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  "body": "\n## Summary\n\nA closed immersion is a [[morphism of schemes|JXRKH6P1Z4]] that identifies its source with a closed subscheme of its target.\nFor affine schemes, closed immersions into $\\operatorname{Spec} A$ are exactly the maps $\\operatorname{Spec} A/I \\to \\operatorname{Spec} A$ for ideals $I$.\nThey make precise the idea of a subvariety cut out by equations, and they include non-reduced subschemes such as a double point.\n\n## Explanations\n\n### L1 human\n\nA closed immersion is the scheme version of \"the set of solutions of some equations inside a bigger space\".\nThe equations $I$ in $A$ cut out $\\operatorname{Spec} A/I$, and every function on it comes from a function on the big space.\nThe equation $x^2 = 0$ on the line cuts out the origin with a double structure.\nThis is a different closed subscheme from the origin cut out by $x = 0$, though both have the same single point.\n",
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