{{For|the concept in differential geometry|Immersion (mathematics)}}
In algebraic geometry, a '''closed immersion''' of schemes is a morphism of schemes <math>f: Z \to X</math> that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''.<ref>Mumford, ''The Red Book of Varieties and Schemes'', Section II.5</ref> The latter condition can be formalized by saying that <math>f^\#:\mathcal{O}_X\rightarrow f_\ast\mathcal{O}_Z</math> is surjective.<ref>{{harvnb|Hartshorne|1977|loc=§II.3}}</ref>
An example is the inclusion map <math>\operatorname{Spec}(R/I) \to \operatorname{Spec}(R)</math> of affine schemes induced by the canonical ring map <math>R \to R/I</math>.
==Other characterizations==
The following are equivalent:
#<math>f: Z \to X</math> is a closed immersion. #For every open affine <math>U = \operatorname{Spec}(R) \subset X</math>, there exists an ideal <math>I \subset R</math> such that <math>f^{-1}(U) = \operatorname{Spec}(R/I)</math> as schemes over ''U''. #There exists an open affine covering <math>X = \bigcup U_j, U_j = \operatorname{Spec} R_j</math> and for each ''j'' there exists an ideal <math>I_j \subset R_j</math> such that <math>f^{-1}(U_j) = \operatorname{Spec} (R_j / I_j)</math> as schemes over <math>U_j</math>. #There is a quasi-coherent sheaf of ideals <math>\mathcal{I}</math> on ''X'' such that <math>f_\ast\mathcal{O}_Z\cong \mathcal{O}_X/\mathcal{I}</math> and ''f'' is an isomorphism of ''Z'' onto the global Spec of <math>\mathcal{O}_X/\mathcal{I}</math> over ''X''.
=== Definition for locally ringed spaces === In the case of locally ringed spaces<ref>{{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}}</ref> a morphism <math>i:Z\to X</math> is a closed immersion if a similar list of criteria is satisfied:
# The map <math>i</math> is a homeomorphism of <math>Z</math> onto its image # The associated sheaf map <math>\mathcal{O}_X \to i_*\mathcal{O}_Z</math> is surjective with kernel <math>\mathcal{I}</math> # The kernel <math>\mathcal{I}</math> is locally generated by sections as an <math>\mathcal{O}_X</math>-module.<ref>{{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}}</ref>
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, <math>i:\mathbb{G}_m\hookrightarrow \mathbb{A}^1</math> where<blockquote><math>\mathbb{G}_m = \text{Spec}(\mathbb{Z}[x,x^{-1}])</math></blockquote>If we look at the stalk of <math>i_*\mathcal{O}_{\mathbb{G}_m}|_0</math> at <math>0 \in \mathbb{A}^1</math> then there are no sections. This implies for any open subscheme <math>U \subset \mathbb{A}^1</math> containing <math>0</math> the sheaf has no sections. This violates the third condition since at least one open subscheme <math>U</math> covering <math>\mathbb{A}^1</math> contains <math>0</math>.
==Properties==
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 <math>X=\bigcup U_j</math> the induced map <math>f:f^{-1}(U_j)\rightarrow U_j</math> is a closed immersion.<ref>{{harvnb|Grothendieck|Dieudonné|1960|loc=4.2.4}}</ref><ref>{{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}}</ref>
If the composition <math>Z \to Y \to X</math> is a closed immersion and <math>Y \to X</math> is separated, then <math>Z \to Y</math> is a closed immersion. If ''X'' is a separated ''S''-scheme, then every ''S''-section of ''X'' is a closed immersion.<ref>{{harvnb|Grothendieck|Dieudonné|1960|loc=5.4.6}}</ref>
If <math>i: Z \to X</math> is a closed immersion and <math>\mathcal{I} \subset \mathcal{O}_X</math> is the quasi-coherent sheaf of ideals cutting out ''Z'', then the direct image <math>i_*</math> 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 <math>\mathcal{G}</math> such that <math>\mathcal{I} \mathcal{G} = 0</math>.<ref>Stacks, Morphisms of schemes. Lemma 4.1</ref>
A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.<ref>Stacks, Morphisms of schemes. Lemma 27.2</ref>
==See also== *Segre embedding *Regular embedding
== Notes == {{reflist}}
== References == *{{EGA|book=I}} *The Stacks Project *{{Hartshorne AG}}
Category:Morphisms of schemes