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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(R/I) \to \operatorname(R) 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(R) \subset X, there exists an ideal I \subset R such that f^(U) = \operatorname(R/I) 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^(U_j) = \operatorname (R_j / I_j) 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^(U_j)\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

See also

*Segre embedding *Regular embedding


Notes





References


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