On stable equivalences with endopermutation source

Markus Linckelmann

doi:10.1016/j.jalgebra.2015.03.014

Abstract

We show that a bimodule between block algebras which has a fusion stable endopermutation module as a source and which induces Morita equivalences between centralisers of nontrivial subgroups of a defect group induces a stable equivalence of Morita type; this is the converse to a theorem of Puig. The special case where the source is trivial has long been known by many authors. The earliest instance of a result deducing a stable equivalence of Morita type from local Morita equivalences with possibly nontrivial endopermutation source is due to Puig, in the context of blocks with abelian defect groups with a Frobenius inertial quotient. The present note is motivated by an application, due to Biland, to blocks of finite groups with structural properties known to hold for hypothetical minimal counterexamples to the Zp⁎-Theorem.

Highlights

IntroductionIf for all nontrivial subgroups Q of P the bimodule MQ induces a Morita equivalence between kCG(Q)eQ and kCH (Q)fQ, M and its dual M ∗ induce a stable equivalence of Morita type between OGb and OHc. The existence of canonical bimodules MQ satisfying Endk(MQ) ∼= (EndO(eQM fQ))(∆Q) in this Theorem is due to Biland [5, Theorem 3.15]
Let p be a prime and O a complete discrete valuation ring having a residue field k of characteristic p; we allow the case O = k
If for all nontrivial subgroups Q of P the bimodule MQ induces a Morita equivalence between kCG(Q)eQ and kCH (Q)fQ, M and its dual M ∗ induce a stable equivalence of Morita type between OGb and OHc

Summary

Introduction

If for all nontrivial subgroups Q of P the bimodule MQ induces a Morita equivalence between kCG(Q)eQ and kCH (Q)fQ, M and its dual M ∗ induce a stable equivalence of Morita type between OGb and OHc. The existence of canonical bimodules MQ satisfying Endk(MQ) ∼= (EndO(eQM fQ))(∆Q) in this Theorem is due to Biland [5, Theorem 3.15]. In the statement of Theorem 1.1, restricting attention to fully centralised subgroups is necessary in order to ensure that A(∆Q) and kCG(Q)eQ are Morita equivalent Another technical difference between the statements of the two theorems is that iM j will be an endopermutation O∆Q-module, while this is not clear for eQM fQ because indecomposable O∆Q-summands with vertices strictly smaller than ∆Q might not be compatible. We will outline how this simplifies the proof in the Remark 5.1 below

Background material on blocks and almost source algebras

On fusion-stable endopermutation modules

Bimodules with fusion-stable endopermutation source