= Tate cohomology of a finite group
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{title2=$\widehat H^r(G,M)$}
= Tate cohomology
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For a finite <group> $G$, Tate cohomology joins <group cohomology> and <group homology> through the norm map $N_G=\sum_{g\in G}g$. For $r\geq1$ it equals ordinary <group cohomology>; for $r\leq-2$ it equals $H_{-r-1}(G,M)$. The middle terms are $\widehat H^0(G,M)=M^G/N_GM$ and $\widehat H^{-1}(G,M)=\ker(N_G)/I_GM$, where $I_GM$ is generated by $gm-m$. For the trivial module $\mathbb Z$, $\widehat H^{-2}(G,\mathbb Z)=G^{\mathrm{ab}}$. With multiplicative coefficients the norm is the product of the translates.
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