Tate cohomology

Summary

Tate cohomology joins the cohomology and the homology of a finite group $G$ into one sequence of groups $\hat H^n(G, M)$ indexed by all integers $n$. It agrees with ordinary cohomology in positive degrees and with homology in degrees at most $-2$. In the two middle degrees it is modified with the norm map. A short exact sequence of modules gives a long exact sequence infinite in both directions. Tate cohomology is a basic tool of class field theory and of Galois cohomology, and for cyclic groups it is periodic of period $2$.