Suppose $f:X \to Y $ is a closed immersion,by definition,$X$ is isomorphic to a closed subscheme of $Y$,so is there any connection between X and $f(X)$ ? When X is isomorphic to $f(X)$ as schemes?
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May I ask how you are defining the structure sheaf on $f(X)$? – Kenny Wong Aug 15 '17 at 17:56
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@KennyWong as a closed set of Y,at least we have the reduced induced structure – Jiabin Du Aug 15 '17 at 23:10
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@KennyWong sorry for so lately reply, about this question,Algebraic geometry by Robin Hartshorne gives an explicit explanation (on page86),and you can refer to it. – Jiabin Du Aug 20 '17 at 06:46
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Sorry for my late reply too. Perhaps we can show that $X$ is isomorphic to this $f(X)$ iff $X$ is a reduced scheme? I haven't thought about this too carefully, but see Georges Elencwajg's anwer here: https://math.stackexchange.com/questions/374944/questions-on-reduced-induced-closed-subscheme – Kenny Wong Aug 20 '17 at 07:31