A morphism of schemes that is separated, universally closed and of finite type. A morphism of schemes is called closed if for any closed the set is closed in , and universally closed if for any base change the morphism is closed. The property of being a proper morphism is preserved under composition, base change and taking Cartesian products. Proper morphisms are closely related to projective morphisms: any projective morphism is proper, and a proper quasi-projective morphism is projective. Any proper morphism is dominated by a projective one (Chow's lemma). See also Complete algebraic variety; Projective scheme.
Proper morphisms have a number of good cohomological properties. 1) If a morphism is proper and if is a coherent sheaf of -modules, then for any the sheaves of -modules are coherent (the finiteness theorem). A similar fact holds for étale cohomology. In particular, if is a complete scheme over a field , then the cohomology spaces are finite-dimensional. 2) For any point , the completion of the -module coincides with
where is the ideal of the subscheme in (the comparison theorem). 3) If is a proper scheme over a complete local ring , then the categories of coherent sheaves on and on its formal completion are equivalent (the algebraization theorem). There are analytic analogues of the first and third properties. For example (see ), for a complete -scheme any coherent analytic sheaf on is algebraizable and
4) Let be a proper morphism, let be a sheaf of finite Abelian groups in the étale topology of , and let be a geometric point of the scheme . Then the fibre of the sheaf at is isomorphic to (the base-change theorem, see ).
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A morphism of schemes is locally of finite type if there exists a covering of by affine open subschemes such that for each there is an open covering by affine subschemes of such that is a finitely-generated algebra over (with respect to the homomorphism of rings which defines ). The morphism is of finite type if the coverings of can be taken finite for all .
A morphism is finite if there exists an affine open covering , , of such that is affine for all , say , and is a finitely-generated -module.
The analytic analogue of property 1) above is called Grauert's finiteness theorem, see Finiteness theorems.
In topology a mapping of topological spaces is said to be proper it for each topological space the mapping is closed. It follows that for every continuous mapping the base-change mapping , , is closed, so that a proper mapping of topological spaces is the same thing as a universally closed mapping. If is locally compact, a continuous mapping is proper if and only if the inverse image of each compact subset of is compact. Sometimes this last property is taken as a definition.
Let be a Noetherian ring which is complete (and separated) with respect to the -adic topology on , i.e. . On one defines a sheaf of topological rings by for . The ringed space is called the formal spectrum of (with respect to ). It is denoted by . A locally Noetherian formal scheme is, by definition, a topologically ringed space which is locally isomorphic to formal spectra of a Noetherian ring. Morphisms of formal schemes are morphisms of the corresponding topologically ringed spaces.
Let be a (locally) Noetherian scheme and a closed subscheme defined by a sheaf of ideas . The formal completion of along , denoted by , is the topologically ringed space . It is a (locally) Noetherian formal scheme.
All this serves to state the following theorem, which is sometimes called the fundamental theorem on proper morphisms: Let be a proper morphism of locally Noetherian schemes, a closed subscheme, the inverse image of . Let and be the formal completions of and along and , respectively. Let be the induced morphism of formal schemes . Then, for every coherent -module on , there are canonical isomorphisms
This theorem can be used to prove the Zariski connectedness theorem (cf. Zariski theorem).
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Proper morphism. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Proper_morphism&oldid=23940