nLab
open subscheme

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

Revised on March 6, 2013 19:26:33 by Zoran Å koda (161.53.130.104)