# 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 ), 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 ).

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