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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $\\Z[G]$ be its group ring.\nA $G$-module is a left [[module|0NZ8JA25WK]] over $\\Z[G]$.\nLet $\\Z$ be the right $\\Z[G]$-module with the trivial action $n \\cdot g = n$.\nLet $P_\\bullet \\to \\Z$ be a [[projective resolution|HK2AA1YSQ2]] of $\\Z$ by right $\\Z[G]$-modules.\nLet $M$ be a $G$-module and let $n \\ge 0$.\nThe $n$-th *homology group* of $G$ with coefficients in $M$ is\n$$H_n(G, M) = H_n(P_\\bullet \\otimes_{\\Z[G]} M) = \\mathrm{Tor}^{\\Z[G]}_n(\\Z, M),$$\nthe homology of the chain complex obtained by the [[tensor product|F0HQXQ0HCD]] with $M$.\n\n## Remarks\n\nThe groups do not depend on the choice of projective resolution, up to canonical isomorphism.\nIn degree $0$, $H_0(G, M) = M_G$ is the module of coinvariants, the quotient of $M$ by the subgroup generated by all $gm - m$.\nIn degree $1$, $H_1(G, \\Z) \\cong G / [G, G]$.\n\nExample: $G = \\Z$ written multiplicatively with generator $t$, so that $\\Z[G] = \\Z[t, t^{-1}]$.\nThe sequence $0 \\to \\Z[G] \\xrightarrow{\\ t - 1\\ } \\Z[G] \\to \\Z \\to 0$ is a free resolution of $\\Z$.\nTensoring with the trivial module $\\Z$ gives the complex $0 \\to \\Z \\xrightarrow{0} \\Z \\to 0$.\nSo $H_0(\\Z, \\Z) = \\Z$, $H_1(\\Z, \\Z) = \\Z$, and $H_n(\\Z, \\Z) = 0$ for $n \\ge 2$.\n\nNon-example: the groups $\\mathrm{Tor}^{\\Z}_n(\\Z, M)$, taken over $\\Z$ instead of $\\Z[G]$, forget the action.\nThey vanish for $n \\ge 1$, while $H_1(\\Z, \\Z) = \\Z$.\n",
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