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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an $R$-[[module|0NZ8JA25WK]].\nFor a [[prime ideal|00MHHZ9SNE]] $\\mathfrak p$, let $M_{\\mathfrak p}$ be the [[localization|68E0F8N94E]] of $M$ at $\\mathfrak p$.\nThe *support* of $M$ is the following subset of the [[spectrum|FB45BHHA2J]] $\\operatorname{Spec} R$:\n$$\\operatorname{Supp} M = \\set{\\mathfrak p \\in \\operatorname{Spec} R : M_{\\mathfrak p} \\ne 0}.$$\n\n## Remarks\n\nIf $M$ is [[finitely generated|05QY1VEBYD]], then $\\operatorname{Supp} M = V(\\operatorname{Ann} M)$, the set of primes that contain the [[annihilator|HB4XDKJTW8]] $\\operatorname{Ann} M = \\set{a \\in R : aM = 0}$.\nIn particular the support of a finitely generated module is closed in the [[Zariski topology|8SQK7RTB63]].\nBy the [[local property|3S58E70Z7Z]] of vanishing, $\\operatorname{Supp} M = \\emptyset$ if and only if $M = 0$.\n\nExample: let $R = \\Z$ and $M = \\Z/6\\Z$.\nFor a prime $p \\ne 2, 3$, the integer $6$ lies outside $p\\Z$, and $m/s = 6m/(6s) = 0$, so $M_{(p)} = 0$.\nAlso $M_{(0)} = 0$ since $6 \\notin (0)$.\nFor $p = 2$, the element $1/1$ is nonzero in $M_{(2)}$, since $u \\cdot 1 \\ne 0$ in $\\Z/6\\Z$ for every odd $u$.\nThe same holds for $p = 3$, so $\\operatorname{Supp} M = \\set{2\\Z, 3\\Z} = V(6\\Z)$.\n\nNon-example of the formula with the annihilator: the $\\Z$-module $\\Q/\\Z$ is not finitely generated, and $\\operatorname{Ann}(\\Q/\\Z) = 0$, so $V(\\operatorname{Ann}) = \\operatorname{Spec} \\Z$.\nBut $(\\Q/\\Z)_{(0)} = 0$, since every element is killed by a nonzero integer.\nSo $(0) \\notin \\operatorname{Supp}(\\Q/\\Z)$, and the support is not $V(\\operatorname{Ann}(\\Q/\\Z))$.\n",
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