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  "body": "\n## Definition\n\nLet $U$ and $X$ be [[schemes|5A35TYZD02]].\nA [[morphism of schemes|JXRKH6P1Z4]] $(f, f^\\#) \\colon U \\to X$ is an *open immersion* if:\n\n1. $f$ is a [[homeomorphism|0QT00PB5G2]] of $U$ onto an [[open subset|02BH3MCBQB]] $V$ of $X$;\n2. the induced map $\\mathcal{O}_X|_V \\to f_* \\mathcal{O}_U$ is an isomorphism of sheaves of rings on $V$.\n\nEquivalently, $f$ is an isomorphism of $U$ onto the open subscheme $(V, \\mathcal{O}_X|_V)$.\n\n## Remarks\n\nExample: for a ring $A$ and $g \\in A$, the morphism $\\operatorname{Spec} A_g \\to \\operatorname{Spec} A$ induced by $A \\to A_g$ is an open immersion onto $D(g)$.\nOn each basic open $D(gh) \\subseteq D(g)$ the sheaf map is the isomorphism $A_{gh} \\cong (A_g)_h$.\nNon-example: for a field $k$, let $f \\colon \\operatorname{Spec} k \\to \\operatorname{Spec} k[x]/(x^2)$ come from $k[x]/(x^2) \\to k$, $x \\mapsto 0$.\nBoth spaces are one point, so $f$ is a homeomorphism onto an open set and clause 1 holds.\nClause 2 fails: the ring map $k[x]/(x^2) \\to k$ is not injective, so it is not an isomorphism.\n",
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