Freyd's generating hypothesis for groups with periodic cohomology

Abstract

Let $G$ be a finite group and let $k$ be a field whose characteristic $p$ divides the order of $G$. Freyd's generating hypothesis for the stable module category of $G$ is the statement that a map between finite-dimensional $kG$-modules in the thick subcategory generated by $k$ factors through a projective if the induced map on Tate cohomology is trivial. We show that if $G$ has periodic cohomology then the generating hypothesis holds if and only if the Sylow $p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions that are equivalent to the GH for groups with periodic cohomology.

11 pages, final version, to appear in Canadian Mathematical Bulletin

References in corpus (1)

• Ghosts in modular representation theory

Cited by in corpus (2)

• Ghost numbers of Group Algebras
• Ghost numbers of group algebras II