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 | References | Similar Articles | Additional Information

Abstract: Freyd's generating hypothesis for the stable module category of a non-trivial finite group is the statement that a map between finitely generated -modules that belongs to the thick subcategory generated by the field 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 when the Sylow -subgroup of has order at least 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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Freyd's generating hypothesis for groups with periodic cohomology.

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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:
https://doi.org/10.1090/S0002-9939-09-09826-8

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.