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  "body": "\n## Definition\n\nLet $R$ be a commutative [[Noetherian ring|0670T59HGE]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be an [[$R$-module|0NZ8JA25WK]].\nThe *$I$-torsion functor* is\n$$\\Gamma_I(M) = \\set{m \\in M : I^n m = 0 \\text{ for some } n \\ge 1}.$$\nIt is a left exact functor from $R$-modules to $R$-modules.\nFor $i \\ge 0$, the *$i$-th local cohomology module* $H^i_I(M)$ is the $i$-th right derived functor of $\\Gamma_I$ at $M$.\nExplicitly, choose an injective resolution $0 \\to M \\to E^0 \\to E^1 \\to \\cdots$ by [[injective modules|85H7PRHDPZ]].\nThen $H^i_I(M)$ is the $i$-th cohomology of the complex $0 \\to \\Gamma_I(E^0) \\to \\Gamma_I(E^1) \\to \\cdots$.\n\n## Remarks\n\nOne has $H^0_I(M) = \\Gamma_I(M)$, and a short exact sequence of modules gives a long exact sequence of local cohomology.\nEquivalently, $H^i_I(M) \\cong \\varinjlim_n \\operatorname{Ext}^i_R(R/I^n, M)$, a direct limit of [[Ext modules|4P7GBWJ4RR]].\nIf $I = (x_1, \\ldots, x_n)$, then $H^i_I(M)$ is also the cohomology of the Čech complex $0 \\to M \\to \\bigoplus_i M_{x_i} \\to \\bigoplus_{i\u003cj} M_{x_i x_j} \\to \\cdots \\to M_{x_1 \\cdots x_n} \\to 0$.\nExample: for $R = K[x]$ and $I = (x)$, the Čech complex of $R$ is $0 \\to R \\to R_x \\to 0$, so $H^0_I(R) = 0$ and $H^1_I(R) = K[x, x^{-1}]/K[x]$.\nThe functor $\\Gamma_I$ is not right exact: for $0 \\to K[x] \\xrightarrow{x} K[x] \\to K \\to 0$ with $I = (x)$ one gets $0 \\to 0 \\to K$, and the last map is not onto.\nIf $(R, \\mathfrak{m})$ is local and $M \\ne 0$ is finitely generated, then $H^i_{\\mathfrak{m}}(M) = 0$ for $i \u003c \\operatorname{depth} M$ and $i \u003e \\dim M$, and it is nonzero for $i = \\operatorname{depth} M$ and $i = \\dim M$.\n",
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