open subscheme

An **open subscheme** of a scheme $(Y,\mathcal{O}_Y)$ is a scheme $(U,\mathcal{O}_Y)$ whose underlying space is the subspace $U$ of $Y$ together with an isomorphism of the structure sheaf $\mathcal{O}_U$ with the restriction $\mathcal{O}_Y|_U$ of the structure sheaf $\mathcal{O}_Y$ to $U$. An isomorphism of a scheme $(X,\mathcal{O}_X)$ and an open subscheme $(U,\mathcal{O}_Y)$ of another scheme $(Y,\mathcal{O}_Y)$ amounts to an open immersion of schemes $(X,\mathcal{O}_X)\hookrightarrow(Y,\mathcal{O}_Y)$.

category: algebraic geometry

Last revised on March 6, 2013 at 19:26:33. See the history of this page for a list of all contributions to it.