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# Ghosts and Strong Ghosts in the Stable Category

Published:2016-08-03
Printed: Dec 2016
• Jon F. Carlson,
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
• Sunil K. Chebolu,
Department of Mathematics, Illinois State University, Normal, IL 61790 USA
• Ján Mináč,
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada
 Format: LaTeX MathJax PDF

## Abstract

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p\gt 0$. 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 variations of ghost 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 ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps which mimic ghosts in high degrees.
 Keywords: Tate cohomology, ghost maps, stable module category, almost split sequence, periodic cohomology
 MSC Classifications: 20C20 - Modular representations and characters 20J06 - Cohomology of groups 55P42 - Stable homotopy theory, spectra

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