Open immersion

Builds on Morphism of schemes Scheme Open set Homeomorphism

Definition

Let $U$ and $X$ be schemes. A morphism of schemes $(f, f^\#) \colon U \to X$ is an open immersion if:

$f$ is a homeomorphism of $U$ onto an open subset $V$ of $X$ ;
the induced map $\mathcal{O}_X|_V \to f_* \mathcal{O}_U$ is an isomorphism of sheaves of rings on $V$ .

Equivalently, $f$ is an isomorphism of $U$ onto the open subscheme $(V, \mathcal{O}_X|_V)$ .

Remarks

Example: for a ring $A$ and $g \in A$ , the morphism $\operatorname{Spec} A_g \to \operatorname{Spec} A$ induced by $A \to A_g$ is an open immersion onto $D(g)$ . On each basic open $D(gh) \subseteq D(g)$ the sheaf map is the isomorphism $A_{gh} \cong (A_g)_h$ . Non-example: for a field $k$ , let $f \colon \operatorname{Spec} k \to \operatorname{Spec} k[x]/(x^2)$ come from $k[x]/(x^2) \to k$ , $x \mapsto 0$ . Both spaces are one point, so $f$ is a homeomorphism onto an open set and clause 1 holds. Clause 2 fails: the ring map $k[x]/(x^2) \to k$ is not injective, so it is not an isomorphism.

Reading path

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Homeomorphism concept
Morphism of schemes concept
Open set concept
Scheme concept
Open immersion definition

Tag	`ZVB7G8TGC3`
URI	`m2://ZVB7G8TGC3`
Kind	definition
Writing	W0 stub
MSC	14A15
Defines	Open immersion
Uses	Morphism of schemes, Scheme, Open set, Homeomorphism

Open immersion

Definition

Remarks

Reading path

Further reading

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