Ghosts and Strong Ghosts in the Stable Module Category

TL;DR

This paper investigates ghost maps in the stable module category of finite groups over a field, exploring their properties, detection methods, and variations, including strong forms of the generating hypothesis and behavior under functors.

Contribution

It introduces and analyzes variations of ghost maps, characterizes groups satisfying a strong generating hypothesis, and shows ghost detection on finite Tate cohomology ranges.

Findings

Ghost maps vanish in specific subgroup cases

Ghosts can be detected on finite Tate cohomology ranges

Certain maps mimic ghosts in high degrees

Abstract

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow $p$-subgroup of $G$ is cyclic of order 2 or 3. In this paper we introduce and study some variations of ghosts maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd's generating hypothesis and show that ghost can be detected on a finite range of Tate cohomology. We also consider maps which mimic ghosts in high degrees.