Finite generation of Tate cohomology

Jon F. Carlson
Department of Mathematics 
University of Georgia 
Athens, GA 30602, USA
jfc@math.uga.edu


Sunil K. Chebolu
Department of Mathematics 
University of Western Ontario 
London, ON N6A 5B7, Canada
schebolu@uwo.ca

Jan Minac
Department of Mathematics
University of Western Ontario
London, ON N6A 5B7, Canada
minac@uwo.ca

Abstract:
Let G be a finite group and let k be a field of characteristic p. If M is a finitely generated indecomposable non-projective kG-module, we conjecture that the Tate cohomology of G with coefficients in M is finitely generated over the Tate cohomology ring of G if and  only if the support variety V_G(M) of M is equal to the entire maximal ideal spectrum V_G(k). We prove various results all of which support this conjecture. It is also shown that all finitely generated kG-modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology.