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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]].\nA *sheaf of $\\mathcal{O}_X$-modules* is a [[sheaf|TB6TM6R4MD]] $\\mathcal{F}$ of [[abelian groups|082GHNPJ53]] on $X$ with the following data and condition.\n\n1. For each open set $U \\subseteq X$, $\\mathcal{F}(U)$ is a [[module|0NZ8JA25WK]] over the ring $\\mathcal{O}_X(U)$.\n2. For open sets $V \\subseteq U$, all $a \\in \\mathcal{O}_X(U)$ and all $s \\in \\mathcal{F}(U)$, one has $(a s)|_V = (a|_V)(s|_V)$.\n\nA morphism of $\\mathcal{O}_X$-modules is a morphism of sheaves $\\varphi \\colon \\mathcal{F} \\to \\mathcal{G}$ such that each $\\varphi_U$ is $\\mathcal{O}_X(U)$-linear.\n\n## Remarks\n\nThe structure sheaf $\\mathcal{O}_X$ is a sheaf of $\\mathcal{O}_X$-modules, with each $\\mathcal{O}_X(U)$ acting on itself by multiplication.\nCondition 2 holds because restriction maps of $\\mathcal{O}_X$ are ring homomorphisms.\nOn $X = \\operatorname{Spec} \\Z$, the constant sheaf $\\underline{\\Z}$ admits no structure of $\\mathcal{O}_X$-module.\nIts sections over $D(2)$ form the group $\\Z$, and clause 1 would need a module structure over $\\mathcal{O}_X(D(2)) = \\Z[1/2]$.\nThis is impossible, since multiplication by $2$ is not invertible on $\\Z$.\nFor each point $x \\in X$, the stalk $\\mathcal{F}_x$ is a module over the local ring $\\mathcal{O}_{X,x}$.\nThe $\\mathcal{O}_X$-modules form an abelian category, with tensor products $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$ and sheaves of homomorphisms.\n",
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