Finite generation of Tate cohomology
Jon F. Carlson Department of Mathematics University of Georgia Athens, GA 30602, USA
Sunil K. Chebolu Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada
Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada
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.
arXiv: Experimental full text search
Greenlees: Tate cohomology in commutative algebra
NCG (Algebra and noncommutative geometry), AAG (Arithmetic algebraic geometry), RT (Groups and representation theory)
Something on Freyd’s generating hypothesis: http://www.math.uiuc.edu/K-theory/0814
Greenlees: Tate cohomology in axiomatic stable homotopy theory
arXiv:1209.4888 Tate and Tate-Hochschild Cohomology for finite dimensional Hopf Algebras from arXiv Front: math.KT by Van C. Nguyen Let A be any finite dimensional Hopf algebra over a field k. We generalize the notion of Tate cohomology for A, which is defined in both positive and negative degrees, and compare it with the Tate-Hochschild cohomology of A that was presented by Bergh and Jorgensen. We introduce cup products that make the Tate and Tate-Hochschild cohomology of A become graded rings. We establish the relationship between these rings, which turns out to be similar to that in the ordinary non-Tate cohomology case. As an example, we explicitly compute the Tate-Hochschild cohomology for a finite dimensional (cyclic) group algebra. In another example, we compute both the Tate and Tate-Hochschild cohomology for a Taft algebra, in particular, the Sweedler algebra H_4.
http://mathoverflow.net/questions/12782/tate-cohomology-via-stable-categories
Salomonsson on products in negative Tate cohomology: http://front.math.ucdavis.edu/1007.3355
nLab page on Tate cohomology