Freyd's generating hypothesis with almost split sequences

Details Freyd's generating hypothesis with almost split sequences Freyd's generating hypothesis with almost split sequences

2008-06-12

arxiv.org/abs/0806.2165v2

Proc. Amer. Math. Soc. 137 (2009) 2575-2580

Mathematics

Representation Theory

6 pages, fixed minor typos, final version, to appear in Proc. Amer. Math. Soc

Scientific paper

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 k factors through a projective 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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