In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by John Tate (1952, p. 297), and are used in class field theory.
Definition If G {\displaystyle G} is a finite group and A {\displaystyle A} a G {\displaystyle G} -module, then there is a natural map N {\displaystyle N} from H 0 ( G , A ) {\displaystyle H_{0}(G,A)} to
H 0 ( G , A ) {\displaystyle H^{0}(G,A)} taking a representative a {\displaystyle a} to ∑ g ∈ G g a {\displaystyle \sum _{g\in G}ga} (the sum over all G {\displaystyle G} -conjugates of a {\displaystyle a} ). The Tate cohomology groups H ^ n ( G , A ) {\displaystyle {\hat {H}}^{n}(G,A)} are defined by:
H ^ n ( G , A ) = H n ( G , A ) {\displaystyle {\hat {H}}^{n}(G,A)=H^{n}(G,A)} for n ≥ 1 {\displaystyle n\geq 1} ,
H ^ 0 ( G , A ) = coker N = {\displaystyle {\hat {H}}^{0}(G,A)=\operatorname {coker} N=} quotient of H 0 ( G , A ) {\displaystyle H^{0}(G,A)} by norms of elements of A {\displaystyle A} ,
H ^ − 1 ( G , A ) = ker N = {\displaystyle {\hat {H}}^{-1}(G,A)=\ker N=} quotient of norm 0 {\displaystyle 0} elements of A {\displaystyle A} by principal elements of A {\displaystyle A} ,
H ^ n ( G , A ) = H − n − 1 ( G , A ) {\displaystyle {\hat {H}}^{n}(G,A)=H_{-n-1}(G,A)} for n ≤ − 2 {\displaystyle n\leq -2} .
Properties If
0 ⟶ A ⟶ B ⟶ C ⟶ 0 {\displaystyle 0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0}
is a short exact sequence of G-modules, then we get the usual long exact sequence of Tate cohomology groups:
⋯ ⟶ H ^ n ( G , A ) ⟶ H ^ n ( G , B ) ⟶ H ^ n ( G , C ) ⟶ H ^ n + 1 ( G , A ) ⟶ H ^ n + 1 ( G , B ) ⋯ {\displaystyle \cdots \longrightarrow {\hat {H}}^{n}(G,A)\longrightarrow {\hat {H}}^{n}(G,B)\longrightarrow {\hat {H}}^{n}(G,C)\longrightarrow {\hat {H}}^{n+1}(G,A)\longrightarrow {\hat {H}}^{n+1}(G,B)\cdots }
If A is an induced G module (meaning, induced from a module for the trivial group) then all Tate cohomology groups of A vanish. The zeroth Tate cohomology group of A is (Fixed points of G on A)/(Obvious fixed points of G acting on A) where by the "obvious" fixed point we mean those of the form ∑ g a {\displaystyle \sum ga} . In other words, the zeroth cohomology group in some sense describes the non-obvious fixed points of G acting on A. The Tate cohomology groups are characterized by the three properties above.
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