# Proper morphism

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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 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

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
How to Cite This Entry:
Proper morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Proper_morphism&oldid=48334
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article