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  "body": "\n## Definition\n\nLet $Z$ and $X$ be [[schemes|5A35TYZD02]].\nA [[morphism of schemes|JXRKH6P1Z4]] $(f, f^\\#) \\colon Z \\to X$ is a *closed immersion* if:\n\n1. $f$ is a [[homeomorphism|0QT00PB5G2]] of $Z$ onto a [[closed subset|0ZCK8DQYN1]] of $X$;\n2. the [[sheaf|TB6TM6R4MD]] map $f^\\# \\colon \\mathcal{O}_X \\to f_* \\mathcal{O}_Z$ is surjective, that is, surjective on every stalk.\n\n## Remarks\n\nExample: for a ring $A$ and an ideal $I$, the morphism $\\operatorname{Spec} A/I \\to \\operatorname{Spec} A$ induced by $A \\to A/I$ is a closed immersion.\nIts image is the closed set $V(I)$, and on the stalk at $\\mathfrak{p} \\supseteq I$ the map is $A_{\\mathfrak p} \\to (A/I)_{\\mathfrak p}$, which is onto.\nEvery closed immersion into $\\operatorname{Spec} A$ is isomorphic to one of these.\nNon-example: let $k$ be a field and let $f \\colon \\operatorname{Spec} k[t] \\to \\operatorname{Spec} k[t^2, t^3]$ come from the inclusion $k[t^2, t^3] \\subseteq k[t]$.\nThen $f$ is a homeomorphism onto the whole space, so clause 1 holds.\nClause 2 fails at the origin: $t$ does not lie in the image of the stalk map.\nNon-example: the open immersion $\\operatorname{Spec} k[x, x^{-1}] \\to \\operatorname{Spec} k[x]$ fails clause 1, since its image $D(x)$ is not closed.\n",
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