Freyd’s generating hypothesis with almost split sequences

Abstract

Freyd’s generating hypothesis for the stable module category of a non-trivial finite group G G is the statement that a map between finitely generated k G kG -modules that belongs to the thick subcategory generated by the field k k factors through a projective module if the induced map on Tate cohomology is trivial. In this paper we show that Freyd’s generating hypothesis fails for k G kG when the Sylow p p -subgroup of G G has order at least 4 4 using almost split sequences. By combining this with our earlier work, we obtain a complete answer to Freyd’s generating hypothesis for the stable module category of a finite group. We also derive some consequences of the generating hypothesis.