Tate cohomology

Tate cohomology fuses homology and cohomology of a finite group into a single sequence indexed by all integers, positive and negative. Ordinary cohomology lives in nonnegative degrees and homology in nonpositive ones; in degree zero they nearly meet but disagree by the norm map. Tate's idea is to splice the two together precisely at that seam, repairing the mismatch with the norm, to produce one elegant theory that runs in both directions and treats the two halves on equal footing.

For a finite group G and a G-module A, the Tate cohomology groups Ĥ^n(G, A) are defined for all n in Z. For n >= 1 they equal ordinary cohomology H^n(G, A); for n <= -2 they equal homology, Ĥ^n(G, A) = H_{-n-1}(G, A). At the seam, Ĥ^0(G, A) = A^G / N(A) is the fixed points modulo the image of the norm N = sum over g of g, and Ĥ^{-1}(G, A) = ker(N) / I·A is the norm-kernel modulo the augmentation submodule. A complete (doubly infinite) resolution makes all this uniform.

Tate cohomology is the natural home of class field theory: the reciprocity isomorphism, the Tate–Nakayama theorem, and the computation of the Brauer group all live here. Its signature feature is periodicity for cyclic groups: if G is cyclic then Ĥ^n(G, A) ≅ Ĥ^{n+2}(G, A) for all n, so the entire doubly infinite sequence is determined by just two groups, Ĥ^0 and Ĥ^{-1}, whose ratio is the Herbrand quotient.