Proper morphism

A morphism of schemes that is separated, universally closed and of finite type. A morphism of schemes $ f : X \rightarrow Y $ is called closed if for any closed $ Z \subset X $ the set $ f ( Z) $ is closed in $ Y $, and universally closed if for any base change $ Y ^ \prime \rightarrow Y $ the morphism $ X \times _ {Y} Y ^ \prime \rightarrow Y ^ \prime $ 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 $ f : X \rightarrow Y $ is proper and if $ F $ is a coherent sheaf of $ O _ {X} $- modules, then for any $ q \geq 0 $ the sheaves of $ O _ {X} $- modules $ R ^ {q} f _ {*} ( F ) $ are coherent (the finiteness theorem). A similar fact holds for étale cohomology. In particular, if $ X $ is a complete scheme over a field $ k $, then the cohomology spaces $ H ^ {q} ( X , F ) $ are finite-dimensional. 2) For any point $ y \in Y $, the completion of the $ O _ {X,y} $- module $ R ^ {q} f _ {*} ( F ) _ {y} $ coincides with

$$ \lim\limits _ {n ^ \leftarrow } \ H ^ {q} ( f ^ { - 1 } ( y) , F / {J ^ {n+} 1 } F ) , $$

where $ J $ is the ideal of the subscheme $ f ^ { - 1 } ( y) $ in $ X $( the comparison theorem). 3) If $ X $ is a proper scheme over a complete local ring $ A $, then the categories of coherent sheaves on $ X $ and on its formal completion $ \widehat{X} $ are equivalent (the algebraization theorem). There are analytic analogues of the first and third properties. For example (see [3]), for a complete $ \mathbf C $- scheme $ X $ any coherent analytic sheaf on $ X ( \mathbf C ) $ is algebraizable and

$$ H ^ {q} ( X , F ) = \ H ^ {q} ( X ( \mathbf C ) , F ^ { \textrm{ an } } ) . $$

4) Let $ f : X \rightarrow Y $ be a proper morphism, let $ F $ be a sheaf of finite Abelian groups in the étale topology of $ X $, and let $ \xi $ be a geometric point of the scheme $ Y $. Then the fibre of the sheaf $ R ^ {q} f _ {*} ( F ) $ at $ \xi $ is isomorphic to $ H ^ {q} ( f ^ { - 1 } ( \xi ) , F \ \mid _ {f ^ { - 1 } ( \xi ) } ) $( the base-change theorem, see [2]).

References

[1]	A. Grothendieck, J. Dieudonné, "Eléments de géométrie 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 $ f: X \rightarrow Y $ is locally of finite type if there exists a covering of $ Y $ by affine open subschemes $ V _ {i} = \mathop{\rm Spec} ( B _ {i} ) $ such that for each $ i $ there is an open covering by affine subschemes $ U _ {ij} = \mathop{\rm Spec} ( A _ {ij} ) $ of $ f ^ { - 1 } ( V _ {i} ) $ such that $ A _ {ij} $ is a finitely-generated algebra over $ B _ {i} $( with respect to the homomorphism of rings $ B _ {i} \rightarrow A _ {ij} $ which defines $ f : U _ {ij} \rightarrow V _ {i} $). The morphism is of finite type if the coverings $ \{ U _ {ij} \} $ of $ f ^ { - 1 } ( V _ {i} ) $ can be taken finite for all $ i $.

A morphism $ f : X \rightarrow Y $ is finite if there exists an affine open covering $ \{ V _ {i} \} $, $ V _ {i} = \mathop{\rm Spec} ( B _ {i} ) $, of $ Y $ such that $ f ^ { - 1 } ( V _ {i} ) $ is affine for all $ i $, say $ f ^ { - 1 } ( V _ {i} ) = \mathop{\rm Spec} ( A _ {i} ) $, and $ A _ {i} $ is a finitely-generated $ B _ {i} $- module.

The analytic analogue of property 1) above is called Grauert's finiteness theorem, see Finiteness theorems.

In topology a mapping of topological spaces $ f: X \rightarrow Y $ is said to be proper it for each topological space $ Z $ the mapping $ f \times id : X \times Z \rightarrow Y \times Z $ is closed. It follows that for every continuous mapping $ g : Z \rightarrow X $ the base-change mapping $ f ^ { \prime } : X \times _ {Y} Z = \{ {( x, z) } : {f( x) = g( z) } \} \rightarrow Z $, $ ( x, z) \mapsto z $, is closed, so that a proper mapping of topological spaces is the same thing as a universally closed mapping. If $ Y $ is locally compact, a continuous mapping $ f: X \rightarrow Y $ is proper if and only if the inverse image of each compact subset of $ Y $ is compact. Sometimes this last property is taken as a definition.

Let $ A $ be a Noetherian ring which is complete (and separated) with respect to the $ I $- adic topology on $ A $, i.e. $ A = \lim\limits _ \leftarrow A / I ^ {n} $. On $ {\mathcal X} = V( I)= \mathop{\rm Spec} ( A /I) \subset \mathop{\rm Spec} ( A) $ one defines a sheaf of topological rings $ {\mathcal O} _ {\mathcal X} $ by $ \Gamma ( D( f) \cap {\mathcal X} , {\mathcal O} _ {\mathcal X} ) = \lim\limits _ \leftarrow A _ {f} / I ^ {n} A _ {f} $ for $ f \in A $. The ringed space $ ( {\mathcal X} , {\mathcal O} _ {\mathcal X} ) $ is called the formal spectrum of $ A $( with respect to $ I $). It is denoted by $ \mathop{\rm Spf} ( A) $. 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 $ X $ be a (locally) Noetherian scheme and $ Y $ a closed subscheme defined by a sheaf of ideas $ {\mathcal I} \subset {\mathcal O} _ {X} $. The formal completion of $ X $ along $ Y $, denoted by $ \widehat{X} $, is the topologically ringed space $ ( Y, \lim\limits _ { {\leftarrow n } } {\mathcal O} _ {X} / {\mathcal I} ^ {n} ) $. 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 $ f : X \rightarrow Y $ be a proper morphism of locally Noetherian schemes, $ Y ^ \prime \subset Y $ a closed subscheme, $ X ^ \prime \equiv X \times _ {Y} Y ^ \prime $ the inverse image of $ Y ^ \prime $. Let $ \widehat{X} $ and $ \widehat{Y} $ be the formal completions of $ X $ and $ Y $ along $ X ^ \prime $ and $ Y ^ \prime $, respectively. Let $ \widehat{f} $ be the induced morphism of formal schemes $ \widehat{X} \rightarrow \widehat{Y} $. Then, for every coherent $ {\mathcal O} _ {X} $- module $ M $ on $ X $, there are canonical isomorphisms

$$ ( R ^ {q} f _ {*} ( M) ) \mid _ {Y ^ \prime } \cong \ R ^ {q} {\widehat{f} } _ {*} ( M \mid _ {X ^ \prime } ) ,\ \ q \geq 0 . $$

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