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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|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|Complete algebraic variety]]; [[Projective scheme|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|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 [[#References|[3]]]), for a complete  $  \mathbf C $-
scheme  $  X $
any [[Coherent analytic sheaf|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|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 [[#References|[2]]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique" ''Publ. Math. IHES'' , '''2–3''' (1961–1963) {{MR|0238860}} {{MR|0217086}} {{MR|0199181}} {{MR|0173675}} {{MR|0163911}} {{MR|0217085}} {{MR|0217084}} {{MR|0163910}} {{MR|0163909}} {{MR|0217083}} {{MR|0163908}} {{ZBL|0203.23301}} {{ZBL|0144.19904}} {{ZBL|0135.39701}} {{ZBL|0136.15901}} {{ZBL|0118.36206}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> "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) {{MR|0354654}} {{MR|0354653}} {{MR|0354652}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck (ed.) et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer (1971) {{MR|0354651}} {{ZBL|1039.14001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
</table>

====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|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|Zariski theorem]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. General topology" , '''Chapt. I, §10''' , Addison-Wesley (1966) (Translated from French) {{MR|0205211}} {{MR|0205210}} {{ZBL|0301.54002}} {{ZBL|0301.54001}} {{ZBL|0145.19302}} </TD></TR></table>