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a scalar multiple of φX. Up to replacing ψ itself by a scalar multiple, we may

suppose ψX = φX. With this assumption, the isomorphism ψ becomes unique.

Then we let φM : Sl(Q,eQ)(M ) → Sl(M ) be the restriction of ψ via the natural

embeddings Sl(Q,eQ)(M ) → Sl(Q,eQ)(X ⊕ M ) and Sl(M ) → Sl(X ⊕ M ).

As a consequence, we have ΦL,M(u) = φM◦ u ◦ φ−1_L for any two objects L, M

of the category OGeM and any u ∈ Homk(Sl(Q,eQ)(X), Sl(Q,eQ)(M )). In other

words, we have a commutative diagram Sl(Q,eQ)(L) φL // SlL,M (Q,eQ)(v)  Sl(L) SlL,M_(v)  Sl(Q,eQ)(M ) φM // Sl(M )

for any two objects L, M of the category OGeM and any morphism of OQ-

modules v : L → M . Thus φ : Sl(Q,eQ) → Sl is exactly what we would like to

call an isomorphism of (Q, eQ)-slash functors, and (ii) is proven.

The statement in (iii) follows from Theorem 3.15 (iii).

3.5

Lifting direct summands

In this section, we will need the following result, which we take from [BR-0000]. Fact 3.22. Let π : A → B be a morphism of O-algebras that are finitely gen- erated as O-modules. Let a be an ideal of the algebra A. The map π induces a one-to-one correspondence between the conjugacy classes of primitive idempo- tents of the algebra A that lie in the ideal a but not in the kernel ker π and the conjugacy classes of primitive idempotents of the algebra π(A) that lie in the ideal π(a).

Lemma 3.23. Let G be a finite group and e be a block of the group algebra OG. LetOGeM be a Brauer-friendly subcategory of the categoryOGeMod, and L be an

object of the subcategory OGeM. Let (Q, eQ) be an e-subpair of the group G and

Sl(Q,eQ):kGeM →kNG(Q,eQ)¯eQMod be a (Q, eQ)-slash functor. If the kNG(Q, eQ)-

module M = Sl(Q,eQ)(L) admits an indecomposable direct summand Y with vertex

vertex subpair (Q, eQ) such that the slashed module Sl(Q,eQ)(X) is isomorphic to

the kNG(Q, eQ)-module Y .

Proof. For any element a ∈ EndOQ(L), we have

br(Q,eQ)◦ Tr G Q(eQa eQ) = X g∈NG(Q,eQ)\G/Q brQ◦ Tr NG(Q,eQ) Ng Q(Q,eQ)(eQ g_e QgageQeQ) = TrNG(Q,eQ) Q ◦ br(Q,eQ)(a)

This computation proves that the Brauer morphism br(Q,eQ) sends the ideal

TrG_Q(EndOQ(L)) of the algebra EndOG(L) onto the ideal Tr

NG(Q,eQ)

Q (Endk(M )) of

the algebra EndkNG(Q,eQ)(M ). If Y is a indecomposable direct summand of the

kNG(Q, eQ)-module M , then there is a primitive idempotent v of the algebra

EndkNG(Q,eQ)(M ) such that Y = vM . If moreover the direct summand Y has ver-

tex Q, then the idempotent v lies in the ideal TrNG(Q,eQ)

Q (Endk(M )). By Lemma

3.22, this primitive idempotent can be lifted to a primitive idempotent u of the algebra EndOG(L) such that u lies in the ideal TrG_Q(EndOQ(L)). Then the sub-

module X = uL is a relatively Q-projective indecomposable direct summand of the OG-module L. Since v = br(Q,eQ)(u), we have Sl(Q,eQ)(X) ' v Sl(Q,eQ)(L) =

vM = Y 6= 0, so (Q, eQ) is a vertex subpair of X.

Theorem 3.24. Let G be a finite group and e be a block of the group alge- bra OG. Let (P, eP, V ) be a fusion-stable endopermutation source triple of the

group G with respect to the block e. Let OGeM be a Brauer-friendly subcate-

gory ofOGeMod such that every indecomposable OGe-module with source triple

(P, eP, V ) is an object of OGeM. Let Sl(P,eP) : kGeM → kNG(P,eP)¯ePMod be

a (P, eP)-slash functor. The slash functor Sl(P,eP) induces a one-to-one cor-

respondence between the isomorphism classes of indecomposable OGe-modules with source triple (P, eP, V ) and the isomorphism classes of indecomposable

kNG(P, eP)¯eP-modules with vertex P and trivial source, i.e., the isomorphism

classes of projective indecomposable k(NG(P, eP)/P )¯eP-modules.

Proof. Let X be an indecomposable OGe-module with source triple (P, eP, V ).

By definition of a source triple, the indecomposable OGe-module X is isomorphic to a direct summand of the OGe-module L = eOGeP⊗OPV . Thanks to Lemma

3.2, we may write L = L1⊕L2, where L1 is a direct sum of indecomposable OGe-

3.5. LIFTING DIRECT SUMMANDS modules with vertices strictly contained in the p-group P . By assumption, L1

is an object of the category OGeM.

We have L = e IndG_N

G(P,eP)L

0_{, where L}0 _{= ON}

G(P, eP)eP ⊗OP V is a direct

sum of indecomposable ONG(P, eP)eP-modules with source triple (P, eP, V ).

It follows from the definition of the Green correspondence that the restriction ResG_NG(P,eP)L1 admits a decomposition

ResG_NG(P,eP)L1 ' L0⊕ L00,

where L00 is a direct sum of indecomposable ONG(P, eP)-modules with vertices

that do not contain the p-group P . This decomposition induces an embedding of NG(P, eP)-interior algebras

φ : EndO(L0) → EndO(L1),

which in turn induces an isomorphism of CG(P )-interior NG(P, eP)-algebras

φP : BrP(EndO(L0)) → Br(P,eP)(EndO(L1)) ' Endk(Sl(P,eP)(L1)).

The CG(P )-interior NG(P, eP)-algebra BrP(EndO(L0)) can be extended to an

NG(P, eP)-interior algebra by pulling back the NG(P, eP)-interior structure of the

algebra Endk(Sl(P,eP)(L1)) through the isomorphism φP. Moreover, by assump-

tion, the endopermutation OP -module V is NG(P, eP)-stable, and BrP(Endk(V )) '

k. Thus, by Lemma 3.14, there is an isomorphism of NG(P, eP)-interior algebras

BrP(EndO(L0)) ' Endk(kNG(P, eP)¯eP ⊗kP k),

hence, by the Skolem-Noether theorem, an isomorphism of kNG(P, eP)¯eP-modules

Sl(P,eP)(L1) ' kNG(P, eP)¯eP ⊗kP k.

Let Y be an indecomposable kNG(P, eP)¯eP-module with vertex P and trivial

source. Then Y is isomorphic to an indecomposable direct summand of the kNG(P, eP)¯eP-module kNG(P, eP)¯eP ⊗kP k, i.e., of Sl(P,eP)(L1).

Let us set A = EndO(L1). As in the proof of Lemma 3.23, we know that the

Brauer morphism br(P,eP) sends the ideal Tr

G

P(AP) of the algebra AG onto the

ideal TrNG(P,eP)

P (Br(P,eP)(A)) of the algebra Br(P,eP)(A)

modules L1 and Sl are relatively P -projective, so the Brauer morphism br(P,eP)

induces an epimorphism

AG → Br(P,eP)(A)

NG(P,eP)_.

By Lemma 3.22, we conclude that the Brauer morphism br(P,eP) induces a one-

to-one correspondence between the conjugacy classes of primitive idempotents of the algebra A that are killed by br(P,eP) and the conjugacy classes of primitive

idempotents of the algebra Br(P,eP)(A). In other words, it induces a one-to-one

correspondence between the isomorphism classes of indecomposable direct sum- mands of L1 and the isomorphism classes of indecomposable direct summands

of Sl(P,eP)(L1). This proves the theorem.

The one-to-one correspondence of Theorem 3.24 may be seen as an instance of the Puig correspondence defined in [Pu-1988a]. With the notation of the lemma, notice that this correspondence depends on the isomorphism class of the slash functor Sl(P,eP). This is consistent with what Thévenaz explains in

[Th-1995, discussion before Example 26.5].