caa a22 4500 606160566 CHVBK 20210128100630.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s10468-014-9476-9 doi (NATIONALLICENCE)springer-10.1007/s10468-014-9476-9 Ghost Numbers of Group Algebras [Elektronische Daten] [J. Christensen, Gaohong Wang] 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. Springer Science+Business Media Dordrecht, 2014 Tate cohomology nationallicence Stable module category nationallicence p -group nationallicence Generating hypothesis nationallicence Ghost map nationallicence Christensen J. Department of Mathematics, University of Western Ontario, N6A 5B7, London, ON, Canada aut Wang Gaohong Department of Mathematics, University of Western Ontario, N6A 5B7, London, ON, Canada aut Algebras and Representation Theory Springer Netherlands 18/1(2015-02-01), 1-33 1386-923X 18:1<1 2015 18 10468 https://doi.org/10.1007/s10468-014-9476-9 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10468-014-9476-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Christensen J. Department of Mathematics, University of Western Ontario, N6A 5B7, London, ON, Canada aut NATIONALLICENCE 700 1- Wang Gaohong Department of Mathematics, University of Western Ontario, N6A 5B7, London, ON, Canada aut NATIONALLICENCE 773 0- Algebras and Representation Theory Springer Netherlands 18/1(2015-02-01), 1-33 1386-923X 18:1<1 2015 18 10468