On stable equivalences with endopermutation source
Markus Linckelmann

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 a converse to 
a theorem of Puig. The special case where the source is trivial has long been known by many 
authors. The earliest instance for 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.