nLab open immersion of schemes

Definition

A morphism $f:X\to Y$ of schemes is an open immersion if the underlying morphism of topological spaces is a homeomorphism onto an open image and the comorphism $f^\sharp : \mathcal{O}_Y\to f_*\mathcal{O}_X$ is an isomorphism of sheaves when restricted to the image of $f$. In other words, an open immersion is a morphism of schemes which decomposes uniquely into an isomorphism of schemes and the identity inclusion of an open subscheme.

Properties

Proposition

Every open immersion of schemes is an étale morphism of schemes

(e.g. Stacks Project, lemma 28.37.9)

Examples

Example

For $R$ a ring, $S \hookrightarrow U(R)$ a multiplicative subset, and $R \longrightarrow R[S^{-1}]$ the projection onto the localization at $S$, then the formal dual map on spectra $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ is an open immersion.

These are the standard opens that define the Zariski topology on algebraic varieties

Related concepts

• closed immersion of schemes

• immersion, open map

• closed immersion of schemes

• Zariski topology

category: algebraic geometry