:''For the same-name concept in differential geometry, see immersion (mathematics).''
In algebraic geometry, a closed immersion of schemes is a morphism of schemes $f:\; Z\; \backslash to\; X$ that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formalized by saying that $f^\backslash \#:\backslash mathcal\_X\backslash rightarrow\; f\_\backslash ast\backslash mathcal\_Z$ is surjective.
An example is the inclusion map $\backslash operatorname(R/I)\; \backslash to\; \backslash operatorname(R)$ induced by the canonical map $R\; \backslash to\; R/I$.

Other characterizations

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

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 $X=\backslash bigcup\; U\_j$ the induced map $f:f^(U\_j)\backslash rightarrow\; U\_j$ is a closed immersion. 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. If $i:\; Z\; \backslash to\; X$ is a closed immersion and $\backslash mathcal\; \backslash subset\; \backslash mathcal\_X$ 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 $\backslash mathcal$ such that $\backslash mathcal\; \backslash mathcal\; =\; 0$. A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.Stacks, Morphisms of schemes. Lemma 27.2

See also

*Segre embedding *Regular embedding

** Notes **

** References **

*
*The Stacks Project
*{{Hartshorne AG
Category:Morphisms of schemes

Other characterizations

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

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 $X=\backslash bigcup\; U\_j$ the induced map $f:f^(U\_j)\backslash rightarrow\; U\_j$ is a closed immersion. 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. If $i:\; Z\; \backslash to\; X$ is a closed immersion and $\backslash mathcal\; \backslash subset\; \backslash mathcal\_X$ 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 $\backslash mathcal$ such that $\backslash mathcal\; \backslash mathcal\; =\; 0$. A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.Stacks, Morphisms of schemes. Lemma 27.2

See also

*Segre embedding *Regular embedding