Closed immersion

The following are equivalent:

1. $f:\; Z\; \backslash to\; X$ is a closed immersion.
2. For every open affine $U\; =\; \backslash operatorname\{Spec\}(R)\; \backslash subset\; X$, there exists an ideal $I\; \backslash subset\; R$ such that $f^\{-1\}(U)\; =\; \backslash operatorname\{Spec\}(R/I)$ as schemes over U.
3. There exists an open affine covering $X\; =\; \backslash bigcup\; U\_j,\; U\_j\; =\; \backslash operatorname\{Spec\}\; R\_j$ and for each j there exists an ideal $I\_j\; \backslash subset\; R\_j$ such that $f^\{-1\}(U\_j)\; =\; \backslash operatorname\{Spec\}\; (R\_j\; /\; I\_j)$ as schemes over $U\_j$.
4. There is a quasi-coherent sheaf of ideals $\backslash mathcal\{I\}$ on X such that $f\_\backslash ast\backslash mathcal\{O\}\_Z\backslash cong\; \backslash mathcal\{O\}\_X/\backslash mathcal\{I\}$ and f is an isomorphism of Z onto the global Spec of $\backslash mathcal\{O\}\_X/\backslash mathcal\{I\}$ over X.

In the case of locally ringed spaces{{Cite web|title=Section 26.4 (01HJ): Closed immersions of locally ringed spaces—The Stacks project|url=https://stacks.math.columbia.edu/tag/01HJ|access-date=2021-08-05|website=stacks.math.columbia.edu}} a morphism $i:Z\backslash to\; X$ is a closed immersion if a similar list of criteria is satisfied

1. The map $i$ is a homeomorphism of $Z$ onto its image
2. The associated sheaf map $\backslash mathcal\{O\}\_X\; \backslash to\; i\_*\backslash mathcal\{O\}\_Z$ is surjective with kernel $\backslash mathcal\{I\}$
3. The kernel $\backslash mathcal\{I\}$ is locally generated by sections as an $\backslash mathcal\{O\}\_X$-module{{Cite web|title=Section 17.8 (01B1): Modules locally generated by sections—The Stacks project|url=https://stacks.math.columbia.edu/tag/01B1|access-date=2021-08-05|website=stacks.math.columbia.edu}}

The only varying condition is the third. It is instructive to look at a counter-example to get a feel for what the third condition yields by looking at a map which is not a closed immersion, $i:\backslash mathbb\{G\}\_m\backslash hookrightarrow\; \backslash mathbb\{A\}^1$ where

$\backslash mathbb\{G\}\_m\; =\; \backslash text\{Spec\}(\backslash mathbb\{Z\}[x,x^\{-1\}])$

A closed immersion is finite and radicial (universally injective). In particular, a closed immersion is universally closed. A closed immersion is stable under base change and composition. The notion of a closed immersion is local in the sense that f is a closed immersion if and only if for some (equivalently every) open covering $X=\backslash bigcup\; U\_j$ the induced map $f:f^\{-1\}(U\_j)\backslash rightarrow\; U\_j$ is a closed immersion.{{harvnb|Grothendieck|Dieudonné|1960|loc=4.2.4}}{{citation|contribution=Part 4: Algebraic Spaces, Chapter 67: Morphisms of Algebraic Spaces|title=The stacks project|title-link=Stacks Project|contribution-url=https://stacks.math.columbia.edu/tag/03H8|publisher=Columbia University|access-date=2024-03-06}}

If the composition $Z\; \backslash to\; Y\; \backslash to\; X$ is a closed immersion and $Y\; \backslash to\; X$ is separated, then $Z\; \backslash to\; Y$ is a closed immersion. If X is a separated S-scheme, then every S-section of X is a closed immersion.{{harvnb|Grothendieck|Dieudonné|1960|loc=5.4.6}}

If $i:\; Z\; \backslash to\; X$ is a closed immersion and $\backslash mathcal\{I\}\; \backslash subset\; \backslash mathcal\{O\}\_X$ is the quasi-coherent sheaf of ideals cutting out Z, then the direct image $i\_*$ from the category of quasi-coherent sheaves over Z to the category of quasi-coherent sheaves over X is exact, fully faithful with the essential image consisting of $\backslash mathcal\{G\}$ such that $\backslash mathcal\{I\}\; \backslash mathcal\{G\}\; =\; 0$.Stacks, Morphisms of schemes. Lemma 4.1

A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.Stacks, Morphisms of schemes. Lemma 27.2

• Segre embedding
• Regular embedding

{{reflist}}

• {{EGA|book=I}}
• The Stacks Project
• {{Hartshorne AG}}

Category:Morphisms of schemes