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  "body": "\n## Definition\n\nA *scheme* is a [[locally ringed space|B7FPT8JTXR]] $(X, \\mathcal{O}_X)$ such that every point $x \\in X$ has an open neighbourhood $U$ for which $(U, \\mathcal{O}_X|_U)$ is an [[affine scheme|B3V3XJ2W8G]].\n\n## Remarks\n\nExample: every affine scheme $\\operatorname{Spec} A$ is a scheme, with $U = X$ for every point.\nExample: the projective line $\\mathbb{P}^1_k$ over a field $k$ is a scheme: it is covered by two open sets, each isomorphic to $\\operatorname{Spec} k[t]$.\nIt is not affine, since its ring of global functions is $k$, while $\\operatorname{Spec} k$ is a single point.\nNon-example: the one-point space with sheaf $\\Z$ is a ringed space but not a scheme.\nIts stalk $\\Z$ is not a local ring, so it is not even locally ringed.\nAn open subset $U$ of a scheme, with the restricted sheaf $\\mathcal{O}_X|_U$, is again a scheme, called an open subscheme.\n",
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