nLab
open subscheme

Contents

Idea

An open subscheme is the analogue of an open subset of a topological space for schemes.

Definition

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).

Open subsets of the underlying topological space of a scheme

Before embarking upon the proof of Proposition , we shall need a few preliminaries. Let AA be a commutative ring. Let SpecASpec A denote the set of prime ideals of AA. Given an element aa of AA, we denote by D A(a)D_{A}(a) the set of prime ideals of AA to which aa does not belong. Either by definition, or by a little basic commutative algebra, we have that {D A(a)|aA}\{ D_{A}(a) | a \in A \} is a basis for the Zariski topology on SpecASpec A.

An affine scheme is by definition the locally ringed space (SpecA,𝒪 SpecA)(Spec A, \mathcal{O}_{Spec A}), where SpecASpec A is the set we have just defined equipped with the Zariski topology, and 𝒪 SpecA\mathcal{O}_{Spec A} is a certain sheaf of rings on this space.

A basic result in commutative algebra is that, for any aAa \in A, (D A(a),𝒪 SpecA|D A(a))(D_{A}(a), \mathcal{O}_{Spec A} | D_{A}(a)) is an affine scheme, isomorphic to (SpecA a,𝒪 SpecA a)(Spec A_{a}, \mathcal{O}_{Spec A_{a}}). Here A aA_{a} is the localisation of AA at aa.

Proposition

Let (X,𝒪 X)(X, \mathcal{O}_{X}) be a scheme, and let UU be an open subset of XX. Then (U,𝒪 X|U)(U, \mathcal{O}_{X} | U), in the same notation as at scheme, is a scheme.

Proof

Let xUx \in U. We must prove that xx has an open neighbourhood WW in UU such that (W,𝒪 U|W)(W, \mathcal{O}_{U} | W) is isomorphic to an affine scheme.

Immediately from the definition of a scheme, there is an open neighbourhood NN of xx in XX such that (N,𝒪 X|N)(N, \mathcal{O}_{X} | N) is isomorphic to an affine scheme, that is to say, a pair (SpecA,𝒪 SpecA)(Spec A, \mathcal{O}_{Spec A}) as defined above, for some commutative ring AA. This isomorphism in particular involves an isomorphism of topological spaces NSpecAN \rightarrow Spec A, which we shall denote by ii.

Since {D A(a)|aA}\{ D_{A}(a) | a \in A \} is a basis for the Zariski topology on SpecASpec A, there is some aAa \in A such that i(x)D A(a)i(x) \in D_{A}(a) and D A(a)i(NU)D_{A}(a) \subset i(N \cap U), noting that since UU is open in XX, NUN \cap U is open in NN.

Now, as we have remarked, (D A(a),𝒪 SpecA|D A(a))(D_{A}(a), \mathcal{O}_{Spec A} | D_{A}(a)) is isomorphic to an affine scheme. Hence (i 1(D A(a)),i *(𝒪 SpecA|D A(a)))(i^{-1}(D_{A}(a)), i^{*}(\mathcal{O}_{Spec A} | D_{A}(a))) is isomorphic to an affine scheme, where i *i^{*} is the inverse image functor from sheaves of commutative rings on D A(a)D_{A}(a) to sheaves of commutative rings on i 1(D A(a))i^{-1}(D_{A}(a)).

But i *(𝒪 SpecA|D A(a))i^{*}(\mathcal{O}_{Spec A} | D_{A}(a)) is isomorphic to 𝒪 U|i 1(D A(a))\mathcal{O}_{U} | i^{-1}(D_{A}(a)). Since i 1(D A(a))i 1(i(NU))=NUUi^{-1}(D_{A}(a)) \subset i^{-1}(i(N \cap U)) = N \cap U \subset U, we conclude that we can take WW to be i 1(D A(a))i^{-1}(D_{A}(a)).

Examples

Example

Let xx be a point of the underlying topological space XX of a scheme (X,𝒪 X)(X, \mathcal{O}_{X}). The set {x}\{ x \} need not be a closed subset of XX, but if it is, then, by Proposition , (X{x},𝒪 X|X{x})(X \setminus \{ x \}, \mathcal{O}_{X} | X \setminus \{ x \}) defines an open subscheme of (X,𝒪 X)(X, \mathcal{O}_{X}).

Last revised on April 29, 2018 at 15:54:11. See the history of this page for a list of all contributions to it.