Finite Generation of Tate Cohomology
نویسندگان
چکیده
Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M , we conjecture that if the Tate cohomology Ĥ ∗ (G, M) of G with coefficients in M is finitely generated over the Tate cohomology ring Ĥ ∗ (G, k), then the support variety VG(M) of M is equal to the entire maximal ideal spectrum VG(k). We prove various results which support this conjecture. The converse of this conjecture is established for modules in the connected component of k in the stable Auslander-Reiten quiver for kG, but it is shown to be false in general. It is also shown that all finitely generated kGmodules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology.
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