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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Freyd's generating hypothesis with almost split sequences


Authors: Jon F. Carlson, Sunil K. Chebolu and Ján Minác
Journal: Proc. Amer. Math. Soc. 137 (2009), 2575-2580
MSC (2000): Primary 20C20, 20J06; Secondary 55P42
Published electronically: February 6, 2009
MathSciNet review: 2497468
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Abstract: Freyd's generating hypothesis for the stable module category of a non-trivial finite group $ G$ is the statement that a map between finitely generated $ kG$-modules that belongs to the thick subcategory generated by the field $ k$ factors through a projective module if the induced map on Tate cohomology is trivial. In this paper we show that Freyd's generating hypothesis fails for $ kG$ when the Sylow $ p$-subgroup of $ G$ has order at least $ 4$ using almost split sequences. By combining this with our earlier work, we obtain a complete answer to Freyd's generating hypothesis for the stable module category of a finite group. We also derive some consequences of the generating hypothesis.


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Additional Information

Jon F. Carlson
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: jfc@math.uga.edu

Sunil K. Chebolu
Affiliation: Department of Mathematics, Illinois State University, Normal, Illinois 61790
Email: schebol@ilstu.edu

Ján Minác
Affiliation: Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada
Email: minac@uwo.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09826-8
PII: S 0002-9939(09)09826-8
Keywords: Tate cohomology, generating hypothesis, stable module category, ghost map, almost split sequence.
Received by editor(s): June 12, 2008
Received by editor(s) in revised form: October 21, 2008
Published electronically: February 6, 2009
Additional Notes: The first author is partially supported by a grant from the NSF
The third author is supported by the NSERC
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.