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Let $X$ be a scheme. Let $Z$ be a closed subscheme of $X$. Let $U$ be an open subscheme of $X$. Then $Y = U \cap Z$ is an open subscheme of $Z$. Can we identify $Y$ with $U\times_X Z$?

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Yes. This doesn't have anything to do with closed subscheme. If $p: Z \to X$ is a morphism of schemes and $U \subset X$ is open subscheme, then the fibre product is $p^{-1}(U)$ with open subscheme structure.

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  • $\begingroup$ Does it still hold if instead of open immersion I assume $U$ is a subspace of $X$ with the sheaf pulled back from $X$? I think $p$ factors through $U$. And here $p^{-1}(U) $ means the sheaf pulled back from that of U. $\endgroup$ – Qixiao Oct 2 '13 at 3:52

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