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Freyd's Generating Hypothesis for Groups with Periodic Cohomology |
Published:2011-05-14
Printed: Mar 2012
Printed: Mar 2012
- Sunil K. Chebolu,
- J. Daniel Christensen,
- Ján Mináč,
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.
| Keywords: | Tate cohomology, generating hypothesis, stable module category, ghost map, principal block, thick subcategory, periodic cohomology |
| MSC Classifications: |
20C20 - Modular representations and characters 20J06 - Cohomology of groups 55P42 - Stable homotopy theory, spectra |