Ghost Numbers of Group Algebras II
- 88 Downloads
We study several closely related invariants of the group algebra k G 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.
KeywordsTate cohomology Stable module category Generating hypothesis Ghost map
Mathematics Subject Classification (2010)Primary 20C20; Secondary 18E30 20J06 55P99
Unable to display preview. Download preview PDF.
- 1.Alperin, J.L.: Local representation theory Cambridge studies in advanced mathematics, vol. 11. Cambridge University Press, Cambridge (1986)Google Scholar
- 3.Benson, D.J.: Cohomology of modules in the principal block of a finite group. New York J. Math. 1, 196–205 (1994/95)Google Scholar
- 7.Carlson, J.F.: Modules and group algebras Lectures in Mathematics, ETH Zürich. Notes by Ruedi Suter. Birkhäuser Verlag, Basel (1996)Google Scholar
- 9.Carlson, J.F., Chebolu, S.K., Mináč, J.: Strong ghost maps in the stable module category. PreprintGoogle Scholar
- 15.Christensen, J.D., Wang, G.: Ghost numbers of group algebras. Algebras and Representation Theory, 1–33. (2014). doi: 10.1007/s10468-014-9476-9
- 16.Freyd, P.: Stable homotopy. In: Proceedings of the conference on categorical algebra (La Jolla, California, 1965), pp 121–172. Springer, New York (1966)Google Scholar