On the Depth of Cohomology Modules

We study the cohomology modules Hi(G,R) of a p‐group G acting on a ring R of characteristic p, for i>0. In particular, we are interested in the Cohen–Macaulay property and the depth of Hi(G,R) regarded as an RG‐module. We first determine the support of Hi(G,R), which turns out to be independent of i. Then we study the Cohen–Macaulay property for H1(G,R). Further results are restricted to the special case that G is cyclic and R is the symmetric algebra of a vector space on which G acts. We determine the depth of Hi(G,R) for i odd and obtain results in certain cases for i even. Along the way, we determine the degrees in which the transfer map TrG R →RG has non‐zero image.