Proper morphism

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 [3]), 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 [2]).

References

[1]	A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" Publ. Math. IHES , 2–3 (1961–1963) MR0238860 MR0217086 MR0199181 MR0173675 MR0163911 MR0217085 MR0217084 MR0163910 MR0163909 MR0217083 MR0163908 Zbl 0203.23301 Zbl 0144.19904 Zbl 0135.39701 Zbl 0136.15901 Zbl 0118.36206
[2]	"Théorie des topos et cohomologie étale des schémas" M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , SGA 4 , Lect. notes in math. , 269; 270; 305 , Springer (1972–1973) MR0354654 MR0354653 MR0354652
[3]	A. Grothendieck (ed.) et al. (ed.) , Revêtements étales et groupe fondamental. SGA 1 , Lect. notes in math. , 224 , Springer (1971) MR0354651 Zbl 1039.14001
[4]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001

Comments

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).

References

[a1]	N. Bourbaki, "Elements of mathematics. General topology" , Chapt. I, §10 , Addison-Wesley (1966) (Translated from French) MR0205211 MR0205210 Zbl 0301.54002 Zbl 0301.54001 Zbl 0145.19302