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

Brauer-friendly modules and slash functors