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  "body": "\n## Definition\n\nLet $G$ be a finite group and let $M$ be a $G$-module, a [[module|0NZ8JA25WK]] over the group ring $\\Z[G]$.\nLet $M^G$ be the submodule of elements fixed by $G$, and let $M_G = M / I_G M$ be the module of coinvariants, where $I_G M$ is generated by all $gm - m$.\nThe norm map $N \\colon M \\to M$ is $N(m) = \\sum_{g \\in G} gm$; it induces a map $\\bar N \\colon M_G \\to M^G$.\nThe *Tate cohomology groups* $\\hat H^n(G, M)$, for $n \\in \\Z$, are:\n\n1. $\\hat H^n(G, M) = H^n(G, M)$ for $n \\ge 1$, the group cohomology.\n2. $\\hat H^0(G, M) = M^G / N(M)$.\n3. $\\hat H^{-1}(G, M) = \\ker(\\bar N \\colon M_G \\to M^G)$.\n4. $\\hat H^{-n-1}(G, M) = H_n(G, M)$ for $n \\ge 1$, the [[group homology|CFAG3VRXEA]].\n\n## Remarks\n\nEquivalently, $\\hat H^n(G, M)$ is the cohomology of $\\Hom_G(P_\\bullet, M)$ for a complete resolution $P_\\bullet$.\nA complete resolution is an exact complex of projective $\\Z[G]$-modules indexed by $\\Z$ that agrees with a projective resolution of $\\Z$ in nonnegative degrees.\n\nExample: let $G$ be [[cyclic|0KVGH0J5B3]] of order $m$ and let $M = \\Z$ with trivial action.\nThen $M^G = M_G = \\Z$ and $N$ is multiplication by $m$.\nSo $\\hat H^0(G, \\Z) = \\Z / m\\Z$ and $\\hat H^{-1}(G, \\Z) = \\ker(m \\colon \\Z \\to \\Z) = 0$.\nFor cyclic $G$, $\\hat H^n(G, M) \\cong \\hat H^{n+2}(G, M)$ for every $n$ and every $M$.\n\nExample: for $M = \\Z[G]$, all groups $\\hat H^n(G, \\Z[G])$ vanish.\nIn degree $0$, both $M^G$ and $N(M)$ equal $\\Z \\cdot \\sum_{g \\in G} g$.\n\nNon-example: for an infinite group $G$, the sum $\\sum_{g \\in G} gm$ is not defined, so the definition needs $G$ finite.\n",
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