NDLI: Ghost Numbers of Group Algebras

Content Provider	Springer Nature Link
Author	Wang, Gaohong Christensen, J. Daniel
Copyright Year	2014
Abstract	Motivated by Freyd’s famous unsolved problem in stable homotopy theory, the generating hypothesis for the stable module category of a finite group is the statement that if a map in the thick subcategory generated by the trivial representation induces the zero map in Tate cohomology, then it is stably trivial. It is known that the generating hypothesis fails for most groups. Generalizing work done for p-groups, we define the ghost number of a group algebra, which is a natural number that measures the degree to which the generating hypothesis fails. We describe a close relationship between ghost numbers and Auslander-Reiten triangles, with many results stated for a general projective class in a general triangulated category. We then compute ghost numbers and bounds on ghost numbers for many families of p-groups, including abelian p-groups, the quaternion group and dihedral 2-groups, and also give a general lower bound in terms of the radical length, the first general lower bound that we are aware of. We conclude with a classification of group algebras of p-groups with small ghost number and examples of gaps in the possible ghost numbers of such group algebras.
Ending Page	33
Page Count	33
Starting Page	1
File Format	PDF
ISSN	1386923X
e-ISSN	15729079
Journal	Algebras and Representation Theory
Issue Number	1
Volume Number	18
Language	English
Publisher	Springer Netherlands
Publisher Date	2014-07-09
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Stable module category Derived categories, triangulated categories Modular representations and characters Associative Rings and Algebras Ghost map p-group Non-associative Rings and Algebras Cohomology of groups Commutative Rings and Algebras Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Tate cohomology Generating hypothesis
Content Type	Text
Resource Type	Article
Subject	Mathematics

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