# Brauer pairs for splendid Rickard complexes

Jadyn V. Breland  Department of Mathematics, University of California Santa
Cruz, Santa Cruz, CA 95064<EMAIL_ADDRESS>and  Sam K. Miller  Department
of Mathematics, University of California Santa Cruz, Santa Cruz, CA 95064
<EMAIL_ADDRESS>
###### Abstract.

We define the notion of a Brauer pair of a chain complex, extending the notion
of a Brauer pair of a $p$-permutation module introduced by Boltje and
Perepelitsky. In fact, the Brauer pairs of a splendid Rickard complex $C$
coincide with the set of Brauer pairs of the corresponding $p$-permutation
equivalence $\Lambda(C)$ induced by $C$. As a result, we derive structural
results for splendid Rickard complexes which are analogous to known structural
properties for $p$-permutation equivalences. In particular, we show splendid
Rickard equivalences induce local splendid Rickard equivalences between
normalizer block algebras as well as centralizer block algebras.

###### Key words and phrases:

Brauer pairs, splendid Rickard complex, $p$-permutation equivalence, block
theory, Broué’s abelian defect group conjecture

###### 2010 Mathematics Subject Classification:

20C05, 20C20, 19A22, 20J05

###### Contents

  1. 1 Introduction
  2. 2 Subgroups of direct products of groups
  3. 3 $p$-permutation modules and the Brauer construction
  4. 4 Block-theoretic preliminaries
  5. 5 Trivial source rings, $p$-permutation equivalences, and splendid Rickard equivalences
  6. 6 Extended tensor products and tensor products of $p$-permutation bimodules
  7. 7 Brauer pairs for $p$-permutation modules and $p$-permutation equivalences
  8. 8 Brauer pairs for splendid Rickard complexes
  9. 9 Structural properties of splendid Rickard complexes

##  1\. Introduction

Broué’s abelian defect group conjecture is one of the hallmark conjectures of
modular representation theory. It predicts an “equivalence” between a block
with abelian defect group, and a corresponding block associated to the
normalizer of the defect group. Moreover, such equivalences are predicted to
induce “local equivalences” as well, in that for these block algebras, there
are corresponding blocks associated to centralizer subgroups which are also
equivalent in the same manner.

Historically, these equivalences were first predicted by Broué to exist as
isotypies, a bouquet of character-theoretic equivalences called “perfect
isometries” for pairs of local subgroups. However, it was suspected that
isotypies were the “shadow” of a categorical equivalence from which all of the
components of the isotypy are induced. Rickard refined this conjecture in [16]
by introducing “splendid” derived equivalences, objects which are now known as
splendid Rickard complexes. Splendid Rickard complexes are chain complexes of
$p$-permutation bimodules which have “diagonal” vertices, and induce
equivalences between the derived categories corresponding to the block
algebras. The key insight of Rickard was that $p$-permutation was the correct
property to require for the existence of equivalences between local subgroups.
In this case, the Brauer construction, a common construction when working with
$p$-permutation modules, induces the local equivalences. Such complexes were
later extended to the language of (almost) source algebras by Linckelmann in
[11].

In [5], Boltje and Xu introduced $p$-permutation equivalences, a type of
equivalence for blocks which exists on the representation ring level. Such
equivalences live between isotypies and splendid Rickard complexes, in the
sense that in order to induce an isotypy from a splendid Rickard equivalence,
one first constructs a corresponding $p$-permutation equivalence by taking the
Lefschetz invariant of the chain complex, i.e. taking an alternating sum of
terms. Much of the theory of splendid Rickard complexes and $p$-permutation
equivalences should be expected to coincide - for instance, to induce local
equivalences between centralizer block algebras for both types of
equivalences, one applies the Brauer construction in the same manner. However,
a splendid Rickard complex has additional structure that a $p$-permutation
equivalence does not - namely, the differentials of the chain complex, and a
grading of components.

The preliminary notion of $p$-permutation equivalence in [5] made restrictive
assumptions, namely that the blocks have a common defect group and that
certain fusion systems are equivalent. The main results were obtained by
assuming that the defect group was abelian. Boltje and Perepelistky in [4]
gave a wider definition for $p$-permutation equivalences, allowing for direct
sums of blocks, removing the assumption of a common defect group, and allowing
the vertices of any indecomposable summand to be “twisted diagonal.” Under
these looser assumptions, it is initially unclear how one would obtain local
equivalences, or if the stricter structural properties of a $p$-permutation
equivalence would hold in the case of block equivalences. The machinery of
Brauer pairs was necessary to resolve these questions, and in fact, many of
the structural invariants, such as blocks having isomorphic fusion systems,
continue to hold. By applying the Brauer construction to a $p$-permutation
equivalence at certain Brauer pairs, one obtains a local $p$-permutation
equivalence, leading to the notion of a Brauer pair of a $p$-permutation
equivalence. If $A$ and $B$ are direct summands of group algebras $kG$ and
$kH$ and $\gamma$ is a $p$-permutation equivalence for $A$ and $B$, we say the
pair $(\Delta(P,\phi,Q),e\otimes f^{*})$, where $\Delta(P,\phi,Q)\leq G\times
H$ is a twisted diagonal subgroup, $e$ a block of $kC_{G}(P)$, and $f$ a block
of $kC_{H}(Q)$, is a $\gamma$-Brauer pair if $e\gamma(\Delta(P,\phi,Q))f\neq
0$.

Boltje and Perepelitisky developed this theory further, deducing the structure
of the poset of Brauer pairs for $p$-permutation equivalences. Moreover, they
showed that the local equivalences induced extend beyond blocks of centralizer
group algebras - they may be extended to stronger equivalences for normalizer
blocks as well. A key observation was that applying the Brauer construction to
a $p$-permutation equivalence does not immediately lead to a centralizer
$p$-permutation equivalence, but rather a module which is “invertible” under
an extended tensor product. This suggests that there is more structure which
is missed by only considering local centralizer $p$-permutation equivalences.

The goal of this paper is adapting much of the theory in [4] to splendid
Rickard complexes. We adopt a similarly flexible definition for splendid
Rickard complexes, allowing for twisted diagonal vertices and for the
equivalence to be between sums of blocks. We then introduce the notion of a
Brauer pair for a splendid Rickard complex. Given a splendid Rickard complex
$C$ for direct summands $A$ and $B$ of group algebras $kG$ and $kH$,
respectively, we say $(\Delta(P,\phi,Q),e\otimes f^{*})$ is a $C$-Brauer pair
if $eC(\Delta(P,\phi,Q))f$ is a noncontractible chain complex. Our main result
is the following, which is Corollary 8.8:

######  Theorem 1.1.

Let $C$ be a splendid Rickard complex between $A$ and $B$ and let
$\gamma:=\Lambda(C)$ be its corresponding $p$-permutation equivalence. Then
$\mathcal{BP}(C)=\mathcal{BP}(\gamma)$. In particular,

  1. (a)

$\mathcal{BP}(C)$ is a $G\times H$-stable ideal in the poset of $A\otimes
B^{*}$-Brauer pairs.

  2. (b)

If $A$ and $B$ are blocks, then any two maximal $C$-Brauer pairs are $G\times
H$-conjugate.

  3. (c)

The following are equivalent for $(\Delta(P,\phi,Q),e\otimes
f^{*})\in\mathcal{B}\mathcal{P}(C)$:

    1. (i)

$(\Delta(P,\phi,Q),e\otimes f^{*})$ is a maximal $C$-Brauer pair.

    2. (ii)

$(P,e)$ is a maximal $A$-Brauer pair.

    3. (iii)

$(Q,f)$ is a maximal $B$-Brauer pair.

  4. (d)

Suppose $A$ and $B$ are blocks and $(\Delta(D,\phi,E),e_{D}\otimes f_{E}^{*})$
is a maximal $C$-Brauer pair. Let $\mathcal{A}$ denote the fusion system of
$A$ associated to $(D,e_{D})$ and let $\mathcal{B}$ denote the fusion system
of $B$ associated to $(E,f_{E})$. Then $\phi:E\to D$ is an isomorphism of the
fusion systems $\mathcal{B}$ and $\mathcal{A}$.

We deduce that splendid Rickard complexes induce equivalences at the
normalizer level as well. This is Corollary 8.9.

######  Theorem 1.2.

Let $C$ be a splendid Rickard complex between blocks $A\subseteq kG$ and
$B\subseteq kH$ and let $(\Delta(P,\phi,Q),e\otimes f^{*})$ be a $C$-Brauer
pair. Set $I=N_{G}(P,e)$ and $J=N_{H}(Q,f)$ and let $\hat{e}$ and $\hat{f}$
denote the unique block idempotents of $k[N_{G}(P)]$ and $k[N_{H}(Q)]$ which
cover $e$ and $f$, respectively. Then,

$\hat{e}k[N_{G}(P)]e\otimes_{kI}\operatorname{Ind}^{I\times J}_{N_{I\times
J}(\Delta(P,\phi,Q))}\big{(}eC(\Delta(P,\phi,Q))f\big{)}\otimes_{kJ}fk[N_{H}(Q)]\hat{f}$

is a splendid Rickard complex for blocks algebras $k[N_{G}(P)]\hat{e}$ and
$k[N_{H}(Q)]\hat{f}$.

Finally, we deduce that the vertices of the bimodules which appear in an
indecomposable splendid Rickard complex $C$ are governed by the Brauer pairs
of $C$. This is Theorem 9.5.

######  Theorem 1.3.

Let $C$ be an indecomposable splendid Rickard complex for block algebras
$A\subseteq kG$ and $B\subseteq kH$. Every trivial source bimodule occurring
in $C$ has a maximal Brauer pair contained in $\mathcal{B}\mathcal{P}(C)$. In
particular, the set consisting of all Brauer pairs for every trivial source
bimodule appearing in $C$ and the set of $C$-Brauer pairs coincide.

The paper is organized as follows. Section 2 introduces some preliminary
definitions involving direct products of groups. Section 3 reviews some
prerequisites on $p$-permutation modules and the Brauer construction. Section
4 is a review of block theory, Brauer pairs, and the fusion system of a block.
Section 5 establishes the definitions of $p$-permutation equivalences and
splendid Rickard complexes we will use for the paper, and proves a few basic
results for splendid Rickard complexes which hold for more restrictive
definitions. Section 6 reviews preliminaries on extended tensor products with
a focus on extended tensor products of $p$-permutation bimodules. Section 7
recalls the definition of a Brauer pair for a $p$-permutation module and a
$p$-permutation equivalence as defined by Perepelitsky and Boltje. Section 8
states and proves the main results of the paper regarding the notion of a
Brauer pair for a splendid Rickard complex. Finally, Section 9 proves as
corollaries a few structural results about the modules which appear in
splendid Rickard complexes, which are analogous to results for $p$-permutation
equivalences.

Notation and conventions: For this paper, $G$ and $H$ are finite groups, $k$
is an algebraically closed field of prime characteristic $p$, and
$\mathcal{O}$ is a complete discrete valuation ring of characteristic 0 with
residue field $k$. The symbol $R$ may be used to denote either $\mathcal{O}$
or $k$. All modules are assumed to be finitely generated, and all chain
complexes are assumed to be bounded. Given two $RG$-modules $M$ and $N$, the
tensor product $M\otimes_{k}N$ is an $RG$-module with $G$-action $g(m\otimes
n)=gm\otimes gn$. We denote by $M^{*}:=\mathrm{Hom}_{R}(M,R)$ the $R$-dual of
$M$, which is an $RG$-module via $G$-action $g\cdot f(m)=f(g^{-1}m)$.
Similarly, if $K$ is a finite group, given a $(RG,RH)$-bimodule $M$ and a
$(RH,RK)$-bimodule $N$, the tensor product $M\otimes_{RH}N$ is a
$(RG,RK)$-bimodule. We denote by $M^{*}$ the $R$-dual of $M$ which is an
$(RH,RG)$-bimodule under the action $h\cdot f(m)\cdot g=f(gmh)$. We write
$s_{p}(G)$ to denote the set of $p$-subgroups of $G$. We refer the reader to
[13, Sections 1.17] for the constructions of tensor products and internal homs
of chain complexes.

##  2\. Subgroups of direct products of groups

We begin by recalling some constructions related to subgroups of direct
products of groups. We refer the reader to [6, Part I, Chapter 2] for more
details.

######  Definition 2.1.

Let $G,H,K$ be finite groups, let $X\leq G\times H$ and $Y\leq H\times K$.

  1. (a)

Denote by $p_{1}:G\times H\to G$ and $p_{2}:G\times H\to H$ the canonical
projections. Setting

$k_{1}(X):=\\{g\mid(g,1)\in X\\}\text{ and }k_{2}(X):=\\{h\mid(1,h)\in H\\},$

we obtain normal subgroups $k_{i}(X)\trianglelefteq p_{i}(X)$ and canonical
isomorphisms $X/(k_{1}(X)\times k_{2}(X))\xrightarrow{\sim}p_{i}(X)/k_{i}(X)$
for $i\in\\{1,2\\}$ induced by the projection maps $p_{i}$. The resulting
isomorphism $\nu_{X}:p_{2}(X)/k_{2}(X)\xrightarrow{\sim}p_{1}(X)/k_{1}(X)$
satisfies $\nu_{X}(hk_{2}(X))=gk_{1}(X)$ if and only if $(g,h)\in X$, where
$(g,h)\in p_{1}(X)\times p_{2}(X)$.

  2. (b)

If $\phi:Q\xrightarrow{\sim}P$ is an isomorphism between subgroups $Q\leq H$
and $P\leq G$, then

$\Delta(P,\phi,Q):=\\{(\phi(q),q)\mid q\in Q\\}$

is a subgroup of $G\times H$. Subgroups of this form will be called twisted
diagonal subgroups of $G\times H$. For $P\leq G$, we set
$\Delta(P):=\Delta(P,\operatorname{id},P)\leq G\times G$. It is easy to verify
that $X\leq G\times H$ is twisted diagonal if and only if $|k_{1}(X)|=1$ and
$|k_{2}(X)|=1$, and that for $(g,h)\in G\times H$, one has

${}^{(g,h)}\Delta(P,\phi,Q)=\Delta({}^{g}P,c_{g}\phi c_{h^{-1}},{}^{h}Q).$

Moreover, the set of twisted diagonal subgroups of $G\times H$ is closed under
taking subgroups.

  3. (c)

The subgroup $X^{\circ}:=\\{(h,g)\in H\times G\mid(g,h)\in X\\}$ is called the
opposite subgroup of $X$.

  4. (d)

The composition of $X$ and $Y$ is defined as

$X\ast Y:=\\{(g,k)\in G\times K\mid\exists h\in H:(g,h)\in X\text{ and
}(h,k)\in Y\\}.$

It is a routine verification that composition is associative. If
$\Delta(P,\phi,Q)\leq G\times H$ and $\Delta(Q,\psi,R)\leq H\times K$ are
twisted diagonal subgroups, then

$\Delta(P,\phi,Q)\ast\Delta(Q,\psi,R)=\Delta(P,\phi\circ\psi,R).$

Note that $(X\ast Y)^{\circ}=Y^{\circ}\ast X^{\circ}$.

The next lemma is routine to verify.

######  Lemma 2.2.

Let $G,H,K$ be finite groups, let $X\leq G\times H$ and $Y\leq H\times K$.

  1. (a)

We have $X\ast
X^{\circ}=\Delta(p_{1}(X))\cdot(k_{1}(X)\times\\{1\\})=\Delta(p_{1}(X))\cdot(\\{1\\}\times
k_{1}(X))=\Delta(p_{1}(X))\cdot(k_{1}(X)\times k_{1}(X))$.

  2. (b)

We have $X\ast X^{\circ}\ast X=X.$

  3. (c)

For any $g\in G,h\in H,k\in K$, we have
${}^{(g,h)}X\ast{}^{(h,k)}Y={}^{(g,k)}(X\ast Y)$.

######  Definition 2.3.

Let $G$ and $H$ be finite groups, let $\Delta(P,\phi,Q)$ be a twisted diagonal
subgroup of $G\times H$, and let $S\leq N_{G}(P)$ and $T\leq N_{H}(Q)$. We
denote by $N_{(S,\phi,T)}$ the subgroup of $S$ consisting of all elements
$g\in S$ such that there exists an element $h\in T$ satisfying $c_{g}\phi
c_{h}=\phi$ as functions from $Q$ to $P$. Note that if $S$ contains
$C_{G}(P)$, then also $N_{(S,\phi,T)}$ also contains $C_{G}(P)$. Moreover, if
$P\leq S$ and $Q\leq T$, then $P\leq N_{(S,\phi,T)}$. We further set
$N_{\phi}:=N_{(N_{G}(P),\phi,N_{H}(Q))}$. Note that $N_{\phi}$ corresponds to
$N_{\phi^{-1}}$ in the literature on fusion systems, see for example in [2].

The following are routine to verify, and will be important in the sequel.

######  Lemma 2.4.

Let $G$ and $H$ be finite groups, let $\Delta(P,\phi,Q)$ be a twisted diagonal
subgroup of $G\times H$, and let $C_{G}(P)\leq S\leq N_{G}(P)$ and
$C_{H}(Q)\leq T\leq N_{H}(Q)$ be intermediate subgroups.

  1. (a)

One has $N_{G\times
G}(\Delta(P))=\Delta(N_{G}(P))\cdot(C_{G}(P)\times\\{1\\})=\Delta(N_{G}(P))\cdot(1\times
C_{G}(P))$.

  2. (b)

For $X=N_{G\times H}(\Delta(P,\phi,Q))$, we have $k_{1}(X)=C_{G}(P)$,
$k_{2}(X)=C_{H}(Q)$, $p_{1}(X)=N_{\phi}$, and $p_{2}(X)=N_{\phi^{-1}}$.

  3. (c)

For $X=N_{S\times T}(\Delta(P,\phi,Q))$, we have $k_{1}(X)=C_{G}(P)$,
$k_{2}(X)=C_{H}(Q)$, $p_{1}(X)=N_{(S,\phi,T)}$, and
$p_{2}(X)=N_{(T,\phi^{-1},S)}$.

##  3\. $p$-permutation modules and the Brauer construction

Throughout this chapter, let $G$ be a finite group and let $(K,\mathcal{O},k)$
be a $p$-modular system large enough for $G$. That is $\mathcal{O}$ and $k$
have $|G|_{p^{\prime}}$ roots of unity. Unless otherwise specified, $R$ will
denote either $\mathcal{O}$ or $k$. The properties listed in this section are
well-known, we refer the reader to [13] or [8] for more details and proofs.

######  Definition 3.1.

An $RG$-module $M$ is a $p$-permutation module if for every $p$-subgroup
$P\leq G$, $\operatorname{Res}^{G}_{P}M$ is a permutation $RG$-module. An
$(RG,RH)$-bimodule $M$ is a $p$-permutation bimodule if it is a
$p$-permutation module when considered as an $R[G\times H]$-module.

######  Theorem 3.2.

  1. (a)

Let $M$ be a $RG$-module. The following are equivalent:

    1. (i)

$M$ is a $p$-permutation module.

    2. (ii)

$M$ is isomorphic to a direct sum of permutation modules.

    3. (iii)

Each indecomposable direct summand of $M$ has trivial source.

  2. (b)

The functor induced by the surjection $\mathcal{O}\to k$,
$\overline{(-)}:{}_{\mathcal{O}G}\mathbf{mod}\to{}_{kG}\mathbf{mod}$ induces a
vertex-preserving bijection from the set of isomorphism classes of
$p$-permutation $\mathcal{O}G$-modules to the set of isomorphism classes of
$p$-permutation $kG$-modules.

  3. (c)

For any two $p$-permutation $\mathcal{O}G$-modules $M,N$, the canonical map
from $\operatorname{Hom}_{\mathcal{O}G}(M,N)$ to
$\operatorname{Hom}_{kG}(\overline{M},\overline{N})$ is surjective.

For this reason, indecomposable $p$-permutation modules are called trivial
source modules. We denote the full subcategory of $p$-permutation $RG$-modules
by ${}_{RG}\textbf{triv}$, and the full subcategory of $p$-permutation
$(RG,RH)$-bimodules by ${}_{RG}\textbf{triv}_{RH}$. Given an idempotent $e\in
Z(RG)$, we may consider ${}_{RGe}\textbf{triv}$ a full subcategory of
${}_{RG}\textbf{triv}$.

######  Definition 3.3.

The Brauer construction is a functor
$(-)(P):{}_{\mathcal{O}G}\textbf{mod}\to{}_{k[N_{G}(P)/P]}\textbf{mod}$
defined on objects by:

$M(P):=M^{P}/\left(\sum_{Q<P}\text{tr}^{P}_{Q}(M^{P})+J(\mathcal{O})M^{P}\right).$

This may be realized as a functor
$(-)(P):{}_{kG}\textbf{mod}\to{}_{k[N_{G}(P)/P]}\textbf{mod}$ as well, via

$M(P):=M^{P}/\sum_{Q<P}\text{tr}^{P}_{Q}(M^{P}).$

The following diagram of functors commutes for any $p$-subgroup $P\in
s_{p}(G)$:

${{}_{\mathcal{O}G}\textbf{mod}}$${{}_{kG}\textbf{mod}}$${{}_{kN_{G}(P)/P}\textbf{mod}}$$\scriptstyle{\overline{(-)}}$$\scriptstyle{-(P)}$$\scriptstyle{-(P)}$

If $P$ is not a $p$-group, then $M(P)=0$ for every $RG$-module $M$. For this
reason, we restrict our attention to taking Brauer quotients at $p$-groups.
The Brauer construction restricts to a functor
$(-)(P):{}_{RG}\mathbf{triv}\to{}_{k[N_{G}(P)/P]}\mathbf{triv}$. The Brauer
construction is an additive functor, but in general it is not exact. Since it
is additive, it can be extended to a functor of chain complexes,
$\operatorname{Ch}({}_{RG}\textbf{mod})\to\operatorname{Ch}({}_{k[N_{G}(P)/P]}\textbf{mod})$
and similarly for ${}_{RG}\mathbf{triv}$.

The following properties follow easily from the definitions.

######  Proposition 3.4.

  1. (a)

Let $M$ be an $RG$-module and $P\in s_{p}(G)$. There is a natural isomorphism
$M(P)^{*}\cong(M^{*})(P)$ of $R[N_{G}(P)/P]$-modules. Similarly, if $M$ is an
$(RG,RH)$-bimodule, then for any $X\in s_{p}(G\times H)$ there is a natural
isomorphism $M(X)^{*}\cong(M^{*})(X^{\circ})$ of $R[N_{H\times
G}(X^{\circ})/X^{\circ}]$-modules.

  2. (b)

Let $M$ be a $RG$-module, $P\in s_{P}(G)$, and $x\in G$. Then there is a
natural isomorphism ${}^{x}(M(P))\cong M({}^{x}P)$ of
$k[{}^{x}(N_{G}(P)/P)]$-modules.

The Brauer construction behaves especially well with respect to
$p$-permutation modules. The following properties are well-known, and are
collected from [13, §5.10-5.11].

######  Proposition 3.5.

  1. (a)

Let $M,N$ be $p$-permutation $RG$-modules and $P\in s_{p}(G)$. Then there is a
natural isomorphism $(M\otimes_{R}N)(P)\cong M(P)\otimes_{k}N(P)$ of
$k[N_{G}(P)/P]$-modules.

  2. (b)

Let $M\cong RX$ be a permutation module and $P\in s_{p}(G)$. Then $M(P)$ has
as a permutation $k$-basis the image of the $P$ fixed points $X^{P}$ under the
quotient map $M^{P}\to M(P)$. In particular, $R[G/P](P)\cong k[N_{G}(P)/P]$ as
$k[N_{G}(P)/P]$-modules.

  3. (c)

Let $M$ be a trivial source $RG$-module. Then $M(P)\neq 0$ if and only if $P$
is contained in a vertex of $M$.

  4. (d)

Let $Q\trianglelefteq P\in s_{p}(G)$ and let $M$ be a $p$-permutation
$RG$-module. Then $M(P)\cong M(Q)(P)$, regarding $M(Q)$ as a
$kN_{G}(Q)$-module. In particular, if $M(P)\neq 0$, then $M(R)\neq 0$ for any
subgroup $R\leq P$.

  5. (e)

For each $P\in s_{p}(G)$, the Brauer construction $M\mapsto M(P)$ induces a
bijection between the set of isomorphism classes of indecomposable
$p$-permutation $RG$-modules with vertex $P$ and the set of isomorphism
classes of indecomposable projective $k[N_{G}(P)/P]$-modules. Moreover, if $M$
is indecomposable with vertex $P$, then $M(P)$ is its Green correspondent.

The Brauer construction also is well-behaved with respect to vertices, and
preserves twisted diagonal vertices.

######  Proposition 3.6.

[4, Lemma 3.7]

  1. (a)

Let $M\in{}_{RG}\mathbf{triv}$ be indecomposable, and let $P\in s_{p}(G)$.
Then each vertex of each indecomposable direct summand of $M(P)$ as
$k[N_{G}(P)]$-module is contained in a vertex of $M$.

  2. (b)

Let $G$ and $H$ be finite groups and let $M\in{}_{RG}\mathbf{triv}_{RH}$ be
indecomposable with twisted diagonal vertices. Let $X\in s_{p}(G\times H)$.
Then each vertex of each indecomposable direct summand of the $k[N_{G\times
H}(X)]$-module $M(X)$ is again twisted diagonal.

######  Definition 3.7.

The canonical map $M^{P}\to M(P)$ is denoted by $\operatorname{Br}^{M}_{P}$,
or just $\operatorname{Br}_{P}$ if the context is clear, and called the Brauer
map. In the case where $M=RG$ and $G$ acts via conjugation, the map
$\operatorname{Br}^{RG}_{P}$ translates to the projection map

$\operatorname{br}_{P}=\operatorname{br}^{G}_{P}:RG\to
kC_{G}(P),\quad\sum_{g\in G}\alpha_{g}\cdot g\mapsto\sum_{g\in
C_{G}(P)}\overline{\alpha_{g}}\cdot g.$

In fact, this is an algebra homomorphism, called the Brauer homomorphism. Note
that $\operatorname{br}_{P}(Z(RG))\subseteq
Z(kC_{G}(P))^{N_{G}(P)}=Z(kN_{G}(P))\cap kC_{G}(P)$.

We refer the reader to [2, Part IV] for more details on Brauer maps. The final
proposition allows us to compute the Brauer construction of a module that has
been cut by a block idempotent.

######  Lemma 3.8.

[4, Lemma 3.8] Let $M\in{}_{RG}\mathbf{triv}$, $P\in s_{p}(G)$, and
$i\in(kG)^{H}$ an idempotent, fixed under the conjugation action of $H\leq G$
with $P\leq H$. Then for each $m\in M^{P}$, one has
$\operatorname{Br}^{M}_{P}(im)=\operatorname{br}^{G}_{P}(i)\operatorname{Br}^{M}_{P}(m)$.
In particular, one obtains a natural isomorphism
$(iM)(P)\xrightarrow{\sim}\operatorname{br}^{G}_{P}(i)M(P)$ of
$kN_{H}(P)$-modules.

##  4\. Block-theoretic preliminaries

Throughout, let $G$ and $H$ be finite groups and assume $(K,\mathcal{O},k)$ is
large enough for $G\times H$, with $k$ algebraically closed. We review some
important preliminaries regarding Brauer pairs and block fusion systems. We
assume the reader is familiar with the definition of a block, and that blocks
of $\mathcal{O}G$ and $kG$ are in one-to-one correspondence induced by the
canonical surjection. If $B$ is a sum of blocks of $kG$, then we write $e_{B}$
for the identity element of $B$.

######  Remark 4.1.

  1. (a)

Every block idempotent of $R[G\times H]=RG\otimes_{R}RH$ is of the form
$e\otimes f$ for uniquely determined block idempotents $e\in Z(RG)$ and $f\in
Z(RH)$. Note that we identify the $R$-algebras $R[G\times H]$ and
$RG\otimes_{R}RH$ with $(g,h)\mapsto g\otimes h$.

  2. (b)

Recall that a Brauer pair of $kG$ is a pair $(P,e)$, with $P\in s_{p}(G)$ and
$e$ is a block idempotent of $kC_{G}(P)$. Note that the block idempotents of
$k[C_{G}(P)]$ coincide with those of $k[PC_{G}(P)]$. Also, $G$ acts by
conjugation on the set of Brauer pairs of $kG$ by
${}^{g}(P,e)=({}^{g}P,{}^{g}e)$. We denote the $G$-stabilizer of $(P,e)$ by
$N_{G}(P,e)$. We have $C_{G}(P)\leq N_{G}(P,e)\leq N_{G}(P)$. For Brauer pairs
$(P,e)$ and $(Q,f)$ of $kG$, one writes $(Q,f)\trianglelefteq(P,e)$ if $Q\leq
P\leq N_{G}(Q,f)$ and $\operatorname{br}_{P}(f)e=e$ (or equivalently,
$\operatorname{br}_{P}(f)e\neq 0$). The transitive closure of the relation
$\trianglelefteq$ on the set of Brauer pairs of $kG$ is denoted by $\leq$. It
is a partial order and is preserved by $G$-conjugation.

If $e$ is a central idempotent of $kG$, then $\operatorname{br}_{P}(e)$ is an
$N_{G}(P)$-stable central idempotent of $kC_{G}(P)$ and also a central
idempotent of $kN_{G}(P)$. If $B$ is a sum of blocks of $kG$, we say that a
Brauer pair $(P,e)$ is a $B$-Brauer pair if $\operatorname{br}_{P}(e_{B})e=e$,
or equivalently if $\operatorname{br}_{P}(e_{B})e\neq 0$. Every Brauer pair is
a $B$-Brauer pair for a unique block $B$ of $kG$, and in this case,
$(\\{1\\},e_{B})\leq(P,e)$. The set of $B$-Brauer pairs is closed under
conjugation and if $(Q,f)\leq(P,e)$ are Brauer pairs, $(Q,f)$ is a $B$-Brauer
pair if and only if $(P,e)$ is. The set of $B$-Brauer pairs is denoted by
$\mathcal{B}\mathcal{P}(B)$.

For a sum $B$ of blocks of $\mathcal{O}G$, we simply define
$\mathcal{B}\mathcal{P}(B)=\mathcal{B}\mathcal{P}(\overline{B})$. Thus Brauer
pairs are by default viewed as pairs $(P,e)$, where $e$ is an idempotent of a
group algebra over $k$. It may sometimes be convenient to lift the idempotent
to the corresponding idempotent over $\mathcal{O}$, and we denote the set of
resulting pairs by $\mathcal{B}\mathcal{P}_{\mathcal{O}}(B)$.

  3. (c)

A defect group of a block $B$ of $kG$ is a subgroup $D$ of $G$, minimal with
respect to the property that $e_{B}\in\operatorname{tr}^{G}_{D}(kG^{D})$.
Equivalently, it is a subgroup $D$ for which $\Delta(D)$ is a vertex of $B$
viewed as a $k[G\times G]$-module. Equivalently, it is a $p$-subgroup $D$ of
$G$, maximal with respect to the property that
$\operatorname{br}_{D}(e_{B})\neq 0$. The defect groups of $B$ form a
$G$-conjugacy class of $p$-subgroups of $G$. If $P$ is a normal $p$-subgroup
of $G$, then $P$ is contained in each defect group of each block $B$, and
$e_{B}$ is contained in the $k$-span of the $p^{\prime}$-elements of
$C_{G}(P)$. Similarly one defines defect groups over $\mathcal{O}$, and the
same structural results hold.

We first state some well-known structural results, see [10] for details.

######  Proposition 4.2.

  1. (a)

For each Brauer pair $(P,e)$ of $kG$ and each $Q\leq P$, there exists a unique
Brauer pair $(Q,f)$ such that $(Q,f)\leq(P,e)$. In particular, if
$(R,g)\leq(P,e)$ are Brauer pairs of $kG$, and $R\leq Q\leq P$, there exists a
unique Brauer pair $(Q,f)$ of $kG$ satisfying $(R,G)\leq(Q,f)\leq(P,e)$.

  2. (b)

Let $(Q,f)$ and $(P,e)$ be Brauer pairs of $kG$. Then $(Q,f)\leq(P,e)$ and
$Q\trianglelefteq P$ if and only if $(Q,f)\trianglelefteq(P,e)$.

  3. (c)

Let $B$ be a block of $kG$. The maximal elements in the poset
$\mathcal{B}\mathcal{P}(B)$ form a single full conjugacy class. Further, a
$B$-Brauer pair $(P,e)$ is a maximal $B$-Brauer pair if and only if $P$ is a
defect group of $B$.

Note that $G\times G$ acts on $kG$ by the $k[G\times G]$-module structure, and
that $G$ acts on $kG$ by $G$-conjugation. These actions are linked by the
diagonal embedding $\Delta:g\mapsto(g,g)$, so that $kG^{\Delta(H)}=kG^{H}$.
The following are easy verifications.

######  Proposition 4.3.

Let $B$ be a sum of blocks of $kG$ and let $(Q,e)$ be a $B$-Brauer pair. Set
$I:=N_{G}(Q,e)$.

  1. (a)

We have $B(\Delta(Q))\cong k[C_{G}(Q)]\operatorname{br}_{Q}(e_{B})$ as
$k[N_{G\times G}(\Delta(Q))]$-modules.

  2. (b)

We have $eB(\Delta(Q))e\cong k[C_{G}(Q)]e$ as $k[N_{I\times
I}(\Delta(Q))]$-modules.

###### Proof.

This is [4, Proposition 4.3], with the unused assumption that $B$ is a block
algebra removed. ∎

######  Remark 4.4.

We next recall the definition of a fusion system of a block. See [2] for more
details, proofs, and the definition of an abstract fusion system and a
saturated fusion system. Let $B$ be a block of $kG$ and let $(P,e)$ be a
maximal $B$-Brauer pair. For $Q\leq P$, denote by $e_{Q}$ the unique block
idempotent of $kC_{G}(Q)$ for which $(Q,e_{Q})\leq(P,e)$.

The fusion system of $B$ associated to $(P,e)$ is the category $\mathcal{F}$
whose objects are subgroups of $P$ and whose morphism set
$\operatorname{Hom}_{\mathcal{F}}(Q,R)$ is the set of group homomorphisms
arising as conjugation maps $c_{g}:Q\to R$ where $g\in G$ satisfies
${}^{g}(Q,e_{Q})\leq(R,e_{R})$. It is well known that $\mathcal{F}$ is a
saturated fusion system on $P$.

For an abstract fusion system $\mathcal{F}$ on a $p$-group $P$, a subgroup
$Q\leq P$ is fully $\mathcal{F}$-centralized (resp. fully
$\mathcal{F}$-normalized) if $|C_{P}(Q)|\geq|C_{P}(Q^{\prime})|$ for all
(resp. $|N_{P}(Q)|\geq N_{P}(Q^{\prime})$) for all subgroups $Q^{\prime}$ of
$P$ which are $\mathcal{F}$-isomorphic to $Q$. Moreover, $Q$ is
$\mathcal{F}$-centric if $C_{P}(Q^{\prime})=Z(Q^{\prime})$ for all
$Q^{\prime}$ that are $\mathcal{F}$-isomorphic to $Q$.

The following theorem of Alperin and Brouè in [1] characterizes
$\mathcal{F}$-centric, fully normalized, and fully centralized subgroups for
block fusion systems. We give the formulation presented in [12].

######  Proposition 4.5.

[12, Theorems 2.4, 2.5] Let $B$ be a block of $kG$, let $(P,e)$ be a maximal
$B$-Brauer pair, and let $\mathcal{F}$ be the fusion system associated to $B$
and $(P,e)$. For every subgroup $Q\leq P$, denote by $e_{Q}$ the unique block
idempotent of $k[C_{G}(Q)]$ for which $(Q,e_{Q})\leq(P,e)$.

  1. (a)

A subgroup $Q$ of $P$ is fully $\mathcal{F}$-centralized if and only if
$C_{P}(Q)$ is a defect group of the block algebra $k[C_{G}(Q)]e_{Q}$. In this
case, $(C_{P}(Q),e_{QC_{P}(Q)})$ is a maximal Brauer pair of
$k[C_{G}(Q)]e_{Q}$. In particular, $Q$ is $\mathcal{F}$-centric if and only if
$Z(Q)$ is a defect group of $k[C_{G}(Q)]e_{Q}$.

  2. (b)

A subgroup $Q$ of $P$ is fully $\mathcal{F}$-normalized if and only if
$N_{P}(Q)$ is a defect group of the block algebra $k[N_{G}(Q,e_{Q})]e_{Q}$. In
this case, $(N_{P}(Q),e_{N_{P}(Q)})$ is a maximal
$k[N_{G}(Q,e_{Q})]e_{Q}$-Brauer pair.

##  5\. Trivial source rings, $p$-permutation equivalences, and splendid
Rickard equivalences

Throughout this section we assume that $G$ and $H$ are finite groups and that
$(K,\mathcal{O},k)$ is a $p$-modular system large enough for $G$ and $H$. We
denote by $R$ either $\mathcal{O}$ or $k$. Let $e$ be a central idempotent of
$RG$ and let $f$ be a central idempotent of $RH$.

######  Definition 5.1.

  1. (a)

We denote by $T(RGe)$ the trivial source ring, which is the Grothendieck group
of the category ${}_{RGe}\textbf{triv}$ with respect to direct sums. The group
$T(RGe)$ is free as an abelian group with standard basis given by the elements
$[M]$, where $M$ runs through a set of representatives of isomorphism classes
of trivial source $RG$-modules. It forms a commutative ring under
multiplication induced by $\otimes_{R}$, denoted $\cdot_{R}$. We always view
$T(RGe)$ as a subgroup of $T(RG)$ in the natural way. Moreover, multiplying a
$p$-permutation $RG$-module by $e$ induces a projection $T(RG)\to T(RGe)$. In
this way, if $b_{1},\dots,b_{k}$ are the blocks of $RG$, then we have a direct
sum decomposition of abelian groups $T(RG)=T(RGb_{1})\oplus\cdots\oplus
T(RGb_{k})$.

Taking duals induces a self-inverse group isomorphism $T(RG)\to T(RG)$. We
write $O(T(RG))$ for the orthogonal unit group of the trivial source ring,
that is, the subgroup of $T(RG)^{\times}$ consisting of all virtual
$p$-permutation modules $\gamma$ whose inverse is its dual, i.e.
$\gamma\cdot_{R}\gamma^{*}=[R]=\gamma^{*}\cdot_{R}\gamma$. This notion does
not extend to idempotents because if $\gamma\in O(T(RGe))$, then
$\gamma^{*}\in O(T(RGe^{*}))$, where $(-)^{*}$ is the map induced by the
involution $g\mapsto g^{-1}$.

  2. (b)

Similarly, we define $T(RGe,RHf):=T(R[G\times H])(e\otimes f^{*})$ where we
identify $R[G\times H]$ with $RG\otimes_{R}RH$ in the natural way. In the case
where $e=f$, $T(RGe,RGe)$ has a ring structure induced by the tensor product
$\otimes_{RGe}$, written $\cdot_{G}$. We denote by $T^{\Delta}(RGe,RHf)$ the
subgroup of $T(RGe,RHf)$ the subgroup generated by all trivial source
$(RGe,RHf)$-bimodules which have twisted diagonal vertices. Taking bimodule
duals induces a group isomorphism $T(RGe,RHf)\to T(RHf,RGe)$, which has
inverse again induced by taking duals. Denote by $O(T(RGe,RGe))$ the subgroup
of $T(RGe,RGe)^{\times}$ consisting of all virtual $p$-permutation bimodules
$\gamma$ whose inverse is its dual, i.e.
$\gamma\cdot_{G}\gamma^{*}=[RGe]=\gamma^{*}\cdot_{G}\gamma$.

Finally, observe that any $M\in T(RGe,RHf)$ induces a group homomorphism
$M\cdot_{H}-:T(RHf)\to T(RGe)$. If $M\in T(RGe,RHf)^{\times}$, it is an
isomorphism.

  3. (c)

The following definition is due to Boltje and Xu in [5]. A $p$-permutation
equivalence between $RGe$ and $RHf$ is an element $\gamma\in
T^{\Delta}(RGe,RHf)$ satisfying

$\gamma\cdot_{H}\gamma^{*}=[RGe]\in T^{\Delta}(RGe,RGe)\text{ and
}\gamma^{*}\cdot_{G}\gamma=[RHf]\in T^{\Delta}(RHf,RHf).$

Denote the set of $p$-permutation equivalences between $RGe$ and $RHf$ by
$T^{\Delta}_{o}(RGe,RHf)$. In the case $e=f$,
$T^{\Delta}_{o}(RGe,RGe)=O(T(RGe,RGe))\cap T^{\Delta}(RGe,RGe)$. We denote
this by $O(T^{\Delta}(RGe,RGe))$ as well. It follows that any $p$-permutation
equivalence between $RGe$ and $RHf$ induces a group isomorphism $T(RGe)\cong
T(RHf)$.

If $\mathcal{A}$ is an additive category, then we denote by
$\operatorname{Ch}^{b}(\mathcal{A})$ the category of bounded chain complexes
in $\mathcal{A}$ and by $K^{b}(\mathcal{A})$ the bounded homotopy category of
chain complexes in $\mathcal{A}$.

######  Definition 5.2.

Let $A$ be a sum of blocks of $RG$ and $B$ a sum of blocks of $RH$. A splendid
Rickard complex for $A$ and $B$ is a bounded chain complex $C$ of
$p$-permutation $(A,B)$-bimodules which satisfy the following conditions:

  1. (a)

For every $n\in\mathbb{Z}$, each indecomposable direct summand of $C_{n}$ has
a twisted diagonal subgroup of $G\times H$ as a vertex.

  2. (b)

$C\otimes_{B}C^{*}\simeq A$ in $K^{b}({}_{A}\mathbf{mod}_{A})$ and
$C^{*}\otimes_{A}C\simeq B$ in $K^{b}({}_{B}\mathbf{mod}_{B})$, where $A$ and
$B$ are considered as chain complexes concentrated in degree $0$.

A splendid Rickard complex $C$ of $(A,B)$-bimodules induces an equivalence of
categories $C\otimes_{B}-:K^{b}({}_{B}\textbf{mod})\to
K^{b}({}_{A}\textbf{mod})$. We call this resulting equivalence a splendid
equivalence between $A$ and $B$.

######  Lemma 5.3.

Let $A$ and $B$ be symmetric $R$-algebras. Suppose that $C$ is a bounded chain
complex of $(A,B)$-bimodules which are projective as right $B$-modules. Then
$C\otimes_{B}C^{*}\simeq 0$ in $K^{b}({}_{A}\mathbf{mod}_{A})$ if and only if
$C\simeq 0$ in $K^{b}({}_{A}\mathbf{mod}_{B})$.

###### Proof.

Since $B$ is symmetric and the terms of $C$ are projective as right
$B$-modules, we have an isomorphism
$C\otimes_{B}C^{*}\cong\mathrm{End}_{B^{op}}(C)$ of chain complexes of
$(A,A)$-bimodules. Therefore, $\mathrm{End}_{B^{op}}(C)$ is contractible by
the hypothesis. The complex $\mathrm{End}_{A\otimes_{k}B^{op}}(C)$ is obtained
from the complex $\mathrm{End}_{B^{op}}(C)$ by taking $A$-fixed points. Since
taking $A$-fixed points is an additive functor,
$\mathrm{End}_{A\otimes_{R}B^{op}}(C)$ is also contractible. Taking homology
we obtain

$0=H_{0}(\mathrm{End}_{A\otimes_{R}B^{op}}(C))\cong\operatorname{End}_{K({}_{A}\mathbf{mod}_{B})}(C).$

We conclude that $C\simeq 0$ in $K({}_{A}\mathbf{mod}_{B})$. The converse is
immediate. ∎

If $A$ and $B$ are blocks of $RG$ and $RH$, then a splendid Rickard complex
for $A$ and $B$ is an indecomposable object in the homotopy category of chain
complexes of $(A,B)$-bimodules.

######  Proposition 5.4.

Suppose that $A$ is a block of $RG$ and $B$ is a block of $RH$. Let $C$ be a
splendid Rickard complex for $A$ and $B$. Then $C$ is indecomposable in
$K^{b}({}_{A}\mathbf{mod}_{B})$. In particular, $C$ has, up to isomorphism, a
unique indecomposable noncontractible direct summand which is a splendid
Rickard equivalence for $A$ and $B$.

###### Proof.

By Lemma 5.3, $C$ is not contractible. Using the Krull-Schmidt theorem in
$\operatorname{Ch}^{b}({}_{A}\mathrm{mod}_{B})$, write
$C=\bigoplus_{i=1}^{n}C_{i}$ where each $C_{i}$ is indecomposable. Then

$A\simeq C\otimes_{B}C^{*}=\bigoplus_{1\leq i,j\leq n}C_{i}\otimes C_{j}^{*}.$

Since $A$ is indecomposable as an $(A,A)$-bimodule, the Krull-Schmidt theorem
in $K^{b}({}_{A}\mathrm{mod}_{A})$ implies that there is a unique pair $(i,j)$
such that $C_{i}\otimes_{B}C_{j}^{*}\simeq A$ and
$C_{k}\otimes_{B}C_{l}^{*}\simeq 0$ for all $(k,l)\neq(i,j)$. Thus, if $i\neq
j$, then $C_{k}\otimes_{B}C_{k}^{*}\simeq 0$ for all $k=1,\ldots,n$ which
forces $C_{k}\simeq 0$ by Lemma 5.3. This implies that $C\simeq 0$, which is a
contradiction. We conclude that $C_{i}\otimes_{B}C_{i}^{*}\simeq A$ and
$C_{k}\simeq 0$ for all $k\neq i$. Thus, $C\simeq C_{i}$ is indecomposable in
$K^{b}({}_{A}\mathbf{mod}_{B})$. Using the other homotopy equivalence, we also
obtain that $B\simeq C^{*}\otimes_{A}C=C_{i}^{*}\otimes_{A}C_{i}$ so that
$C_{i}$ is a splendid Rickard equivalence for $A$ and $B$. ∎

The notion of a “splendid” (split endomorphism two-sided tilting complex
induced from diagonal subgroups) Rickard complex is due to Jeremy Rickard. In
Rickard’s paper [16] such complexes are referred to as splendid equivalences,
however the nomenclature “Rickard complex” had been used even prior to the
publishing of [16], see for instance [9]. Rickard’s original definition
assumed that $G$ and $H$ shared a common Sylow $p$-subgroup, and that the
vertices of the modules in a splendid Rickard complex are contained inside the
diagonal subgroup generated by the common Sylow. We remove that assumption
here, instead asserting only that the modules are twisted diagonal.

Rickard proved that under his original definition, in order for a complex
between two _blocks_ to induce a splendid equivalence, it is sufficient for
only one of the homotopy equivalences to be satisfied. This argument extends
to the more general definition given here. Note that the following result
assumes $A$ and $B$ are single blocks, but the statement and argument can be
modified to work for sums of blocks as well.

######  Proposition 5.5.

Assume that $A$ and $B$ are blocks of $RG$ and $RH$, respectively. Let $C$ be
a bounded chain complex of $p$-permutation $(A,B)$-bimodules which satisfies
condition 5.2(a). If $C\otimes_{B}C^{*}\simeq A$ in
$K^{b}({}_{A}\mathbf{mod}_{A})$, then $C$ is a splendid Rickard complex.

###### Proof.

An indecomposable $(A,B)$-bimodule $M$ with a twisted diagonal vertex $X$ is
projective as a left $A$-module and a right $B$-module. Indeed, by the Mackey
Formula and the Krull-Schmidt theorem, every indecomposable direct summand of
$\mathrm{Res}_{G\times 1}^{G\times H}(M)$ is projective relative to $(G\times
1)\cap{}^{(g,h)}X$ for some $(g,h)\in G\times H$. Since ${}^{(g,h)}X$ is also
twisted diagonal, this intersection is trivial. This shows that $M$ is
projective as a left $A$-module. A similar argument shows that $M$ is
projective as a right $B$-module. Hence, the result follows from [13,
4.14.16]. ∎

Boltje and Xu proved that $p$-permutation equivalences lie between isotypies
and splendid equivalences in the following way. Given a bounded chain complex
$C$ of $p$-permutation modules, the Lefschetz invariant of $C$ is

$\Lambda(C):=\sum_{n\in\mathbb{Z}}(-1)^{n}[C_{n}]\in T(RG).$

######  Theorem 5.6.

[5, Theorem 1.5] Let $C$ be a splendid Rickard complex between $A$ and $B$.
Then $\gamma:=\Lambda(C)$ is a $p$-permutation equivalence between $RGe$ and
$RHf$.

######  Remark 5.7.

Under Rickard’s original definition, if $C$ is a splendid Rickard complex for
$A$ and $B$, for any $Q\leq P$ and blocks $e_{Q}$ and $f_{Q}$ corresponding to
the blocks of $kC_{G}(Q)$ and $kC_{H}(Q)$ determined by $A$ and $B$, the chain
complex $e_{Q}C(\Delta Q)f_{Q}$ is a splendid Rickard complex for
$kC_{G}(Q)e_{Q}$ and $kC_{H}(Q)f_{Q}$. Such equivalences are regarded as
“local” equivalences, equivalences between blocks of local centralizer
subgroups.

##  6\. Extended tensor products and tensor products of $p$-permutation
bimodules

In this section, we explain some of the tensor products that will arise in the
sequel. Throughout, $G,H,K$ are finite groups, and $R$ denotes a commutative
ring.

######  Definition 6.1.

Let $X\leq G\times H$ and $Y\leq H\times K$. Furthermore, let
$M\in{}_{RX}\mathbf{mod}$ and $N\in{}_{RY}\mathbf{mod}$. Since $k_{1}(X)\times
k_{2}(X)\leq X$, $M$ can be viewed as $(R[k_{1}(X)],R[k_{2}(X)])$-bimodule.
Similarly, $N$ can be viewed as $(R[k_{1}(Y)],R[k_{2}(Y)])$-bimodule, so
$M\otimes_{R[k_{1}(X)\cap k_{2}(Y)]}N$ is a
$(R[k_{1}(X)],R[k_{2}(Y)])$-bimodule. Note that $k_{1}(X)\times k_{2}(Y)\leq
X\ast Y$, and that this bimodule structure can be extended to a $R[X\ast
Y]$-module structure such that for $(g,k)\in X\ast Y,$ $m\in M$, $n\in N$, we
have

$(g,k)\cdot(m\otimes n)=(g,h)m\otimes(h,k)n,$

where $h\in H$ is chosen such that $(g,h)\in X$ and $(h,k)\in Y$. Note this is
independent of choice of $h$. This construction was first used in [7]. We
denote this extended tensor product by
$M\underset{RH}{\overset{X,Y}{\otimes}}N\in{}_{R[X\ast Y]}\mathbf{mod}$, and
obtain a bifunctor
$-\underset{RH}{\overset{X,Y}{\otimes}}-:{}_{RX}\mathbf{mod}\times{}_{RY}\mathbf{mod}\to{}_{R[X\ast
Y]}\mathbf{mod}$.

Some quick calculations determine that this construction is associative in the
following way: if $L$ is a finite group, $Z\leq K\times L$, and
$P\in{}_{RZ}\mathbf{mod}$, then

$(M\underset{RH}{\overset{X,Y}{\otimes}}N)\underset{RK}{\overset{X\ast
Y,Z}{\otimes}}P\xrightarrow{\sim}M\underset{RH}{\overset{X,Y\ast
Z}{\otimes}}(N\underset{RK}{\overset{Y,Z}{\otimes}}P),\quad(m\otimes n)\otimes
p\mapsto m\otimes(n\otimes p).$

defines a natural isomorphism of $R[X\ast Y\ast Z]$-modules which is natural
in $M,N$ and $K$.

Moreover, for $g\in G$, $h\in H$, $k\in K$, we have a natural isomorphism

${}^{(g,k)}(M\underset{RH}{\overset{X,Y}{\otimes}}N)\cong{}^{(g,h)}M\underset{RH}{\overset{{}^{(g,h)}X,{}^{(h,k)}Y}{\otimes}}{}^{(h,k)}N$

of $R[{}^{(g,k)}(X\ast Y)]$-modules.

The following theorem of Serge Bouc describes an interaction between additive
induction, standard tensor products, and extended tensor products.

######  Theorem 6.2.

[7] Let $X\leq G\times H$, $Y\leq H\times K$, $M\in{}_{RX}\mathbf{mod}$,
$N\in{}_{RY}\mathbf{mod}$. There is a natural isomorphism

$\operatorname{Ind}^{G\times H}_{X}(M)\otimes_{RH}\operatorname{Ind}^{H\times
K}_{Y}(N)\cong\bigoplus_{t\in[p_{2}(X)\backslash
H/p_{1}(Y)]}\operatorname{Ind}^{G\times
K}_{X\ast{}^{(t,1)}Y}(M\mathop{\otimes}\limits_{RH}^{X,{}^{(t,1)}Y}{}^{(t,1)}N)$

of $(RG,RK)$-bimodules.

We next turn our attention to extended tensor products of $p$-permutation
modules in conjunction with the Brauer construction. In this case, the tensor
products are particularly well-behaved.

######  Lemma 6.3.

[4, Lemma 7.2] Let $X\leq G\times H$ and $Y\leq H\times K$.

  1. (a)

If $M\in{}_{\mathcal{O}X}\mathbf{triv}$ and
$N\in{}_{\mathcal{O}Y}\mathbf{triv}$, then
$M\overset{X,Y}{\underset{\mathcal{O}H}{\otimes}}N\in{}_{\mathcal{O}[X\ast
Y]}\mathbf{triv}$.

  2. (b)

If $M\in{}_{\mathcal{O}X}\mathbf{triv}$ and
$N\in{}_{\mathcal{O}Y}\mathbf{triv}$ are indecomposable with twisted diagonal
vertices, then each indecomposable direct summand of the $\mathcal{O}[X\ast
Y]$-module $M\overset{X,Y}{\underset{\mathcal{O}H}{\otimes}}N$ has twisted
diagonal vertices.

######  Remark 6.4.

This construction originally appears in [3]. For a fixed twisted diagonal
$p$-subgroup $\Delta(P,\sigma,R)\leq G\times K$, denote by
$\Gamma_{H}(P,\sigma,R)$ the set of triples $(\phi,Q,\psi)$ where $Q\leq H$
and $\psi:R\xrightarrow{\sim}Q$ and $\phi:Q\xrightarrow{\sim}P$ are
isomorphisms with $\phi\circ\psi=\sigma$. The group $H$ acts on
$\Gamma_{H}(P,\sigma,R)$ by

${}^{h}(\phi,Q,\psi):=(\phi c_{h}^{-1},{}^{h}Q,c_{h}\psi).$

Note that $\operatorname{stab}_{H}(\phi,Q,\psi)=C_{H}(Q)$. For
$M\in{}_{\mathcal{O}G}\mathbf{mod}_{\mathcal{O}H}$,
$N\in{}_{\mathcal{O}H}\mathbf{mod}_{\mathcal{O}K}$, and
$(\phi,Q,\psi)\in\Gamma_{H}(P,\sigma,R)$, one has a
$(k[C_{G}(P)],k[C_{K}(R)])$-bimodule homomorphism

$\displaystyle\Phi_{(\phi,Q,\psi)}:M(\Delta(P,\phi,Q))\otimes_{k[C_{H}(Q)]}N(\Delta(Q,\psi,R))$
$\displaystyle\to(M\otimes_{\mathcal{O}H}N)(\Delta(P,\sigma,R))$
$\displaystyle\operatorname{Br}_{\Delta(P,\phi,Q)}(m)\otimes\operatorname{Br}_{\Delta(Q,\psi,R)}(n)$
$\displaystyle\mapsto\operatorname{Br}_{\Delta(P,\sigma,R)}(m\otimes n),$

(see [3, 3.1]) which is natural in $M$ and $N$. The following theorem
characterizes the tensor product on the right-hand side for a particular class
of $p$-permutation modules.

######  Theorem 6.5.

[3, Theorem 3.3] Let $\Delta(P,\sigma,R)\in s_{p}(G\times K)$ be a twisted
diagonal $p$-subgroup, let $\Gamma=\Gamma_{H}(P,\sigma,R)$ be as above, and
let $\widetilde{\Gamma}\subseteq\Gamma$ be a set of representatives of the
$H$-orbits of $\Gamma$. Furthermore, let
$M\in{}_{\mathcal{O}G}\mathbf{triv}_{\mathcal{O}H}$ and
$N\in{}_{\mathcal{O}H}\mathbf{triv}_{\mathcal{O}K}$ be $p$-permutation
bimodules whose indecomposable direct summands have twisted diagonal vertices.
Then the direct sum of the homomorphisms $\Phi_{(\phi,Q,\psi)}$ for
$(\phi,Q,\psi)\in\widetilde{\Gamma}$ yields an isomorphism

$\Phi:\bigoplus_{(\phi,Q,\psi)\in\widetilde{\Gamma}}M(\Delta(P,\phi,Q))\otimes_{k[C_{H}(Q)]}N(\Delta(Q,\psi,R))\to(M\otimes_{\mathcal{O}H}N)(\Delta(P,\sigma,R))$

of $(k[C_{G}(P)],k[C_{K}(R)])$-bimodules which is natural in $M$ and $N$.

Note that not only $C_{G}(P)\times C_{K}(R)$, but also the bigger group
$N_{G\times K}(\Delta(P,\sigma,R))$ acts on the right-hand side of the
isomorphism above. Therefore, we may consider the extended tensor product
structure on the right-hand side and attempt to transport it over to the left.
This was done by Boltje and Perepelitsky in the following way. First, note the
domain of the homomorphism $\Phi_{(\phi,Q,\psi)}$ carries a $k[N_{G\times
H}(\Delta(P,\phi,Q))\ast N_{H\times K}(\Delta(Q,\psi,R))]$-module structure
via the extended tensor product construction. This extends the
$k[C_{G}(P)\times C_{K}(R)]$-module structure from before, since
$k_{1}(N_{G\times H}(\Delta(P,\phi,Q)))=C_{G}(P)$ and $k_{2}(N_{G\times
H}(\Delta(P,\phi,Q)))=C_{H}(Q)=k_{1}(N_{H\times K}(\Delta(Q,\psi,R)))$, and
$k_{2}(N_{H\times K}(\Delta(Q,\psi,R)))=C_{K}(R)$. It follows that
$\Phi_{(\phi,Q,\psi)}$ actually defines a homomorphism

$\Phi_{(\phi,Q,\psi)}:M(\Delta(P,\phi,Q))\overset{N_{G\times
H}(\Delta(P,\phi,Q)),N_{H\times
K}(\Delta(Q,\psi,R))}{\underset{kH}{\otimes}}N(\Delta(Q,\psi,R))\to(M\otimes_{\mathcal{O}H}N)(\Delta(P,\sigma,R))$

of $k[N_{G\times H}(\Delta(P,\phi,Q))\ast N_{H\times
K}(\Delta(Q,\psi,R))]$-bimodules which is natural in $M$ and $N$.

######  Corollary 6.6.

[4, Corollary 7.4] Let
$M\in{}_{\mathcal{O}G}\mathbf{triv}_{\mathcal{O}H},N\in{}_{\mathcal{O}H}\mathbf{triv}_{\mathcal{O}K}$,
$\Delta(P,\sigma,R)\in s_{p}(G\times K)$, and
$\widetilde{\Gamma}\subseteq\Gamma=\Gamma_{H}(P,\sigma,R)$ be as before.

  1. (a)

The group $N_{G\times K}(\Delta(P,\sigma,R))\times H$ acts on $\Gamma$ via

${}^{((g,k),h)}(\phi,Q,\psi):=(c_{g}\phi c_{h}^{-1},{}^{h}Q,c_{h}\psi
c_{k}^{-1}).$

For the induced action of $N_{G\times K}(\Delta(P,\sigma,R))$ on the
$H$-orbits $[\phi,Q,\psi]_{H}$ of $\Gamma$, one has

$\operatorname{stab}_{N_{G\times
K}(\Delta(P,\sigma,R))}([\phi,Q,\psi]_{H})=N_{G\times H}(\Delta(P,\phi,Q))\ast
N_{H\times K}(\Delta(Q,\psi,R))$

for each $(\phi,Q,\psi)\in\Gamma$.

  2. (b)

Let $\widehat{\Gamma}\subseteq\widetilde{\Gamma}$ be a set of representatives
of the $N_{G\times K}(\Delta(P,\sigma,R))\times H$-orbits of $\Gamma$. The
homomorphisms $\Phi_{(\phi,Q,\psi)}$ for each
$(\phi,Q,\psi)\in\widehat{\Gamma}$ induce an isomorphism

$\Phi:\bigoplus_{\gamma=(\phi,Q,\psi)\in\widehat{\Gamma}}\operatorname{Ind}_{X(\gamma)\ast
Y(\gamma)}^{N_{G\times
K}(\Delta(P,\sigma,R))}\left(M(\Delta(P,\phi,Q))\overset{X(\gamma),Y(\gamma)}{\underset{kH}{\otimes}}N(\Delta(Q,\psi,R))\right)\xrightarrow{\sim}(M\otimes_{\mathcal{O}H}N)(\Delta(P,\sigma,R))$

of $k[N_{G\times K}(\Delta(P,\sigma,R))]$-modules which is natural in $M$ and
$N$, with $X(\gamma):=N_{G\times H}(\Delta(P,\phi,Q))$ and
$Y(\gamma):=N_{H\times K}(\Delta(Q,\psi,R))$ for any
$\gamma=(\phi,Q,\psi)\in\Gamma$.

This theorem has a block-wise version as well. First, observe that for any
twisted diagonal subgroup $\Delta(P,\phi,Q)$ of $G\times H$, one has
$C_{G\times H}(\Delta(P,\phi,Q))=C_{G}(P)\times C_{H}(Q)$, and that for Brauer
pairs $(P,e),(Q,f)$ for $kG$ and $kH$ respectively, the pair
$(\Delta(P,\phi,Q),e\otimes f^{*})$ is a Brauer pair for $k[G\times
H]=kG\otimes_{k}kH$.

######  Theorem 6.7.

[4, Theorem 7.5] Let $(P,e)\in\mathcal{BP}(kG)$ and
$(R,d)\in\mathcal{BP}(kK)$. Suppose $\sigma:R\xrightarrow{\sim}P$ is an
isomorphism and let $C_{G}(P)\leq S\leq N_{G}(P,e)$ and $C_{K}(R)\leq T\leq
N_{K}(R,d)$ be intermediate subgroups. Further, let
$\Omega:=\Omega_{H}((P,e),\sigma,(R,d))$ denote the set of triples
$(\phi,(Q,f),\psi)$, where $(Q,f)\in\mathcal{BP}(kH)$ and
$\psi:R\xrightarrow{\sim}Q$ and $\phi:Q\xrightarrow{\sim}P$ are isomorphisms
such that $\sigma=\phi\circ\psi$. Finally, let
$M\in{}_{\mathcal{O}G}\mathbf{triv}_{\mathcal{O}H}$ and
$N\in{}_{\mathcal{O}H}\mathbf{triv}_{\mathcal{O}K}$ be $p$-permutation modules
all of whose indecomposable direct summands have twisted diagonal vertices.

  1. (a)

The group $N_{G\times K}(\Delta(P,\sigma,R))\times H$ acts on $\Omega$ via

${}^{((g,k),h)}(\phi,(Q,f),\psi)=(c_{g}\phi c_{h}^{-1},{}^{h}(Q,f),c_{h}\psi
c_{k}^{-1})$

and $\operatorname{stab}_{H}(\phi,(Q,f),\psi)=C_{H}(Q)$.

  2. (b)

Let $\widetilde{\Omega}\subseteq\Omega$ be a set of representatives of the
$H$-orbits of $\Omega$. The restrictions $\Phi_{(\phi,(Q,f),\psi)}$ of
$\Phi_{(\phi,Q,\psi)}$ to
$eM(\Delta(P,\phi,Q))f\otimes_{k[C_{H}(Q)]}fN(\Delta(Q,\psi,R))d$ define an
isomorphism

$\displaystyle\Phi:\bigoplus_{(\phi,(Q,f),\psi)\in\widetilde{\Omega}}eM(\Delta(P,\phi,Q))f\otimes_{k[C_{H}(Q)]}fN(\Delta(Q,\psi,R))d$
$\displaystyle\cong e(M\otimes_{\mathcal{O}H}N)(\Delta(P,\sigma,R))d$
$\displaystyle e\mathrm{Br}_{\Delta(P,\phi,Q)}(m)f\otimes
f\mathrm{Br}_{\Delta(Q,\psi,R)}(n)d$ $\displaystyle\mapsto
e\mathrm{Br}_{\Delta(P,\sigma,R)}(m\otimes n)d$

of $(k[C_{G}(P)]e,k[C_{K}(R)]d)$-bimodules which is natural in $M$ and $N$.

  3. (c)

Let $\widehat{\Omega}\subseteq\widetilde{\Omega}\subseteq\Omega$ be a set of
representatives of the $N_{S\times T}(\Delta(P,\sigma,R))\times H$ orbits of
$\Omega.$ The homomorphisms
$\Phi_{(\phi,(Q,f),\psi)}:eM(\Delta(P,\phi,Q))f\otimes_{k[C_{H}(Q)]}fN(\Delta(Q,\psi,R))d\to
e(M\otimes_{\mathcal{O}H}N)(\Delta(P,\sigma,R))d$ for
$(\phi,(Q,f),\psi)\in\widehat{\Omega}$ induce an isomorphism

$\Phi:\bigoplus_{\omega\in\hat{\Omega}}\operatorname{Ind}^{N_{S\times
T}(\Delta(P,\sigma,R))}_{X(\omega)\ast
Y(\omega)}(M(\omega)\mathop{\otimes}_{kH}^{X(\omega),Y(\omega)}N(\omega))\cong
e(M\otimes_{\mathcal{O}H}N)(\Delta(P,\sigma,R))d$

of $N_{S\times T}(\Delta(P,\sigma,R))$-modules which is natural in $M$ and
$N$. Here, for $\omega=(\phi,(Q,f),\psi)\in\widehat{\Omega}$, we set
$X(\omega):=N_{S\times H}(\Delta(P,\phi,Q),e\otimes
f^{*}),Y(\omega):=N_{H\times T}(\Delta(Q,\psi,R),f\otimes
d^{*}),M(\omega):=eM(\Delta(P,\phi,Q))f,$ and
$N(\omega):=fN(\Delta(Q,\psi,R))d$.

  4. (d)

Assume that $G=K$, $(P,e)=(R,d)$, $S=T$, $\sigma=\text{id}_{P}$. Let
$\Lambda:=\Lambda_{H}(P)$ be the set of pairs $(\phi,(Q,f))$ where $(Q,f)$ is
a Brauer pair of $kH$ and $\phi:Q\to P$ is an isomorphism. The group $S\times
H$ acts on $\Lambda$ via

${}^{(g,h)}(\phi,(Q,f))=(c_{g}\phi c_{h}^{-1},{}^{h}(Q,f)),$

for $(g,h)\in S\times H$ and $(\phi,(Q,f))\in\Lambda.$ Let
$\hat{\Lambda}\subseteq\widetilde{\Lambda}\subseteq\Lambda$ be such that
$\widetilde{\Lambda}$ (resp. $\hat{\Lambda}$) is a set of representatives of
the $H$-orbits (resp. $S\times H$-orbits) of $\Lambda$. Then one has a natural
isomorphism

$e(M\otimes_{\mathcal{O}H}N)(\Delta(P))e\cong\bigoplus_{(\phi,(Q,f))\in\widetilde{\Lambda}}eM(\Delta(P,\phi,Q))f\otimes_{k[C_{H}(Q)]}fN(\Delta(Q,\phi^{-1},P))e$

of $(k[C_{G}(P)]e,k[C_{G}(P)]e)$-bimodules and a natural isomorphism

$e(M\otimes_{\mathcal{O}H}N)(\Delta(P))e\cong\bigoplus_{\lambda=(\phi,(Q,f))\in\widehat{\Lambda}}\operatorname{Ind}^{N_{S\times
S}(\Delta(P))}_{\Delta(I(\lambda))(C_{G}(P)\times
1)}(M(\lambda)\mathop{\otimes}_{kH}^{X(\lambda),Y(\lambda)}N(\lambda))$

of $k[N_{S\times S}(\Delta(P))](e\otimes e^{*})$-modules. Here, for
$\lambda=(\phi,(Q,f))\in\Lambda$, we set $X(\lambda):=N_{S\times
H}(\Delta(P,\phi,Q),e\otimes f^{*})$, $Y(\lambda):=N_{H\times
S}(\Delta(Q,\phi^{-1},P),f\otimes e^{*})$,
$M(\lambda):=eM(\Delta(P,\phi,Q))f,$ $N(\lambda):=fN(\Delta(Q,\phi^{-1},P))e$,
and $I(\lambda):=N_{(S,\phi,N_{H}(Q,f))}$.

Note that the previous theorems also hold over base ring $k$ replacing
$\mathcal{O}$.

##  7\. Brauer pairs for $p$-permutation modules and $p$-permutation
equivalences

Let $A$ and $B$ be sums of blocks of $kG$ and $kH$, respectively. Boltje and
Perepelitsky in [4] introduced the notion of a Brauer pair of a
$p$-permutation equivalence $\gamma$ (to be precise, they initially defined
the notion for an element $\gamma\in T^{\Delta}(A,B)$ for which
$\gamma\cdot_{B}\gamma^{*}=[A]$).

Because a $p$-permutation equivalence can be written as a difference of
bimodules whose indecomposable direct summands have twisted diagonal vertices,
applying the Brauer construction at a subgroup $X\leq G\times H$ to a
$p$-permutation equivalence will only be nonzero when $X$ is twisted diagonal.
However, this is not a sufficient criteria for the result to be nonzero, and
this motivates the definitions given in this section. We review the
definitions and results in [4].

######  Definition 7.1.

[4, Definition 5.1] Let $M\in{}_{kG}\textbf{triv}$. We call a Brauer pair
$(P,e)$ of $kG$ an $M$-Brauer pair if $M(P,e):=e\cdot M(P)\neq 0$, where $M$
has $k[N_{G}(P,e)]e$-module structure. We denote the set of Brauer pairs of
$M$ by $\mathcal{BP}(M)$.

######  Proposition 7.2.

[4, Proposition 5.3, 5.4] Let $M,N\in{}_{kG}\mathbf{triv}$.

  1. (a)

$\mathcal{BP}(M)$ is a $G$-stable ideal in the poset $\mathcal{BP}(kG)$, i.e.
it is stable under $G$-conjugation and if $(Q,f)\leq(P,e)$ are Brauer pairs of
$kG$ such that $(P,e)$ is an $M$-Brauer pair, then also $(Q,f)$ is an
$M$-Brauer pair.

  2. (b)

Assume $M$ is indecomposable. Then the maximal $M$-Brauer pairs are precisely
the $M$-Brauer pairs $(P,e)$ where $P$ is a vertex of $M$. Moreover, any two
maximal $M$-Brauer pairs are $G$-conjugate.

  3. (c)

Let $M$ and $N$ be indecomposable $p$-permutation $kG$-modules, suppose that
$(P,e)\in\mathcal{BP}(kG)$ is both a maximal $M$-Brauer pair and a maximal
$N$-Brauer pair, and set $I:=N_{G}(P,e)$. Then $M(P,e)$ and $N(P,e)$ are
indecomposable $p$-permutation $kIe$-modules. Moreover, $M\cong N$ if and only
if $M(P,e)\cong N(P,e)$ as $kIe$-modules.

One may also define Brauer pairs for virtual $p$-permutation modules.

######  Definition 7.3.

[4, Definition 9.6] Let $\omega\in T(kG)$. A Brauer pair $(P,e)$ of $kG$ is
called an $\omega$-Brauer pair if $\omega(P,e)=e\omega(P)\neq 0$ in
$T(k[N_{G}(P,e)])$. The set of $\omega$-Brauer pairs is denoted
$\mathcal{BP}(\omega)$.

Let $\gamma\in T^{\Delta}(A,B)$ be a $p$-permutation equivalence for $A$ and
$B$ and let $\Delta(P,\phi,Q)\subset G\times H$ be a twisted diagonal
subgroup. Via restriction, we can also view $\gamma(\Delta(P,\phi,Q))\in
T(k[N_{G\times H}(\Delta(P,\phi,Q))])$ as an element of the corresponding
Grothendieck group of $C_{G}(P)\times C_{H}(Q)=C_{G\times
H}(\Delta(P,\phi,Q))$.

######  Remark 7.4.

  1. (a)

Let $\Delta(P,\phi,Q)$ be a twisted diagonal $p$-subgroup of $G\times H$. If
$(P,e)$ is a Brauer pair of $kG$ and $(Q,f)$ is a Brauer pair of $kH$, then
$(\Delta(P,\phi,Q),e\otimes f^{*})$ is a Brauer pair of $k[G\times H]$, and
conversely, if $(\Delta(P,\phi,Q),e\otimes f^{*})$ is a Brauer pair of
$k[G\times H]$, then $(P,e)$ is a Brauer pair of $kG$ and $(Q,f)$ is a Brauer
pair of $kH$. In this case, with $I:=N_{G}(P,e)$ and $J:=N_{H}(Q,f)$, one has
$N_{G\times H}(\Delta(P,\phi,Q),e\otimes f^{*})=N_{I\times
J}(\Delta(P,\phi,Q))$ and $e\gamma(P,\phi,Q)f\in T^{\Delta}(k[N_{I\times
J}(\Delta(P,\phi,Q))](e\otimes f^{*}))$. Moreover, $(\Delta(P,\phi,Q),e\otimes
f^{*})$ is an $A\otimes B^{*}$-Brauer pair if and only if $(P,e)$ is an
$A$-Brauer pair and $(Q,f)$ is a $B$-Brauer pair. Denote the set of $A\otimes
B^{*}$-Brauer pairs $(X,d)$ where $X\leq G\times H$ is a twisted diagonal
subgroup by $\mathcal{BP}^{\Delta}(A,B).$

  2. (b)

Let $\Delta(P^{\prime},\phi^{\prime},Q^{\prime})\leq\Delta(P,\phi,Q)$ be
twisted diagonal subgroups of $G\times H$, and let
$(\Delta(P^{\prime},\phi^{\prime},Q^{\prime}),e^{\prime}\otimes f^{\prime*})$
and $(\Delta(P,\phi,Q),e\otimes f^{*})$ be $k[G\times H]$-Brauer pairs. Then
$(\Delta(P^{\prime},\phi^{\prime},Q^{\prime}),e^{\prime}\otimes
f^{\prime*})\leq(\Delta(P,\phi,Q),e\otimes f^{*})$ if and only if
$(P^{\prime},e^{\prime})\leq(P,e)$ and $(Q^{\prime},f^{\prime})\leq(Q,f)$.

The following proposition gives convenient reformulations of being a
$\gamma$-Brauer pair.

######  Proposition 7.5.

[4, Proposition 10.8] Let $(\Delta(P,\phi,Q),e\otimes
f^{*})\in\mathcal{BP}^{\Delta}(A,B)$ and set $I:=N_{G}(P,e)$ and
$J:=N_{H}(Q,f)$. The following are equivalent:

  1. (a)

$(\Delta(P,\phi,Q),e\otimes f^{*})\in\mathcal{BP}(\gamma)$, i.e.
$e\gamma(P,\phi,Q)f\neq 0$ in $T(k[N_{I\times J}(\Delta(P,\phi,Q))](e\otimes
f^{*}))$.

  2. (b)

$e\gamma(P,\phi,Q)f\neq 0$ in $T(k[C_{G}(P)]e,k[C_{H}(Q)]f)$.

  3. (c)

$(\Delta(P,\phi,Q),e\otimes f^{*})\in\mathcal{BP}(M)$ for some indecomposable
module $M\in{}_{A}\mathbf{triv}_{B}$ appearing in $\gamma$.

######  Proposition 7.6.

[4, Corollary 10.9] Let $(P,e)$ be an $A$-Brauer pair and define $\Lambda$ as
in Theorem 6.7 and let $\Lambda_{B}\subseteq\Lambda$ be the set of pairs
$(\psi,(Q,f))$ such that $(Q,f)$ is a $B$-Brauer pair. There exists a unique
$H$-orbit of pairs $(\psi,(Q,f))\in\Lambda_{B}$ with the property that
$(\Delta(P,\psi,Q),e\otimes f^{*})$ is a $\gamma$-Brauer pair.

######  Proposition 7.7.

[4, Theorem 10.10] Suppose $\gamma$ is a $p$-permutation equivalence for $A$
and $B$. Let $\mathcal{I}$ denote the set of primitive idempotents in $Z(A)$
and let $\mathcal{J}$ denote the set of primitive idempotents in $Z(B)$.

  1. (a)

For every $e\in\mathcal{I}$, there exists a unique $f_{e}\in\mathcal{J}$ such
that $e\gamma f_{e}\neq 0$. The assignment $e\mapsto f_{e}$ establishes a
bijection $f:\mathcal{I}\to\mathcal{J}$.

  2. (b)

We have $\gamma=\sum_{e\in\mathcal{I}}e\gamma f_{e}$. Moreover, for every
$e\in\mathcal{I}$, $e\gamma f_{e}$ is a $p$-permutation equivalence for $kGe$
and $kGf_{e}$.

  3. (c)

For every $e\in\mathcal{I}$, the $e\gamma f_{e}$-Brauer pairs are precisely
the $\gamma$-Brauer pairs which contain $(1,e\otimes f_{e}^{*})$. Thus, we
have $\mathcal{BP}(\gamma)=\coprod_{e\in\mathcal{I}}\mathcal{BP}(e\gamma
f_{e})$.

###### Proof.

For claims (a) and (b), see [4, Theorem 10.10]. For (c), let
$\lambda:=(\Delta(P,\phi,Q),b\otimes c^{*})$ be a Brauer pair for $k[G\times
H]$. Then

$(e\gamma f_{e})(\lambda)=b(e\gamma
f_{e})(\Delta(P,\phi,Q))c=b\operatorname{Br}_{P}(e)\gamma(\Delta(P,\phi,Q))\operatorname{Br}_{Q}(f_{e})c.$

The condition that $b\operatorname{Br}_{P}(e)\neq 0$ and
$\operatorname{Br}_{Q}(f_{e})c\neq 0$ is equivalent to
$b\operatorname{Br}_{P}(e)=b$ and $\operatorname{Br}_{Q}(f_{e})c=c$ and also
to $(1,e\otimes f_{e}^{*})\leq\lambda$. Considering the above equation, it
follows immediately that $\lambda$ is a $e\gamma f_{e}$-Brauer pair if and
only if $\lambda$ is a $\gamma$-Brauer pair and $(1,e\otimes
f_{e}^{*})\leq\lambda$. ∎

######  Theorem 7.8.

[4, Theorem 10.11]

  1. (a)

The set of $\gamma$-Brauer pairs forms a $G\times H$-stable ideal in the poset
of $A\otimes B^{*}$-Brauer pairs.

  2. (b)

If $A$ and $B$ are blocks, then any two maximal $\gamma$-Brauer pairs are
$G\times H$-conjugate.

  3. (c)

For $(\Delta(P,\phi,Q),e\otimes f^{*})\in\mathcal{BP}(\gamma)$, the following
are equivalent:

    1. (i)

$(\Delta(P,\phi,Q),e\otimes f^{*})$ is a maximal $\gamma$-Brauer pair;

    2. (ii)

$(P,e)$ is a maximal $A$-Brauer pair;

    3. (iii)

$(Q,f)$ is a maximal $B$-Brauer pair.

Finally, it was proved that applying the Brauer construction to
$p$-permutation equivalences at Brauer pairs induces local $p$-permutation
equivalences on multiple levels.

######  Proposition 7.9.

[4, Proposition 11.1] Let $(\Delta(P,\phi,Q),e\otimes f^{*})$ be a
$\gamma$-Brauer pair and set $I:=N_{G}(P,e)$ and $J:=N_{H}(Q,f)$. For every
$g\in I$ there exists a unique element $hC_{H}(Q)\in J/C_{H}(Q)$ such that
$c_{g}\circ\phi=\phi\circ c_{h}:Q\to P$. Similarly for every $h\in J$, there
exists a unique element $gC_{G}(P)\in I/C_{G}(P)$ such that
$c_{g}\circ\phi=\phi\circ c_{h}$. These associations define mutually inverse
group isomorphisms between $I/C_{G}(P)$ and $J/C_{H}(Q)$. The isomorphism
$J/C_{H}(Q)\cong I/C_{G}(P)$ restricts to an isomorphism $QC_{H}(Q)/Q\cong
PC_{G}(P)/P,hC_{H}(Q)\mapsto\phi(h)C_{G}(P)$, for $h\in Q$. The group
$Y:=N_{G\times H}(\Delta(P,\phi,Q),e\otimes f^{*})$ satisfies
$p_{1}(Y)=I,p_{2}(Y)=J,k_{1}(Y)=C_{G}(P),k_{2}(Y)=C_{H}(Q)$.

######  Theorem 7.10.

[4, Theorem 11.4] Let $(\Delta(P,\phi,Q),e\otimes
f^{*})\in\mathcal{BP}(\gamma)$ be a $\gamma$-Brauer pair and set
$I:=N_{G}(P,e)$, $J:=N_{H}(Q,f)$. Suppose $C_{G}(P)\leq S\leq I$ and
$C_{H}(Q)\leq T\leq J$ are intermediate groups related via the isomorphism in
the previous proposition, and set $Y:=N_{S\times H}(\Delta(P,\phi,Q),e\otimes
f^{*})=N_{S\times T}(\Delta(P,\phi,Q))$. Then the element
$e\gamma(P,\phi,Q)f\in T(kY(e\otimes f^{*}))$ after restriction to $Y$
satisfies

$k[C_{G}(P)]e=e\gamma(P,\phi,Q)f\cdot^{Y,Y^{\circ}}_{H}f\gamma(P,\phi,Q)^{*}e\in
T(k[N_{S\times S}(\Delta(P))](e\otimes e^{*})).$

Moreover, the element

$\overline{\gamma}:=\operatorname{Ind}_{Y}^{S\times T}(e\gamma(P,\phi,Q)f)\in
T^{\Delta}(kSe,kTf)$

satisfies $\overline{\gamma}\cdot_{T}\overline{\gamma}^{*}=[kSe]\in
T^{\Delta}(kSe,kSe)$. In particular, if $\gamma$ is a $p$-permutation
equivalence between $A$ and $B$, then $\overline{\gamma}$ is a $p$-permutation
equivalence between $kSe$ and $kTf$.

######  Corollary 7.11.

[4, Proposition 11.5] Let $(\Delta(D,\phi,E),e_{D}\otimes f^{*}_{E})$ be a
maximal $\gamma$-Brauer pair. In particular, $D$ is a defect group of $A$ and
$E$ is a defect group of $B$. For any subgroup $P\leq D$, let $(P,e_{P})$
denote the unique $A$-Brauer pair with $(P,e_{P})\leq(D,e_{D})$ and for any
subgroup $Q\leq E$ let $(Q,f_{Q})$ denote the unique $B$-Brauer pair with
$(Q,f_{Q})\leq(E,f_{E})$. Let $Q\leq E$ and set $P:=\phi(Q)$. Then
$(\Delta(P,\phi,Q),e_{P}\otimes f^{*}_{Q})$ is a $\gamma$-Brauer pair.

  1. (a)

Set $\gamma^{\prime}:=e_{P}\gamma(P,\phi,Q)f_{Q}\in
T^{\Delta}(k[C_{G}(P)]e_{P},k[C_{H}(Q)]f_{Q}).$ $\gamma^{\prime}$ is a
$p$-permutation equivalence between $k[C_{G}(P)]e_{P}$ and $k[C_{H}(Q)]f_{Q}$.
The $k[C_{G}(P)]e_{P}\otimes k[C_{H}(Q)]f^{*}_{Q}$-Brauer pair
$(\Delta(C_{D}(P),\phi,C_{E}(Q)),e_{PC_{D}(P)}\otimes f^{*}_{QC_{E}(Q)})$ is a
$\gamma^{\prime}$-Brauer pair. It is a maximal $\gamma^{\prime}$-Brauer pair
if and only if $P$ is fully $\mathcal{A}$-centralized if and only if $Q$ is
fully $\mathcal{B}$-centralized.

  2. (b)

Set $I,J$ as before, and $\gamma^{\prime\prime}:=\operatorname{Ind}^{I\times
J}_{N_{I\times J}(\Delta(P,\phi,Q)}(e_{P}\gamma(P,\phi,Q)f_{Q})\in
T^{\Delta}(kIe_{P},kJf_{Q})$. $\gamma^{\prime\prime}$ is a $p$-permutation
equivalence between $kIe_{P}$ and $kJf_{Q}$. The $kIe_{P}\otimes
kJf_{Q}^{*}$-Brauer pair
$(\Delta(N_{D}(P),\phi,N_{E}(Q)),e_{N_{D}(P)}\otimes f^{*}_{N_{H}(Q)})$ is a
$\gamma^{\prime\prime}$-Brauer pair. It is a maximal
$\gamma^{\prime\prime}$-Brauer pair if and only if $P$ is fully
$\mathcal{A}$-normalized if and only if $Q$ is fully $\mathcal{B}$-normalized.

##  8\. Brauer pairs for splendid Rickard complexes

Each splendid Rickard complex $C$ induces a $p$-permutation equivalence
$\gamma:=\Lambda(C)$, and therefore has an associated set of $\gamma$-Brauer
pairs. As with $p$-permutation equivalences, in general applying the Brauer
construction at an arbitrary subgroup to $C$ will not yield a local splendid
Rickard complex. We may define a Brauer pair for a splendid Rickard complex in
the same way as we have for $p$-permutation equivalences, treating
contractible complexes as zero objects. In fact, the Lefschetz invariant of a
contractible complex is zero in the corresponding Grothendieck group, but the
converse of the statement rarely holds.

We will show that applying the Brauer construction at the Brauer pairs for the
corresponding $p$-permutation equivalence $\gamma$ do indeed yield local
splendid equivalences. Moreover, we demonstrate that applying the Brauer
construction at any non-Brauer pair results in a contractible complex. The
main result of this paper is the corollary of the two above results, which is
that the Brauer pairs for a splendid Rickard complex and its corresponding
$p$-permutation equivalence are the same.

For this section, suppose $A,B$ are direct summands of group algebras $kG$ and
$kH$ respectively. Let $C$ be a splendid Rickard complex inducing a splendid
equivalence between $A$ and $B$, and set $\gamma:=\Lambda(C)$. Given a Brauer
pair $\omega=(\Delta(P,\phi,Q),e\otimes f^{*})$, denote by $\omega^{\circ}$
the “opposite” Brauer pair $(\Delta(Q,\phi^{-1},P),f\otimes e^{*}).$

######  Definition 8.1.

Let $C$ be a bounded chain complex of $p$-permutation $(A,B)$-bimodules and
set $\omega=(\Delta(P,\phi,Q),e\otimes f^{*})\in\mathcal{BP}^{\Delta}(A,B)$.
We say that $\omega$ is a Brauer pair for $C$ if and only if
$C(\omega):=eC(\Delta(P,\phi,Q))f\not\simeq 0$ as a complex of $k[N_{I\times
J}(\Delta(P,\phi,Q))](e\otimes f^{*})$-modules where $I=N_{G}(P,e)$ and
$J=N_{G}(Q,f)$. Denote the set of $C$-Brauer pairs by $\mathcal{BP}(C)$.

Let $C$ be a splendid Rickard complex. The goal of this section is to
determine the set $\mathcal{B}\mathcal{P}(C)$ of $C$-Brauer pairs. Notice that
the Lefschetz invariant of $C(\omega)$ is $\gamma(\omega)$. Since the
Lefschetz invariant of a contractible complex is zero, it is immediate that
$\mathcal{BP}(\gamma)\subseteq\mathcal{BP}(C)$. However, a noncontractible
complex can have Lefschetz invariant equal to zero, so it is not clear whether
or not these sets are equal.

We will make liberal use of the following standard homological algebra facts.

######  Proposition 8.2.

  1. (a)

Let $\mathcal{A},\mathcal{B}$ be abelian categories, and let
$F,G:\mathcal{A}\to\mathcal{B}$ be naturally isomorphic additive functors. The
induced functors
$\overline{F},\overline{G}:\operatorname{Ch}(\mathcal{A})\to\operatorname{Ch}(\mathcal{B})$
are naturally isomorphic.

  2. (b)

Let $\mathcal{A},\mathcal{B},\mathcal{C}$ be abelian categories, and let
$F,G:\mathcal{A}\times\mathcal{B}\to\mathcal{C}$ be naturally isomorphic
bifunctors which are additive in each component. The induced functors
$\overline{F},\overline{G}:Ch^{b}(\mathcal{A})\times Ch^{b}(\mathcal{B})\to
Bi^{b}(\mathcal{C})$ are naturally isomorphic, where $Bi^{b}(\mathcal{C})$
denotes the category of bounded $\mathcal{C}$-bicomplexes. Postcomposing with
the functor corresponding to taking the total complex yields a natural
isomorphism $\overline{F},\overline{G}:Ch^{b}(\mathcal{A})\times
Ch^{b}(\mathcal{B})\to\operatorname{Ch}^{b}(\mathcal{C})$.

In particular, the isomorphisms given in Theorem 6.7 and Theorem 6.2 extend to
chain complex isomorphisms.

First, we wish to reduce to the case that $A$ and $B$ are blocks. In order to
do this, we will establish an analogue of Proposition 7.7 for splendid Rickard
complexes. The main tool is the following result.

######  Lemma 8.3.

Suppose that $C$ is a splendid Rickard equivalence between sums of blocks $A$
and $B$ and that $(P,e)$ is an $A$-Brauer pair. Let
$\Lambda_{B}\subseteq\Lambda$ be the set of pairs $(\phi,(Q,f))$ where $(Q,f)$
is a $B$-Brauer pair and $\phi:Q\to P$ is a group isomorphism. There exists a
unique $H$-orbit of pairs $(\phi,(Q,f))\in\Lambda_{B}$ with the property that
$eC(\Delta(P,\phi,Q))f$ is not contractible as a complex of
$(k[C_{G}(P)]e,k[C_{H}(Q)]f)$-bimodules.

###### Proof.

Since $C$ is a splendid Rickard equivalence, we have a homotopy equivalence
$C\otimes_{B}C^{*}\simeq A$ of complexes of $(A,A)$-bimodules. We apply the
functor $-(\Delta(P),e\otimes e^{*})$ to this equation. Using Proposition 9.3,
Theorem 6.7(d), and Proposition 3.4, we obtain an isomorphism

$\displaystyle k[C_{G}(P)]e$ $\displaystyle\cong A(\Delta(P),e\otimes e^{*})$
$\displaystyle\simeq\left(C\otimes_{kH}C^{*}\right)(\Delta(P),e\otimes e^{*})$
$\displaystyle\cong\bigoplus_{(\phi,(Q,f))\in\widetilde{\Lambda}}eC(\Delta(P,\phi,Q))f\otimes_{k[C_{H}(Q)]}fC^{*}(\Delta(Q,\phi^{-1},P))e$
$\displaystyle\cong\bigoplus_{(\phi,(Q,f))\in\widetilde{\Lambda}\cap\Lambda_{B}}eC(\Delta(P,\phi,Q))f\otimes_{k[C_{H}(Q)]}(eC(\Delta(P,\phi,Q))f)^{*}.$

in the homotopy category of complexes of
$(k[C_{G}(P)]e,k[C_{G}(P)]e)$-bimodules. The change in index is justified
since $eC(P,\phi,Q)f=0$ whenever $(Q,f)$ is not a $B$-Brauer pair. Since
$k[C_{G}(P)]e$ is an indecomposable bimodule, the corresponding complex
concentrated in degree zero is indecomposable when viewed as an object of the
homotopy category. Using the Krull-Schmidt theorem, we conclude that there is
a unique pair $(\phi,(Q,f))\in\widetilde{\Lambda}\cap\Lambda_{B}$ such that

$eC(\Delta(P,\phi,Q))f\otimes_{k[C_{H}(Q)]}(eC(\Delta(P,\phi,Q))f)^{*}\simeq
k[C_{G}(P)]e.$

It follows from Lemma 5.3 that $eC(\Delta(P,\phi,Q))f\not\simeq 0$ as a
complex of $(k[C_{G}(P)]e,k[C_{H}(Q)]f)$-bimodules. This proves the existence
part of the statement. In order to prove uniqueness, suppose that
$(\phi^{\prime},(Q^{\prime},f^{\prime}))\in\widetilde{\Lambda}\cap\Lambda_{B}$
is any other pair. Set
$C^{\prime}:=eC(\Delta(P,\phi^{\prime},Q^{\prime}))f^{\prime}$ so that we have
$C^{\prime}\otimes_{k[C_{H}(Q^{\prime})]}C^{\prime*}\simeq 0$. Thus,
$C^{\prime}$ is a complex of
$(k[C_{G}(P)]e,k[C_{H}(Q^{\prime})]f^{\prime})$-bimodules satisfying the
hypothesis of Lemma 5.3. From this, it follows that
$eC(\Delta(P,\phi^{\prime},Q^{\prime}))f^{\prime}\simeq 0$, as desired. ∎

The next result shows that a splendid Rickard equivalence $C$ determines a
bijection between the block direct summands of $A$ and $B$. Moreover, $C$ is a
direct sum (in the homotopy category) of splendid Rickard equivalences for
each pair of corresponding blocks and the set of Brauer pairs for $C$ breaks
up as the disjoint union of the sets of Brauer pairs for each summand.

######  Theorem 8.4.

Suppose $A$ and $B$ are sums of blocks and that $C$ is a splendid Rickard
complex for $A$ and $B$. Let $f:\mathcal{I}\to\mathcal{J}$ be the bijection
determined by $\gamma:=\Lambda(C)$ from Proposition 7.7.

  1. (a)

For every $e\in\mathcal{I}$, the element $f_{e}\in\mathcal{J}$ is unique with
the property that $eCf_{e}\not\simeq 0$.

  2. (b)

We have $C=\bigoplus_{e\in\mathcal{I}}eCf_{e}$ in
$K^{b}({}_{A}\mathbf{mod}_{B})$. Moreover, for every $e\in\mathcal{I}$,
$eCf_{e}$ is a splendid Rickard complex for $kGe$ and $kGf_{e}$.

  3. (c)

For every $e\in\mathcal{I}$, the $eCf_{e}$-Brauer pairs are precisely the
$C$-Brauer pairs which lie above $(1,e\otimes f_{e}^{*})$. We have
$\mathcal{BP}(C)=\coprod_{e\in\mathcal{I}}\mathcal{BP}(eCf_{e})$.

###### Proof.

The Lefchetz invariant of $eCf_{e}$ is $e\gamma f_{e}$, which is nonzero by
Proposition 7.7(a). Therefore, $eCf_{e}\not\simeq 0$. By applying Lemma 8.3 to
the splendid Rickard complex $C$ and the $A$-Brauer pair $(1,e)$, we obtain
that there exists a unique $f\in\mathcal{J}$ such that $eCf\not\simeq 0$. By
uniqueness, $f=f_{e}$. This proves (a).

Using (a), we have a direct sum decomposition

$C=\bigoplus_{e\in\mathcal{I}}eCf_{e}$

in the homotopy category of chain complexes of $(A,A)$-bimodules. Since $kGe$
and $kHf_{e}$ are blocks, it suffices by Proposition 5.5 to show just one of
the homotopy equivalences. Multiplying the above equation by $e$ yields
$eC=eCf_{e}$. Thus, we obtain

$\displaystyle kGe=eAe$ $\displaystyle\simeq
e(C\otimes_{B}C^{*})e=eCf_{e}\otimes_{kGf_{e}}(eCf_{e})^{*}.$

This completes the proof of (b). Finally, let
$\lambda=(\Delta(P,\phi,Q),b\otimes c^{*})$ be a $k[G\times H]$-Brauer pair.
Then using Lemma 3.8, we have

$(eCf_{e})(\lambda)=b(eCf_{e})(\Delta(P,\phi,Q))c=b\operatorname{Br}_{P}(e)C(\Delta(P,\phi,Q))\operatorname{Br}_{Q}(f_{e})c.$

As in the proof of Proposition 7.7, it is easily deduced from this equation
that $\lambda$ is a $eCf_{e}$-Brauer pair if and only if $\lambda$ is a
$C$-Brauer pair satisfying $(1,e\otimes f_{e}^{*})\leq\lambda$. Since every
$C$-Brauer pair of $k[G\times H]$ lies above a unique block, we conclude that
the union is disjoint. ∎

Therefore, it suffices to consider only splendid Rickard complexes for block
algebras. From here, assume $A$ and $B$ are block algebras. We next prove a
chain complex-theoretic version of Theorem 7.10.

######  Theorem 8.5.

Suppose that $A$ and $B$ are blocks and $C$ is a splendid Rickard complex for
$A$ and $B$, and set $\gamma:=\Lambda(C)$. Set
$\lambda:=(\Delta(P,\phi,Q),e\otimes f^{*})$ be a $\gamma$-Brauer pair and set
$I:=N_{G}(P,e),J:=N_{H}(Q,f)$. Suppose $C_{G}(P)\leq S\leq I$ and
$C_{H}(Q)\leq T\leq J$ are intermediate groups related via the isomorphism
$I/C_{G}(P)\to J/C_{H}(Q)$ defined in Proposition 7.9, and set $Y:=N_{S\times
H}(\Delta(P,\phi,Q),e\otimes f^{*})$. Then the chain complex
$C(\lambda):=eC(\Delta(P,\phi,Q))f\in\mathrm{Ch}^{b}({}_{kY(e\otimes
f^{*})}\mathbf{triv})$ satisfies

$k[C_{G}(P)]e\simeq
C(\lambda)\mathop{\otimes}\limits_{kH}^{Y,Y^{\circ}}C(\lambda)^{*}\text{ in
}K^{b}({}_{k[N_{S\times S}(\Delta(P))](e\otimes e^{*})}\mathbf{triv})$

and

$k[C_{H}(Q)]f\simeq
C(\lambda)^{*}\mathop{\otimes}\limits_{kG}^{Y^{\circ},Y}C(\lambda)\text{ in
}K^{b}({}_{k[N_{T\times T}(\Delta(Q))](f\otimes f^{*})}\mathbf{triv}).$

In particular, $C(\lambda)$ is a splendid Rickard complex for $k[C_{G}(P)]e$
and $k[C_{H}(Q)]f$, and the chain complex

$\operatorname{Ind}_{Y}^{S\times
T}(C(\lambda))\in\mathrm{Ch}^{b}({}_{kSe}\mathbf{triv}_{kTf})$

is a splendid Rickard complex for $kSe$ and $kTf$.

###### Proof.

We show that $k[C_{G}(P)]e\simeq
C^{\prime}\mathop{\otimes}\limits_{kH}^{Y,Y^{\circ}}C^{\prime*}$, and the dual
homotopy equivalence follows similarly.

First, observe that the Lefchetz invariant of the chain complex
$C(\lambda)\mathop{\otimes}\limits_{kH}^{Y,Y^{\circ}}C(\lambda)^{*}$ is equal
to
$\gamma(\lambda)\mathop{\cdot}\limits_{H}^{Y,Y^{\circ}}\gamma(\lambda)^{*}$.
Since $\lambda\in\mathcal{BP}(\gamma)$, Theorem 7.10 implies that
$\gamma(\lambda)\mathop{\cdot}\limits_{H}^{Y,Y^{\circ}}\gamma(\lambda)^{*}$ is
nonzero. This implies that
$C(\lambda)\mathop{\otimes}\limits_{kH}^{Y,Y^{\circ}}C(\lambda)^{*}$ is not
contractible as a complex of $k[N_{S\times S}(\Delta(P)](e\otimes
e^{*})$-modules.

Next, we apply the Brauer construction at $\Delta(P)\leq G\times G$ to the
homotopy equivalence of $(A,A)$-bimodules $C\otimes_{B}C^{*}\simeq A$. Using
Lemma 3.8, we obtain a homotopy equivalence

$kC_{G}(P)e\simeq e[C\otimes_{B}C^{*}](\Delta(P))e,$

of chain complexes of $k[N_{S\times S}(\Delta(P))](e\otimes e^{*})$-modules.
Using the notation in Theorem 6.7(d), choose a set of $S\times H$ orbit
representatives $\widehat{\Lambda}\subseteq\Lambda$ which contains
$(\phi,(Q,f))$. Using Theorem 6.7 together with Proposition 8.2 we obtain a
homotopy equivalence

$\displaystyle k[C_{G}(P)]e$ $\displaystyle\simeq
e(C\otimes_{B}C^{*})(\Delta(P))e$
$\displaystyle\cong\bigoplus_{\tau=(\psi,(R,b))\in\widehat{\Lambda}}\operatorname{Ind}^{N_{S\times
S}(\Delta(P))}_{\Delta(I(\tau))(C_{G}(P)\times
1)}(C(\tau)\mathop{\otimes}\limits_{kH}^{X(\tau),X(\tau)^{\circ}}C(\tau)^{*})$

of chain complexes of $k[N_{S\times S}(\Delta(P))](e\otimes e^{*})$-modules.
Notice that when $\tau=(\phi,(Q,f))$, we have $X(\tau)=Y$,
$C(\tau)=C(\lambda)$, and, by Proposition 7.9, $I(\tau):=N_{(S,\phi,J)}=S$. It
follows that from the latter that

$\Delta(I(\tau))(C_{G}(P)\times 1)=\Delta(S)(C_{G}(P)\times 1)=N_{S\times
S}(\Delta(P)).$

Thus, $C(\lambda)\otimes_{H}^{Y,Y^{\circ}}C(\lambda)^{*}$ is a noncontractible
direct summand of the complex on the right hand side. On the other hand,
$k[C_{G}(P)]e$ is an indecomposable $k[N_{S\times S}(\Delta(P))]$-bimodule,
hence indecomposable and noncontractible as a chain complex. By the Krull-
Schmidt theorem, precisely one of the terms in the direct sum must be
noncontractible. We conclude that $k[C_{G}(P)]e\simeq
C(\lambda)\otimes_{H}^{Y,Y^{\circ}}C(\lambda)^{*}$.

Therefore,

$\operatorname{Ind}^{S\times S}_{N_{S\times
S}(\Delta(P))}(k[C_{G}(P)]e)\simeq\operatorname{Ind}^{S\times S}_{N_{S\times
S}(\Delta(P))}(C(\lambda)\otimes_{kH}^{Y,Y^{\circ}}C^{*}(\lambda^{\circ}))$

Observe $kC_{G}(P)$ is a transitive $k[N_{S\times S}(\Delta(P))]$-permutation
module with $\Delta(S)$ stabilizing $1\in C_{G}(P)$, hence

$kC_{G}(P)\cong\operatorname{Ind}_{\Delta(S)}^{N_{S\times S}(\Delta(P))}(k),$

and therefore

$\operatorname{Ind}^{S\times S}_{N_{S\times S}(\Delta(P))}(k[C_{G}(P)]e)\cong
e(\operatorname{Ind}^{S\times S}_{N_{S\times S}(\Delta(P))}(kC_{G}(P)))e\cong
e\operatorname{Ind}^{S\times S}_{\Delta(S)}(k)e\cong kSe.$

So we have

$\operatorname{Ind}^{S\times S}_{N_{S\times
S}(\Delta(P))}(C(\lambda)\otimes_{kH}^{Y,Y^{\circ}}C^{*}(\lambda^{\circ}))\simeq\operatorname{Ind}^{S\times
S}_{N_{S\times S}(\Delta(P))}(k[C_{G}(P)]e)\cong kSe.$

We show that the left-hand side is isomorphic to

$\operatorname{Ind}_{Y}^{S\times
T}(C(\lambda))\otimes_{kT}(\operatorname{Ind}^{S\times T}_{Y}C(\lambda))^{*}.$

First, notice that $Y=N_{S\times J}(\Delta(P,\phi,Q))$ using the definitions.
Then Lemma 2.4(c) implies that $p_{1}(Y)=N_{(S,\phi,J)}$ and
$p_{2}(Y)=N_{(J,\phi^{-1},S)}$. Since $\lambda\in\mathcal{BP}(\gamma)$ and $S$
and $T$ are related under the isomorphism $I/C_{G}(P)\to J/C_{G}(Q)$,
Proposition 7.9 implies that $p_{1}(Y)=N_{(S,\phi,J)}=S$ and
$p_{2}(Y)=N_{(J,\phi^{-1},S)}=T$. In particular, $Y\leq S\times T$ so the
claim makes sense.

Next, we have the identifications

$(\operatorname{Ind}_{Y}^{S\times
T}C(\lambda))^{*}=(\operatorname{Ind}_{Y}^{S\times
T}(eC(P,\phi,Q)f))^{*}\cong\operatorname{Ind}_{Y^{\circ}}^{T\times
S}(fC^{*}(Q,\phi^{-1},P)e).$

Then applying Theorem 6.2, we have

$\operatorname{Ind}^{S\times
T}_{Y}(C(\lambda))\otimes_{kT}\operatorname{Ind}^{T\times
S}_{Y^{o}}(C^{*}(\lambda^{\circ}))\cong\bigoplus_{t\in[p_{2}(Y)\backslash
T/p_{1}(Y^{\circ})]}\operatorname{Ind}^{S\times
S}_{Y\ast{}^{(t,1)}Y^{\circ}}\big{(}eC(P,\phi,Q)f\otimes_{kT}^{Y,{}^{(t,1)}Y}fC^{*}(Q,\phi^{-1},P)e\big{)}.$

However, $p_{2}(Y)=p_{1}(Y^{\circ})=T$, therefore the indexing set in the
direct sum has cardinality one, and the corresponding term is
$\operatorname{Ind}^{S\times
S}_{Y}\big{(}C(\lambda)\otimes_{kT}^{Y,Y^{\circ}}C^{*}(\lambda^{\circ})\big{)}$.
We conclude that

$kSe\simeq\operatorname{Ind}^{S\times
S}_{Y}\left(C(\lambda)\otimes_{kT}^{Y,Y^{\circ}}C^{*}(\lambda^{\circ})\right)\cong\operatorname{Ind}^{S\times
T}_{Y}(C(\lambda))\otimes_{kT}(\operatorname{Ind}_{Y}^{S\times
T}(C(\lambda)))^{*}.$

The second homotopy equivalence follows similarly, and therefore
$\operatorname{Ind}^{S\times T}_{Y}(C(\lambda))$ is a splendid Rickard complex
for $kSe$ and $kTf$. ∎

We next show that in addition, if $\omega:=(\Delta(P,\phi,Q),e\otimes f^{*})$
is not a $\gamma$-Brauer pair, then $C(\omega)\simeq 0$ as a complex of
$k[N_{G\times H}(\omega)]$-modules.

######  Lemma 8.6.

  1. (a)

Let $C$ be a complex of $(A,B)$-bimodules and let $C^{\prime}$ be a complex of
$(B,B^{\prime})$-bimodules for some $k$-algebras $A,B,B^{\prime}$. If
$C^{*}\otimes_{A}C\simeq B$ and $C\otimes_{B}C^{\prime}\simeq 0$, then
$C^{\prime}\simeq 0$.

  2. (b)

Let $H\leq G$ and let $C$ be a chain complex of $kH$-modules. Then
$\operatorname{Ind}^{G}_{H}C\simeq 0$ if and only if $C\simeq 0$.

###### Proof.

  1. (a)

We have

$C^{\prime}\cong
B\otimes_{B}C^{\prime}\simeq(C^{*}\otimes_{A}C)\otimes_{B}C^{\prime}=C^{*}\otimes_{A}(C\otimes_{B}C^{\prime})\simeq
C^{*}\otimes_{A}0\cong 0.$

.

  2. (b)

The Mackey formula is natural and therefore extends to chain complexes. We
have

$\operatorname{Res}^{G}_{H}\operatorname{Ind}^{G}_{H}C\cong\bigoplus_{g\in[H\backslash
G/H]}\operatorname{Ind}^{H}_{H\cap{}^{g}H}\operatorname{Res}^{H}_{H\cap{}^{g}H}C.$

Suppose $\operatorname{Ind}^{G}_{H}C\simeq 0$. Then the left-hand side is
contractible since restriction preserves contractibility. All summands on the
right-hand side are contractible as well, since direct summands of
contractible chain complexes are again contractible. Since there is a coset
representative in $[H\backslash G/H]$ belonging to $H$, $C$ is isomorphic to a
direct summand of $\operatorname{Res}^{G}_{H}\operatorname{Ind}^{G}_{H}C$, and
thus is contractible. The reverse direction is clear since
$\operatorname{Ind}_{H}^{G}$ is an additive functor.

∎

######  Theorem 8.7.

Suppose $C$ is a splendid Rickard complex for $A$ and $B$, with
$\gamma=\Lambda(C)$. If $\omega:=(\Delta(R,\psi,Q),d\otimes
f^{*})\in\mathcal{BP}^{\Delta}(A,B)$ is not a $\gamma$-Brauer pair, then
$C(\omega):=dC(\Delta(R,\psi,Q))f$ is contractible as a chain complex of
$k[N_{G\times H}(\Delta(R,\psi,Q),d\otimes f^{*})](d\otimes f^{*})$-modules.
In particular, $\omega$ is not a $C$-Brauer pair.

###### Proof.

There exists an $A$-Brauer pair $(P,e)$ and an isomorphism $\phi:Q\to P$ such
that $\lambda:=(\Delta(P,\phi,Q),e\otimes f^{*})$ is a $\gamma$-Brauer pair.
Set $I=N_{G}(P,e),\ J=N_{H}(Q,f)$, $K=N_{G}(R,d)$, $X=N_{G\times H}(\lambda)$,
and $Y=N_{G\times H}(\omega)^{\circ}$. By Theorem 8.5,
$\operatorname{Ind}_{X}^{I\times J}(C(\lambda))$ is a splendid Rickard complex
for $kIe$ and $kJf$. In order to prove the claim, it suffices to show that

$\operatorname{Ind}_{X}^{I\times
J}(C(\lambda))\mathop{\otimes}\limits_{kJ}\operatorname{Ind}_{Y}^{J\times
K}(C(\omega)^{*})\simeq 0.$

If so, Lemma 8.6(a) implies that $\operatorname{Ind}_{Y}^{J\times
K}(C(\omega)^{*})\simeq 0$. Then Lemma 8.6(b) implies that
$C(\omega)^{*}\simeq 0$. By taking duals, we obtain $C(\omega)\simeq 0$, as
desired.

In order to prove the above claim, we apply Theorem 6.2 to obtain

$\operatorname{Ind}_{X}^{I\times
J}(C(\lambda))\mathop{\otimes}\limits_{kJ}\operatorname{Ind}_{Y}^{J\times
K}(C(\omega)^{*})\cong\bigoplus_{t\in[p_{2}(X)\backslash
J/p_{1}(Y)]}\operatorname{Ind}_{X\ast{}^{(t,1)}Y}^{I\times
K}\left(C(\lambda)\mathop{\otimes}\limits_{kJ}^{X,{}^{(t,1)}Y}{}^{(t,1)}(C(\omega)^{*})\right).$

Thus, it suffices to show that every complex on the right hand side is
contractible. Let $t\in J$. Set $\omega_{t}={}^{(1,t)}\omega$. Notice that
$\omega_{t}^{\circ}=(\Delta(Q,c_{t}\circ\psi^{-1},R),f\otimes d^{*})$ since
$t\in J$. For ease of notation, write

$\lambda\circ\omega_{t}^{\circ}:=(\Delta(P,\phi\circ
c_{t}\circ\psi^{-1},R),e\otimes d^{*}).$

Let $\Omega$ denote the set of triples $(\sigma,(S,a),\tau)$ where $(S,a)$ is
a $kH$-Brauer pair and $\tau:R\to S$ and $\sigma:S\to P$ are isomorphisms for
which $\sigma\circ\tau=\phi\circ c_{t}\circ\psi^{-1}$. Choose a set
$\hat{\Omega}$ of $N_{I\times K}(\Delta(P,\phi\circ
c_{t}\circ\psi^{-1},R)\times H$-orbit representatives of $\Omega$ which
contains $(\phi,(Q,f),c_{t}\circ\psi^{-1})$. Then Theorem 6.7(c) gives an
isomorphism

$\bigoplus_{\nu\in\hat{\Omega}}\operatorname{Ind}_{X(\nu)\ast
Y(\nu)}^{N_{I\times K}(\Delta(P,\phi\circ
c_{t}\circ\psi^{-1},R))}\left(C(\nu)\mathop{\otimes}\limits_{kH}^{X(\nu),Y(\nu)}C^{*}(\nu)\right)\cong(C\otimes_{kH}C^{*})(\lambda\circ\omega_{t}^{\circ}),$

where for $\nu=(\sigma,(S,a),\tau))$, we set $X(\nu):=N_{I\times
H}(\Delta(P,\sigma,S),e\otimes a^{*})$, $Y(\nu):=N_{H\times
K}(\Delta(S,\tau,R),a\otimes d^{*})$, $C(\nu):=eC(\Delta(P,\sigma,S))a$ and
$C^{*}(\nu):=aC^{*}(\Delta(S,\tau,R))d$. In particular, when
$\nu=(\phi,(Q,f),c_{t}\circ\psi^{-1})$, we have $X(\nu)=X$,
$Y(\nu)={}^{(t,1)}Y$, $C(\nu)=C(\lambda)$ and, by Proposition 3.4,
$C^{*}(\nu)\cong{}^{(t,1)}(C(\omega)^{*})$. Thus, since
$(\phi,(Q,f),c_{t}\circ\psi^{-1})\in\hat{\Omega}$, we conclude that

$\operatorname{Ind}_{X\ast{}^{(t,1)}Y}^{N_{I\times K}(\Delta(P,\phi\circ
c_{t}\circ\psi^{-1},R))}\left(C(\lambda)\mathop{\otimes}_{kH}^{X,{}^{(t,1)}Y}{}^{(t,1)}(C(\omega)^{*})\right)$

is homotopy equivalent to a direct summand of

$(C\otimes_{kH}C^{*})(\lambda\circ\omega^{\circ})\simeq
A(\lambda\circ\omega_{t}^{\circ}).$

We claim $A(\lambda\circ\omega_{t}^{\circ})=eA(\Delta(P,\phi\circ
c_{t}\circ\psi^{-1},R))d=0$. Suppose to the contrary that
$A(\lambda\circ\omega_{t}^{\circ})\neq 0$. Then
$\lambda\circ\omega_{t}^{\circ}$ is a Brauer pair for $A$. However, the Brauer
pairs of any block algebra (regarded as bimodule) are up to $G\times
G$-conjugacy given by pairs of the form $(\Delta(S),e_{S}\otimes e_{S}^{*})$
for some $S\leq G$ and some block $e_{S}$. So there exist $(g_{1},g_{2})\in
G\times G$ for which

$(\Delta(P,\phi\circ c_{t}\circ\psi^{-1},R),e\otimes
d^{*})={}^{(g_{1},g_{2})}(\Delta(S),e_{S}\otimes
e_{S}^{*})=(\Delta({}^{g_{1}}S,c_{g_{1}g_{2}^{-1}},{}^{g_{2}}S),{}^{g_{1}}e_{S}\otimes{}^{g_{2}}e_{S}^{*}).$

It is a straightforward computation that

${}^{g_{1}}e_{S}=e,{}^{g_{2}}e_{S}=d,P={}^{g_{1}}S,R={}^{g_{2}}S,\phi\circ
c_{t}=c_{g_{1}g_{2}^{-1}}\circ\psi.$

In this case,

$\displaystyle{}^{(g_{2}g_{1}^{-1},1)}\lambda$
$\displaystyle={}^{(g_{2}g_{1}^{-1},1)}(\Delta(P,\phi,Q),e\otimes f^{*})$
$\displaystyle=(\Delta({}^{g_{2}g_{1}^{-1}}P,c_{g_{2}g_{1}^{-1}}\circ\phi,Q),{}^{g_{2}g_{1}^{-1}}e\otimes
f^{*})$ $\displaystyle=(\Delta(R,\psi\circ c_{t^{-1}},Q),d\otimes
f^{*})=\omega_{t}.$

This implies that $\omega_{t}$ is a $\gamma$-Brauer pair since it is conjugate
to the $\gamma$-Brauer pair $\lambda$. However, $\omega$ and $\omega_{t}$ are
also conjugate, so we conclude that $\omega$ is a $\gamma$-Brauer pair, a
contradiction. Thus $A(\lambda\circ\omega_{t}^{\circ})=0$. Since direct
summands of contractible complexes are contractible, we obtain

$\operatorname{Ind}_{X\ast{}^{(t,1)}Y}^{N_{I\times K}(\Delta(P,\phi\circ
c_{t}\circ\psi^{-1},R))}\left(C(\lambda)\mathop{\otimes}\limits_{kJ}^{X,{}^{(t,1)}Y}{}^{(t,1)}(C(\omega)^{*})\right)\simeq
0.$

Finally, by applying the functor $\operatorname{Ind}_{N_{I\times
K}(\Delta(P,\phi\circ c_{t}\circ\psi^{-1},R))}^{I\times K}(-)$ to this
equation, we obtain

$\operatorname{Ind}_{X\ast{}^{(t,1)}Y}^{I\times
K}\left(C(\lambda)\mathop{\otimes}\limits_{kJ}^{X,{}^{(t,1)}Y}{}^{(t,1)}(C(\omega)^{*})\right)\simeq
0$

for all $t\in J$. This completes the proof.

∎

As a corollary of the preceding theorem, we obtain that the Brauer pairs for a
splendid Rickard equivalence and the Brauer pairs for its Lefschetz invariant
coincide.

######  Corollary 8.8.

Let $C$ be a splendid Rickard complex between $A$ and $B$ and let
$\gamma:=\Lambda(C)$ be its corresponding $p$-permutation equivalence. Then
$\mathcal{BP}(C)=\mathcal{BP}(\gamma)$. In particular,

  1. (a)

$\mathcal{BP}(C)$ is a $G\times H$-stable ideal in the poset of $A\otimes
B^{*}$-Brauer pairs.

  2. (b)

If $A$ and $B$ are blocks, then any two maximal $C$-Brauer pairs are $G\times
H$-conjugate.

  3. (c)

The following are equivalent for $(\Delta(P,\phi,Q),e\otimes
f^{*})\in\mathcal{B}\mathcal{P}(C)$:

    1. (i)

$(\Delta(P,\phi,Q),e\otimes f^{*})$ is a maximal $C$-Brauer pair.

    2. (ii)

$(P,e)$ is a maximal $A$-Brauer pair.

    3. (iii)

$(Q,f)$ is a maximal $B$-Brauer pair.

  4. (d)

Suppose $A$ and $B$ are blocks and $(\Delta(D,\phi,E),e_{D}\otimes f_{E}^{*})$
is a maximal $C$-Brauer pair. Let $\mathcal{A}$ denote the fusion system of
$A$ associated to $(D,e_{D})$ and let $\mathcal{B}$ denote the fusion system
of $B$ associated to $(E,f_{E})$. Then $\phi:E\to D$ is an isomorphism of the
fusion systems $\mathcal{B}$ and $\mathcal{A}$.

###### Proof.

By Proposition 7.7 and Theorem 8.4, we may assume that $A$ and $B$ are blocks.
Then the fact that $\mathcal{BP}(C)\subseteq\mathcal{BP}(\gamma)$ follows from
Theorem 8.7. The other inclusion
$\mathcal{BP}(\gamma)\subseteq\mathcal{BP}(C)$ was already established at the
beginning of this section. Claims (a)-(c) above follow immediately from [4,
Theorem 10.11]. Claim (d) follows from [4, Theorem 11.2]. ∎

Another easy corollary we obtain is that splendid Rickard complexes induce
local normalizer splendid equivalences.

######  Corollary 8.9.

Let $C$ be a splendid Rickard complex between blocks $A\subseteq kG$ and
$B\subseteq kH$ and let $(\Delta(P,\phi,Q),e\otimes f^{*})$ be a $C$-Brauer
pair. Set $I=N_{G}(P,e)$ and $J=N_{H}(Q,f)$ and let $\hat{e}$ and $\hat{f}$
denote the unique block idempotents of $k[N_{G}(P)]$ and $k[N_{H}(Q)]$ which
cover $e$ and $f$, respectively. Then,

$\hat{e}k[N_{G}(P)]e\otimes_{kI}\operatorname{Ind}^{I\times J}_{N_{I\times
J}(\Delta(P,\phi,Q))}\big{(}eC(\Delta(P,\phi,Q))f\big{)}\otimes_{kJ}fk[N_{H}(Q)]\hat{f}$

is a splendid Rickard complex for blocks algebras $k[N_{G}(P)]\hat{e}$ and
$k[N_{H}(Q)]\hat{f}$.

###### Proof.

Given any Brauer pair $(P,e)$ of $kG$, we have a Morita equivalence
${}_{k[N_{G}(P)]\hat{e}}\mathbf{mod}\to{}_{kIe}\mathbf{mod}$, where $\hat{e}$
is the unique block of $kN_{G}(P)$ covering $e$. Explicitly,
$\hat{e}=\operatorname{tr}^{N_{G}(P)}_{I}(e)$, and the Morita equivalence is
induced by the $(kIe,k[N_{G}(P)]\hat{e})$-bimodule $e[N_{G}(P)]\hat{e}$ (see
[15]). The result now follows since

$\operatorname{Ind}^{I\times J}_{N_{I\times
J}(\Delta(P,\phi,Q))}\big{(}eC(\Delta(P,\phi,Q))f\big{)}$

is a splendid equivalence for $kIe$ and $kJf$. ∎

##  9\. Structural properties of splendid Rickard complexes

In this section, we deduce an analogous result to the following theorem of
Boltje and Perepelitsky.

######  Theorem 9.1.

[4, Theorem 14.1, 14.3] Let $\gamma$ be a $p$-permutation equivalence between
blocks $A$ and $B$ of $kG$ and $kH$, respectively. Let $N$ be an
indecomposable $(A,B)$-bimodule appearing in $\gamma$. Then
$\mathcal{B}\mathcal{P}(N)\subseteq\mathcal{B}\mathcal{P}(\gamma)$. Moreover,
there is a unique indecomposable $(A,B)$-bimodule $M$ appearing in $\gamma$
with multiplicity $\pm 1$ whose vertex is of the form $\Delta(D,\phi,E)$ where
$D$ is a defect group of $A$. If $M\not\cong N$, then
$\mathcal{B}\mathcal{P}(N)\subset\mathcal{B}\mathcal{P}(M)$.

The module $M$ as above is called the maximal module of $\gamma$. We begin
proving the analogous statement with a block-wise reformulation of a theorem
in [14].

We first state an oft-used lemma. The proof of the following folklore lemma is
from correspondence from P. Webb.

######  Lemma 9.2.

Let $C$ be a chain complex of $kG$-modules and suppose that for some
$i\in\mathbb{Z}$, we have decompositions of $kG$-modules

$\displaystyle C_{i}$ $\displaystyle=U_{i}\oplus V_{i}$ $\displaystyle
C_{i-1}$ $\displaystyle=U_{i-1}\oplus V_{i-1}$

Let $\eta_{i}:U_{i}\to C_{i}$ be inclusion and $\pi_{i-1}:C_{i-1}\to U_{i-1}$
be projection. Suppose that $\pi_{i-1}d_{i}\eta_{i}:U_{i}\to U_{i-1}$ is an
isomorphism. Then there is a direct sum decomposition $C\cong D\oplus W$ as
complexes, where

$W\cong 0\to U_{i}\xrightarrow{\operatorname{id}}U_{i}\to 0$

is contractible.

###### Proof.

The map $d_{i}\eta_{i}:U_{i}\to C_{i-1}$ is injective since
$\pi_{i-1}d_{i}\eta_{i}$ is, and we define

$W_{i-1}=d_{i}\eta_{i}(U_{i})\cong U_{i}.$

Now $C_{i-1}=W_{i-1}\oplus V_{i-1}$ because $W_{i-1}$ projects isomorphically
onto $U_{i-1}$. We see that

$0\to U_{i}\xrightarrow{d_{i}\mid_{U_{i}}}W_{i-1}\to 0$

is a contractible subcomplex of $C$. This shows that we may assume

$\displaystyle C_{i}$ $\displaystyle=U_{i}\oplus V_{i}$ $\displaystyle
C_{i-1}$ $\displaystyle=U_{i-1}\oplus V_{i-1}$

where $0\to U_{i}\xrightarrow{d_{i}\mid_{U_{i}}}U_{i-1}\to 0$ is a
contractible subcomplex of $C$, adjusting the rest of the notation to fit the
assumption. We define

$X_{i}=\\{(-(d_{i}\mid_{U_{i}})^{-1}\pi^{i-1}d_{i}(x),x)\mid x\in
V_{i}\\}\subseteq C_{i},$

and $C_{i}=U_{i}\oplus X_{i}$ because $X_{i}$ projects isomorphically to
$V_{i}$. We calculate

$\displaystyle d_{i}(X_{i})$
$\displaystyle=\\{d_{i}(-d_{i}\mid_{U_{i}})^{-1}d_{i}(x)+d_{i}(x)\mid x\in
V_{i}\\}$ $\displaystyle=\\{-\pi_{i-1}d_{i}(x)+d_{i}(x)\mid x\in V_{i}\\}$
$\displaystyle\subseteq V_{i-1}$

Thus $C$ is the direct sum of the complexes

$0\to U_{i}\xrightarrow{d_{i}\mid_{U_{i}}}U_{i-1}\to 0$

and

$\dots\to
C_{i+1}\xrightarrow{\pi_{X_{i}}d_{i+1}}X_{i}\xrightarrow{d_{i}\mid_{X_{i}}}V_{i-1}\xrightarrow{d_{i-1}\mid_{V_{i-1}}}C_{i-2}\to\cdots.$

∎

######  Lemma 9.3.

Let $M$ and $N$ be $p$-permutation $kG$-modules and let $f:M\to N$ be a
$kG$-module homomorphism. Assume that $N=N_{1}\oplus\cdots\oplus N_{k}$ is a
decomposition into indecomposable summands. For each $1\leq i\leq k$, let
$(P_{i},e_{i})$ be a maximal $N_{i}$-Brauer pair. Then $f$ is split surjective
if and only if $e_{i}\cdot f(P_{i})$ is surjective for all $1\leq i\leq k$.

###### Proof.

The forward implication is immediate. We induct on the number of
indecomposable summands of $N$. For the base case, suppose that $N$ is
indecomposable. Suppose that $(P,e)$ is a maximal $N$-Brauer pair and that
$e\cdot f(P)$ is surjective. By Proposition 7.2, $P$ is a vertex of $N$ and
therefore $N(P)$ is indecomposable. Since $(P,e)$ is an $N$-Brauer pair,
$eN(P)\neq 0$ which implies that $eN(P)=N(P)$. Since $e\cdot f(P)$ is
surjective, it follows that $f(P)$ is surjective as well. By [13, 5.10.7], $f$
is split surjective.

For the inductive step, suppose $N=N_{1}\oplus\cdots\oplus N_{k}$. Consider
the projection of $f$ onto $N_{1}$, then by the base case, it follows that
this is split surjective. Let $M=M_{1}\oplus M^{\prime}$, with the inclusion
of $M_{1}$ into $M$ composed with $f$ and projection onto $N_{1}$ an
isomorphism, and $M^{\prime}$ a complement of $M_{1}$. It follows by Lemma 9.2
that we may find a corresponding direct sum decomposition of $N=N_{1}\oplus
N^{\prime}$, with $f=f_{1}\oplus f^{\prime}$ satisfying $f_{1}:M_{1}\to N_{1}$
an isomorphism and $f^{\prime}:M^{\prime}\to N^{\prime}$ a homomorphism. Since
$N^{\prime}$ is $p$-permutation with one less indecomposable summand, the
inductive hypothesis completes the proof.

∎

######  Theorem 9.4.

Let $C$ be a chain complex of $p$-permutation $kG$-modules, and let
$\mathcal{X}$ be a subset of $\mathcal{B}\mathcal{P}(kG)$ which is $G$-stable
and closed (under $\leq$). The following are equivalent:

  1. (a)

For all $(P,e)\not\in\mathcal{X}$, $e\cdot C(P)$ is acyclic.

  2. (b)

There exists a chain complex $D$ with $C\simeq D$ such that every trivial
source module occurring up to isomorphism in $D$ has a maximal Brauer pair
contained in $\mathcal{X}$.

If $C$ is indecomposable and satisfies (a), then $C$ itself has the property
in (b).

###### Proof.

(b) implies (a) is straightforward. Let $n$ be the minimum integer for which
$C_{n}\neq 0$. For each nonzero $C_{n}$, write $C_{n}=X_{n}\oplus Y_{n}$,
where $X_{n}$ consists of all indecomposable summands which have a maximal
pair contained in $\mathcal{X}$. Thus, $Y_{n}$ consists of all indecomposable
summands of $C_{n}$ which have a maximal Brauer pair which is not contained in
$\mathcal{X}$. We construct a chain complex $D$ identical to $C$, except in
degree $n+1$, where we set $D_{n+1}=X_{n}\oplus C_{n+1}$, and add the identity
map on $X_{n}$ to the differential of $D_{n+1}$, $d_{n+1}$. Notice that
$f\cdot X_{n}(Q)=0$ for all $(Q,f)\not\in\mathcal{X}$ since $\mathcal{X}$ is a
$G$-stable ideal. Therefore, $f\cdot C(Q)\cong f\cdot D(Q)$. But since $f\cdot
C(Q)$ is acyclic for all $(Q,f)\not\in\mathcal{X}$ by assumption, so is
$f\cdot D(Q)$. Therefore $d_{n+1}$ composed with projection onto $Y_{n}$ is
split surjective by Lemma 9.3, and by construction, it follows that $d_{n+1}$
is split surjective. Thus, we have that $D\cong
D^{\prime}\oplus(C_{n}\xrightarrow{\sim}C_{n})$, where $D^{\prime}$ has lowest
nonzero degree $n+1$.

Note that $C$ is homotopy equivalent to a shift of the mapping cone of the
chain complex homomorphism $D^{\prime}\to X_{n}$. Moreover, if
$(Q,f)\not\in\mathcal{X}$, $f\cdot C(Q)\cong f\cdot D(Q)\simeq f\cdot
D^{\prime}(Q)$, and these complexes are by assumption acyclic. We now perform
an inductive argument as follows. If the length of $C$ is one, then (b) holds,
since there is only one nonzero term which must vanish after applying the
Brauer construction at all $(Q,f)\not\in\mathcal{X}$, and the rest follows by
the previous lemma.

Now we assume (a) implies (b) holds for complexes of length at most $n$, and
suppose $C$ has length $n+1$. If the length of $C$ is $n+1$, then $D^{\prime}$
has length $n$. Since $f\cdot D^{\prime}(Q)$ is also acyclic for all
$(Q,f)\not\in\mathcal{X}$, $D^{\prime}$ contains only modules with maximal
Brauer pair contained in $\mathcal{X}$. Since $C$ is homotopy equivalent to a
shift of the mapping cone $D^{\prime}\to X_{n}$, and both $D^{\prime}$ and
$X_{n}$ (as complex) are homotopy equivalent to complexes with all degrees
containing only trivial source modules with maximal Brauer pairs contained in
$\mathcal{X}$, we conclude $C$ also satisfies this property, by invariance of
the mapping cone under homotopy equivalences.

Lastly, suppose that $C$ is indecomposable. Then there exists a contractible
complex $C^{\prime}$ such that $C\oplus C^{\prime}\cong D$. Hence, $C$ is
isomorphic to a direct summand of $D$ and therefore $C$ has the desired
property. ∎

######  Corollary 9.5.

Let $C$ be an indecomposable splendid Rickard complex for block algebras
$A\subseteq kG$ and $B\subseteq kH$. Every trivial source bimodule occurring
in $C$ has a maximal Brauer pair contained in $\mathcal{B}\mathcal{P}(C)$. In
particular, the set consisting of all Brauer pairs for every trivial source
bimodule appearing in $C$ and the set of $C$-Brauer pairs coincide.

###### Proof.

By Corollary 8.8, $\mathcal{B}\mathcal{P}(C)$ is a $G\times H$-stable ideal in
$\mathcal{B}\mathcal{P}(k[G\times H])$. Thus, by the preceding theorem, every
trivial source bimodule occuring in $C$ has a maximal Brauer pair contained in
$\mathcal{B}\mathcal{P}(C)$. Denoting the set of all Brauer pairs which occur
in any trivial source bimodule in $C$ by $\mathcal{X}$, it follows that
$\mathcal{X}\subseteq\mathcal{B}\mathcal{P}(C)$. On the other hand, if
$(\Delta(P,\phi,Q),e\otimes f^{*})$ is a $C$-Brauer pair, then
$eC(\Delta(P,\phi,Q))f\neq 0$. Thus, there exists a trivial source bimodule
$M$ appearing in $C$ such that $(\Delta(P,\phi,Q),e\otimes
f^{*})\in\mathcal{B}\mathcal{P}(M)$. Thus,
$\mathcal{B}\mathcal{P}(C)\subseteq\mathcal{X}$. ∎

By Proposition 5.4, the preceding result applies to any splendid Rickard
equivalence between blocks after removing contractible direct summands.
Moreover, by using Theorem 8.4 one obtains an obvious analogue for a splendid
Rickard equivalence between sums of blocks. Finally, we note that Corollary
7.11 also holds for $C$-Brauer pairs as well.

Acknowledgements: The authors would like to thank their advisor, Robert
Boltje, for the many hours of guidance and support which made this paper
possible. They would also like to thank Peter Webb for communicating and
providing a proof of Lemma 9.2.

## References

  * [1] J. Alperin and M. Broué.  Local methods in block theory.  Annals of Math, 110(2):143–157, 1979.
  * [2] M. Aschbacher, R. Kessar, and B. Oliver.  Fusion Systems in Algebra and Topology.  Cambridge University Press, Cambridge, 2011.
  * [3] R. Boltje and S. Danz.  A ghost ring for the left-free double burnside ring and an application to fusion systems.  Advances in Math, 229:1688–1733, 2012.
  * [4] R. Boltje and P. Perepletsky.  $p$-permutation equivalences between blocks of group algebras.  arXiv:2007.09253 [math.GR], 2020.
  * [5] R. Boltje and B. Xu.  On $p$-permutation equivalences: betwen rickard equivalences and isotypies.  Transations of the American Mathematical Society, 360:5067–5087, 2008.
  * [6] S. Bouc.  Biset Functors for Finite Groups.  Springer-Verlag, Berlin, 2010.
  * [7] S. Bouc.  Bisets as categories and tensor product of induced bimodules.  Applied Categorical Structures, 18:517–521, 2010.
  * [8] M. Broué.  On scott modules and $p$-permutation modules: An approach through the brauer morphism.  Proceedings of the American Mathematical Society, 93(3):401–408, 1985.
  * [9] M. Broue.  Rickard equivalences and block theory.  In International Conference on Group Theory “Groups 93”, 1993\.
  * [10] M. Broué and L. Puig.  Charaters and local structure in $g$-algebras.  Journal of Algebra, 63:306–317, 1980.
  * [11] M. Linckelmann.  On derived equivalences and local structure between blocks of finite groups.  Turkish Journal of Mathematics, 22(1):93–108, 1998.
  * [12] M. Linckelmann.  Simple fusion systems and the solomon 2-group.  Journal of Algebra, 296:385–401, 2006.
  * [13] M. Linckelmann.  The Block Theory of Finite Group Algebras, Volume 1.  Cambridge University Press, Cambridge, 2018.
  * [14] S. K. Miller.  Endotrivial complexes.  arXiv:2309.12138 [math.RT], 2023.
  * [15] H. Nagao and Y. Tsushima.  Representations of Finite Groups.  Academic Press, Boston, 1989.
  * [16] J. Rickard.  Splendid equivalences: Derived categories and permutation modules.  Proceedings of the London Mathematical Society, 72:331–358, 1996\.