Let \((X, \OO_X)\) be a scheme and let \(U \subseteq X\) be any open subset. Show that \(\qty{U, \ro{\OO_X}{U}}\) is a scheme. We call this the induced scheme structure on the open set \(U\), and we refer to \(\qty{U, \ro{\OO_X}{U}}\) as an open subscheme of \(X\).
Solution
1. The restricted ringed space\[ (U,\mathcal O_X|_U) \]is a locally ringed space.
For a point \(P\in U\), the stalk of the restricted sheaf is naturally \[ (\mathcal O_X|_U)_P\cong\mathcal O_{X,P}. \] Since \(X\) is a scheme, \(\mathcal O_{X,P}\) is a local ring. Hence every stalk of the restricted sheaf is local.
2. Fix \(P\in U\). Choose an affine open neighborhood\[ P\in V\subseteq X, \qquad V\cong\operatorname{Spec}A. \]Then\[ U\cap V \]is an open neighborhood of \(P\) in \(V\).
The affine neighborhoods form a cover of the scheme \(X\), so such a \(V\) exists. Since both \(U\) and \(V\) are open in \(X\), their intersection is open, and it contains \(P\).
3. There exists\[ f\in A \]such that\[ P\in D_V(f)\subseteq U\cap V. \]
The distinguished opens \[ D_V(f), \qquad f\in A, \] form a basis for the Zariski topology on \[ V=\operatorname{Spec}A. \] Since \(U\cap V\) is an open neighborhood of \(P\), some distinguished basic neighborhood of \(P\) is contained in it.
4. With its induced structure sheaf,\[ D_V(f) \]is an affine scheme:\[ \boxed{ \left(D_V(f),\mathcal O_X|_{D_V(f)}\right) \cong \operatorname{Spec}A_f. } \]
On the affine open \(V\), the restricted structure sheaf is its usual affine structure sheaf. Hartshorne II.2.1 identifies every distinguished open of an affine scheme with the spectrum of the corresponding localization: \[ \left(D_V(f),\mathcal O_V|_{D_V(f)}\right) \cong \operatorname{Spec}A_f. \] Since \[ \mathcal O_V=\mathcal O_X|_V, \] this is the displayed isomorphism.
5. The distinguished affine opens obtained in <1>3–<1>4 cover \(U\).
For every point \(P\in U\), steps <1>2–<1>4 construct an affine open neighborhood \[ P\in D_V(f)\subseteq U. \] Taking all such neighborhoods gives an open affine cover of \(U\).
6. Therefore\[ \boxed{(U,\mathcal O_X|_U)\text{ is a scheme}.} \]
By 1 it is a locally ringed space, and by 5 it admits an open cover by affine schemes. This is exactly the definition of a scheme.
7. Q.E.D.
Step 6 proves that every open subset of a scheme carries the induced open-subscheme structure.