sl(M ) is a kHē P -module. We set χ * Sl(M ) = Sl(M ) as a k-vector space, and we endow χ * Sl(M ) with the action · χ of the groupH defined by h · χ m = h · χ(h)m for any h ∈H and m ∈ Sl(M ), where the single dot stands for the preexisting action of the groupH on the k-vector space Sl(M ). If L, M are two objects of OGe M and u : L → M is a morphism of OP -modules, then we set χ * Sl L,M (u) = Sl L,M (u), considered as a k-linear map χ * Sl(L) → χ * Sl(M ). This defines another (P, e P )-slash functor χ * Sl : OGe M → kHē P Mod. Notice that the slash functors Sl and χ * Sl might be isomorphic.
Theorem 15. Let G be a finite group, e be a block of the group G, and OGe M be a Brauerfriendly category of OGe-modules. Let (P, e P ) be an e-subpair of the group G, and H be a subgroup of G such that P C G (P ) H N G (P, e P ).  Proof. We know from Lemma 14 (i) that there exists a (P, e P )-slash functor Sl (P,e P ) : OGe M → kC G (P )ē P Mod. We may assume that this slash functor is nonzero, i.e., that there exists an object X of OGe M such that Sl (P,e P ) (X) = 0.
Let L and M be two objects of the category OGe M. The O-module Hom O (e P L, e P M ) has a natural structure of O(H × H)-module, so the Brauer quotient Br ∆P (Hom O (e P L, e P M )) admits a natural structure of k(C G (P ) × C G (P ))∆N G (P, e P )-module. If L = M , the Brauer quotient Br ∆P (End O (e P M )) End k (Sl (P,e P ) (M )) is more precisely a C G (P )-interior N G (P, e P )algebra, as defined in [12].
The main result of [9] implies that End k (Sl (P,e P ) (M )) may be extended (non-uniquely) to an H-interior algebra. This defines an extension of the slashed module Sl (P,e P ) (M ) to a kHmodule, which is unique up to twisting by a linear character χ :H/C G (P ) → k × . Let us choose, once and for all, such an extension for the slashed module Sl (P,e P ) (X).
Then, for any object M of OGe M, the slashed module Sl (P,e P ) (M ) may admit several extensions to a kH-module. Each one of them defines a structure of k(H ×H)-module on Hom k (Sl (P,e P ) (X), Sl (P,e P ) (M )). By definition, the slash functor Sl (P,e P ) induces an isomorphism of k(C G (P ) × C G (P ))-modules Br ∆P (Hom O (e P X, e P M ))− → Hom k (Sl (P,e P ) (X), Sl (P,e P ) (M )).
Since the slashed module Sl (P,e P ) (X) is nonzero, there is only one extension of Sl (P,e P ) (M ) to a kH-module such that the above map is a morphism of k(C G (P ) × C G (P ))∆H-module. We denote this kHe P -module by Sl (P,e P ) (M ), in italics. This defines a (P, e P )-slash functor Let Sl : OGe M −→ kHē P Mod be another (P, e P )-slash functor. Then, by Lemma 14 (ii), the restriction ResH C G (P ) Sl is isomorphic to the slash functor Sl (P,e P ) . By transport of structure, we may suppose that ResH C G (P ) Sl = Sl (P,e P ) , i.e., the only difference between Sl (P,e P ) and Sl is the choice of an extension of the slashed module Sl (P,e P ) (X) to a kH-module. Up to twisting Sl (P,e P ) by a linear character, we can obtain Sl (P,e P ) (X) = Sl(X), which implies Sl (P,e P ) = Sl. This completes the proof of the theorem.
In general, there are several (P, e P )-slash functors on a given Brauer-friendly category, none of which can be singled out. However, in the case of the category OGe Perm of p-permutation modules, there is a canonical choice : the Brauer functor Br (P,e P ) .

The essential image of a slash functor
In this section, we prove that the essential image of a slash functor is, as expected, a Brauerfriendly category. This enables us to compose slash functors, and to prove that such a composition results in another slash functor. These results are prepared by two technical lemmas, which will also be used in the next section. Lemma 16. Let H be a finite group and P be a normal p-subgroup of H. Let R be a p-subgroup of H that contains P , and V be an OR-module. Suppose that the restriction Res H P Ind H R V is an endopermutation OP -module. The C H (P )-interior H-algebra Br ∆P (Ind H R End O (V )) admits a (non-unique) extension to an H-interior algebra. Once this extension has been chosen, there is a natural isomorphism of H-interior algebras where we write Ind H R for the induction of interior algebras (see [16, §16]). Proof. Let us write S = End O (V ), an R-interior matrix algebra over O that can also be seen as a O(R × R)-module. We may assume that Br ∆P (S) = 0, since the conclusion of the lemma is trivial otherwise. On the one hand, the H-interior algebra A = Ind H R S is defined, as an The map φ : S → A, s → 1 H ⊗ s ⊗ 1 H is an embedding of algebras. It induces an isomorphism of R-algebras S→ αAα, where α = φ(1) is an idempotent of the algebra A R . By [9], the C H (P )-interior H-algebra A P = Br ∆P (A) admits an extension to a H-interior algebra, which makes A P a k(H × H)-module. Consider the idempotent α P = br P (α) ∈ (A P ) R , and the R-interior subalgebra α P A P α P . The decomposition 1 A P = g∈H/R g α P of the unity into mutually orthogonal idempotents brings a decomposition of the k(R × R)-module A P : On the other hand, the Brauer quotient S P = Br ∆P (S) is a matrix algebra over k, with a natural R-interior structure that also makes it a k(R × R)-module. The H-interior algebra The embedding ψ : S P → B, s → 1 H ⊗s⊗1 H induces an isomorphism of R-algebras S P→ βBβ, where β = ψ(1) ∈ B R . The decomposition 1 B = g∈H/R g β brings a decomposition of the k(R × R)-module B: The Brauer functor Br ∆P sends the map φ : S → A to a morphism of algebras φ P : S P → A P , which restricts to an isomorphism of R-algebras φ P : S P→ α P A P α P . By uniqueness of the R-interior structure on S P , φ P is an isomorphism of R-interior algebras. Thus φ P : S P → A P is a morphism of k(R × R)-modules. By the universal property of induced modules, there exists a unique morphism of k(H × H)-modules Φ : B → A P such that Φ • ψ = φ P .
By construction, we have Φ(β) = α P and Φ induces an isomorphism of R-interior algebras βBβ→ α P A P α P . Since Φ is a morphism of k(H × H)-modules, it follows from the above decompositions of B and A P that Φ is an isomorphism of k(R×R)-modules. Then the definition of induced interior algebras implies that Φ is an isomorphism of H-interior algebras.
Lemma 17. With the assumptions of Lemma 16, let i be an idempotent of the algebra (OH) R . Write i P = br P (i). The isomorphism Φ induces an isomorphism of H-interior algebras Proof. With the notations of the proof of Lemma 16, we consider the idempotent u = Tr H R (iαi) of the algebra A H , and the idempotent u P = br P (u) = Tr H R (i P α P i P ) of the algebra (A P ) H . A direct computation in the induced algebra A yields uAu = OHi ⊗ OR S ⊗ OR iOH, so that u P A P u P Br ∆P (OHi ⊗ OR S ⊗ OR iOH).
Then we consider the idempotent v = Tr H R (i P ⊗ 1 Br ∆P (S) ⊗ i P ) = Tr H R (i P βi P ) of the algebra B H . The same computation brings vBv = kHi P ⊗ kR Br ∆P (S) ⊗ kR i P kH.
Since the morphism Φ sends β to α P and commutes with the relative trace map, we obtain Φ(v) = u P . So Φ induces an isomorphism of H-interior algebras Φ i : vBv → u P A P u P .
We can now prove the main result of this section.
Lemma 18. Let G be a finite group, e be a block of the algebra OG, and OGe M be a Brauerfriendly category of OGe-modules. Let (P, e P ) be an e-subpair of the group G, H be a subgroup of G such that P C G (P ) H N G (P, e P ), and Sl (P,e P ) : OGe M → kHē P Mod be a (P, e P )-slash functor. Then there exists a Brauer-friendly category kHē P M of kHē Pmodules that contains the essential image of Sl (P,e P ) .
Proof. We let S be the set of source triples (Q, e Q , V ) of the group G for which there exists a source triple (R, e R , W ) of an indecomposable direct summand of an object of OGe M such that (P, e P ) (Q, e Q ) (R, e R ) and V is a direct summand of the OQ-module Res R Q W . We letS be the set of triples (Q,ē Q ,V ) for which there exists an element (Q, e Q , V ) of the set S such that Q = Q/P ,ē Q = br P (e Q ), and the kQ-moduleV is a capped indecomposable direct summand of a P -slashed module V [P ]. We denote by OHe P M (resp. kHē P M) the full subcategory of OHe P Mod (resp. kHē P Mod) of which the objects are the direct sums of indecomposable modules with source triples in S (resp.S). These are Brauer-friendly categories.
We know from Lemma 15 that there exists a (P, e P )-slash functor Sl (P,e P ) : OHe P M → kHē P Mod. Let M be an object of the category OGe M. By Lemma 9, the OHe P -module e P M admits the decomposition e P M = L ⊕ L , where L is a Brauer-friendly OHe P -module and L is a direct sum of indecomposable OHe P -modules with vertices that do not contain the p-subgroup P . It follows from Lemma 15 (ii) that the slashed modules Sl (P,e P ) (M ) and Sl (P,e P ) (L) are isomorphic, up to twisting by a linear character of the groupH/C G (P ). Notice that such a twist preserves the source triples of an indecomposable kHē P -module. Let (Q,ē Q ,V ) be a source triple of an indecomposable direct summand of Sl (P,e P ) (M ), hence of Sl (P,e P ) (L). To study this triple, we may first suppose that the OHe P -module L is indecomposable. Then we may suppose, as in the proof of Theorem 3, that L = OHi ⊗ OR W , with (R, e R , W ) a source triple of the group H that lies in the set S, and i a primitive idempotent of the algebra (OH) R such thatē R br R (i) = 0. With this assumption, we know from Lemma 17 that the slashed module Sl (P,e P ) (L) is isomorphic to the kHē P -module L P = kHbr P (i)⊗ kQ/P W [P ]. Then we deduce from Theorem 3 that the subpair (R,ē R ) is contained in (Q/P, br P (e Q )), and that W is isomorphic to a direct summand of the kR-module Res Lemma 18 enables us to study the transitivity of slash functors.
Lemma 19. Let G be a finite group, e be a block of the group G, and OGe M be a Brauer-friendly category of OGe-modules.
Let X be an indecomposable OGe-module with source triple (P, e P , V ). Let X be its Green correspondent, an object of the category OHe P M. Up to twisting the slash functor Sl (P,e P ) by a linear character of the groupH/C G (P ), we may assume that the slashed modules Sl (P,e P ) (X) and Sl (P,e P ) (X ) are isomorphic. Once this assumption has been made for X, the same is necessarily true for any other indecomposable OGe-module with source triple (P, e P , V ).
By definition of a source triple, X is isomorphic to a direct summand of the OHe P -module L = OHe P ⊗ OP V . A P -slashed module relative to the capped indecomposable endopermutation OP -module V is the k-vector space k. Thus, by Lemma 17, there is an isomorphism of kHē P -modules Sl (P,e P ) (L) kHē P ⊗ kP k kHē P .
Since L is an endo-p-permutation OH-module, the Brauer map br By a classical result about the lifting of idempotents such as [16, Theorem 3.1], the epimorphism β induces a one-to-one correspondence between the conjugacy classes of primitive idempotents of the algebra End OH (L) and the conjugacy classes of primitive idempotents of End kH (Sl (P,e P ) (L)). In other words, the mapping X → Sl (P,e P ) (X ) induces a one-to-one correspondence between the isomorphism classes of indecomposable direct summands of the OH-module L and the isomorphism classes of indecomposable direct summands of the kHmodule Sl (P,e P ) (L), i.e., between the isomorphism classes of indecomposable OHe P -modules with source triple (P, e P , V ) and the isomorphism classes of indecomposable projective kHē Pmodules. Then the above remark concerning Green correspondents completes the proof.
The one-to-one correspondence of Theorem 20 may be seen as an instance of the Puig correspondence defined in [10]. An indecomposable Brauer-friendly OGe-module X is characterised by a quadruple ((P, e P ), V,X), as described at the beginning of this section. The correspondence X ↔X is usually not canonical; it depends on the isomorphism class of the slash functor Sl (P,e P ) . This is consistent with what Thévenaz explains in [16,before Example 26.5].