Freyd's generating hypothesis for groups with periodic cohomology

Details Freyd's generating hypothesis for groups with periodic cohomology Freyd's generating hypothesis for groups with periodic cohomology

2007-10-17

arxiv.org/abs/0710.3356v5

Mathematics

Representation Theory

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

Scientific paper

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.

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