Cohomology of permutation representation in negative dimensions (Q1911640)

Let the finite group \(G\) act on the finite set \(X\). The author splices together a standard chain complex for \(X\) and a standard cochain complex for \(X\) to get an acyclic \(G\)-complex \(C_*(X; G)\), infinite in both directions. Then if \(A\) is a \(G\)-module he defines \(H^n(X;G,A)\) to be the cohomology in dimension \(n\) of \(\text{Hom}_G(C_*(X;G),A)\), but he does not give any qualitative properties that might motivate the construction. Note that one gets Tate cohomology by taking \(X = G\) with action by left multiplication. The author makes boundary maps explicit in terms of stabilizer groups and orbit representatives.