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:''For the same-name concept in differential geometry, see immersion (mathematics).'' In algebraic geometry, a closed immersion of schemes is a morphism of schemes $f: Z \to X$ 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 $f^\#:\mathcal_X\rightarrow f_\ast\mathcal_Z$ is surjective. An example is the inclusion map $\operatorname\left(R/I\right) \to \operatorname\left(R\right)$ induced by the canonical map $R \to R/I$.

Other characterizations

The following are equivalent: #$f: Z \to X$ is a closed immersion. #For every open affine $U = \operatorname\left(R\right) \subset X$, there exists an ideal $I \subset R$ such that $f^\left(U\right) = \operatorname\left(R/I\right)$ as schemes over ''U''. #There exists an open affine covering $X = \bigcup U_j, U_j = \operatorname R_j$ and for each ''j'' there exists an ideal $I_j \subset R_j$ such that $f^\left(U_j\right) = \operatorname \left(R_j / I_j\right)$ as schemes over $U_j$. #There is a quasi-coherent sheaf of ideals $\mathcal$ on ''X'' such that $f_\ast\mathcal_Z\cong \mathcal_X/\mathcal$ and ''f'' is an isomorphism of ''Z'' onto the global Spec of $\mathcal_X/\mathcal$ 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 $X=\bigcup U_j$ the induced map $f:f^\left(U_j\right)\rightarrow U_j$ is a closed immersion. If the composition $Z \to Y \to X$ is a closed immersion and $Y \to X$ is separated, then $Z \to Y$ is a closed immersion. If ''X'' is a separated ''S''-scheme, then every ''S''-section of ''X'' is a closed immersion. If $i: Z \to X$ is a closed immersion and $\mathcal \subset \mathcal_X$ 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 $\mathcal$ such that $\mathcal \mathcal = 0$. A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.Stacks, Morphisms of schemes. Lemma 27.2

*Segre embedding *Regular embedding

Notes

References

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