Algebras and Representation Theory

, Volume 18, Issue 3, pp 849–880 | Cite as

Ghost Numbers of Group Algebras II

Article

Abstract

We study several closely related invariants of the group algebra kG of a finite group. The basic invariant is the ghost number, which measures the failure of the generating hypothesis and involves finding non-trivial composites of maps each of which induces the zero map in Tate cohomology (“ghosts”). The related invariants are the simple ghost number, which considers maps which are stably trivial when composed with any map from a suspension of a simple module, and the strong ghost number, which considers maps which are ghosts after restriction to every subgroup of G. We produce the first computations of the ghost number for non-p-groups, e.g., for the dihedral groups at all primes, as well as many new bounds. We prove that there are close relationships between the three invariants, and make computations of the new invariants for many families of groups.

Keywords

Tate cohomology Stable module category Generating hypothesis Ghost map 

Mathematics Subject Classification (2010)

Primary 20C20; Secondary 18E30 20J06 55P99 

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Western OntarioLondonCanada

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