Freyd's Generating Hypothesis for Groups with Periodic Cohomology

Chebolu, Sunil K. ; Christensen, J. Daniel ; Mináč, Ján

Canadian mathematical bulletin, Tome 55 (2012) no. 1, pp. 48-59

Résumé

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 $\text{GH}$ for groups with periodic cohomology.

Détail

DOI : 10.4153/CMB-2011-090-5

Mots-clés : 20C20, 20J06, 55P42, Tate cohomology, generating hypothesis, stable module category, ghost map, principal block, thick subcategory, periodic cohomology

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