The generating hypothesis for the stable module category of a $p$-group

David J. Benson, Sunil K. Chebolu, J. Daniel Christensen, Jan Minac

Published 2006-11-13, updated 2007-01-19Version 3

Freyd's generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial finite p-group G if and only if G is either C_2 or C_3. We also give various conditions which are equivalent to the generating hypothesis.