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Freyd's Generating Hypothesis for Groups with Periodic Cohomology

  Published:2011-05-14
 Printed: Mar 2012
  • Sunil K. Chebolu,
    Department of Mathematics, Illinois State University, Normal, IL 61761, U.S.A.
  • J. Daniel Christensen,
    Department of Mathematics, University of Western Ontario, London, ON N6A 5B7
  • Ján Mináč,
    Department of Mathematics, University of Western Ontario, London, ON N6A 5B7
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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 Tate cohomology, generating hypothesis, stable module category, ghost map, principal block, thick subcategory, periodic cohomology
MSC Classifications: 20C20, 20J06, 55P42 show english descriptions Modular representations and characters
Cohomology of groups
Stable homotopy theory, spectra
20C20 - Modular representations and characters
20J06 - Cohomology of groups
55P42 - Stable homotopy theory, spectra
 

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