Freyd's generating hypothesis with almost split sequences

TL;DR

This paper investigates Freyd's generating hypothesis in the stable module category of finite groups, demonstrating its failure for groups with Sylow p-subgroups of order at least 4 using almost split sequences, and providing a complete characterization.

Contribution

It proves the failure of Freyd's generating hypothesis for certain finite groups and completes the classification using almost split sequences and prior results.

Findings

Freyd's generating hypothesis fails for groups with Sylow p-subgroups of order ≥ 4

Complete classification of the hypothesis's validity for finite groups

Derived consequences of the generating hypothesis in modular representation theory

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 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.