Open Immersions

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Joined: Jan 12, 2011

by iarnab_kundu, Dec 17, 2018, 5:27 PM

Definition- Let $U\subset X$ be an an open subset of a scheme $X$ . Then $U$ has a canonical sub-scheme structure given by restricting the structure sheaf of $X$ to $U$ , that is $\mathcal{O}_U=\mathcal{O}_X |_U$ . We denote this as $U\hookrightarrow X$
Definition- Given a map of schemes $V\to X$ , we say that it is an open immersion if, firstly on the topological spaces induces a homeomorphism into an open subset $U\subset X$ , and secondly it factors as $V\to U\hookrightarrow X$ .

Proposition- Suppose $f:Y\to X$ be a morphism of schemes, such that the image $f(Y)\subset U$ where $U$ is an open subset of $X$ . Then it factors via $f':Y\to U$ and $i:U\hookrightarrow X$ .

algebraic geometry

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Cracking the Nut Open

Open Immersions

by iarnab_kundu, Dec 17, 2018, 5:27 PM

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