The generating hypothesis for the stable module category of a $p$-group

TL;DR

This paper investigates Freyd's generating hypothesis within the stable module category of finite p-groups, establishing it holds only for groups of order 2 or 3, and provides equivalent conditions for the hypothesis.

Contribution

It proves the generating hypothesis holds exclusively for cyclic groups of order 2 or 3 and identifies conditions equivalent to the hypothesis.

Findings

The hypothesis holds only for G ≅ C_2 or C_3.

Provides conditions equivalent to the generating hypothesis.

Characterizes when the hypothesis is valid for finite p-groups.

Abstract

Freyd's generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial finite p-group G if and only if G is either C_2 or C_3. We also give various conditions which are equivalent to the generating hypothesis.