Abstract: Given a finite group G, a theorem of Benson, Carlson and Robinson states: there exists a positive integer r=r(G) such that for any commutative ring R of coefficients and any RG-module M, if the Tate cohomology HH^n(G, M) = 0 for r + 1 consecutive values of n then HH^n(G, M) = 0 for all n positive and negative.
The proof gives an explicit construction of an r that works. Namely, suppose e_1, e_2, ..., e_c are homogenous generators of the cohomology ring HH^*(G,Z). Then one can choose r as the sum of the deg(e_i -1). Alternately, if the group G has a faithful complex representation of degree n, one can take r=(n-1)2. In neither case is the r one obtains expected to be best possible. In recent joint work with Cohen and Nakano we used homological methods and algebraic topology to compute, in many cases, Young module cohomology HH^i(S_n, Y^l) for the symmetric group S_n and a Young module Y^l (l a partition). We discuss two related problems. The first is to compute the smallest degree i >= 0 such that HH^i(S_d, Y^l) is nonzero. This computation leads to some Young modules that have nonzero cohomology, but nevertheless have very large vanishing ranges. Thus the second problem is an attempt to use Young modules to realize the ``maximal gap", in the sense of the BCR result, and thus obtain lower bounds for the ideal value of r(S_n).