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A morphism of schemes that is separated, universally closed and of finite type. A morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754501.png" /> is called closed if for any closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754502.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754503.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754504.png" />, and universally closed if for any [[Base change|base change]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754505.png" /> the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754506.png" /> 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]].
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Proper morphisms have a number of good cohomological properties. 1) If a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754507.png" /> is proper and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754508.png" /> is a [[Coherent sheaf|coherent sheaf]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p0754509.png" />-modules, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545010.png" /> the sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545011.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545012.png" /> are coherent (the finiteness theorem). A similar fact holds for étale cohomology. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545013.png" /> is a complete scheme over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545014.png" />, then the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545015.png" /> are finite-dimensional. 2) For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545016.png" />, the completion of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545017.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545018.png" /> coincides with
+
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545019.png" /></td> </tr></table>
+
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]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545020.png" /> is the ideal of the subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545022.png" /> (the comparison theorem). 3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545023.png" /> is a proper scheme over a complete local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545024.png" />, then the categories of coherent sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545025.png" /> and on its formal completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545026.png" /> are equivalent (the algebraization theorem). There are analytic analogues of the first and third properties. For example (see [[#References|[3]]]), for a complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545027.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545028.png" /> any [[Coherent analytic sheaf|coherent analytic sheaf]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545029.png" /> is algebraizable and
+
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 _ {*} ( ) _ {y} $
 +
coincides with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545030.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n ^  \leftarrow  } \
 +
H  ^ {q} ( f ^ { - 1 } ( y) , F / {J  ^ {n+} 1 } F  ) ,
 +
$$
  
4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545031.png" /> be a proper morphism, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545032.png" /> be a sheaf of finite Abelian groups in the étale topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545033.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545034.png" /> be a geometric point of the [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545035.png" />. Then the fibre of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545036.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545037.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545038.png" /> (the base-change theorem, see [[#References|[2]]]).
+
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
  
====References====
+
$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" ''Publ. Math. IHES'' , '''2–3''' (1961–1963) {{MR|0238860}} {{MR|0217086}} {{MR|0199181}} {{MR|0173675}} {{MR|0163911}} {{MR|0217085}} {{MR|0217084}} {{MR|0163910}} {{MR|0163909}} {{MR|0217083}} {{MR|0163908}} {{ZBL|0203.23301}} {{ZBL|0144.19904}} {{ZBL|0135.39701}} {{ZBL|0136.15901}} {{ZBL|0118.36206}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> "Théorie des topos et cohomologie étale des schémas" M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , ''SGA 4'' , ''Lect. notes in math.'' , '''269; 270; 305''' , Springer (1972–1973) {{MR|0354654}} {{MR|0354653}} {{MR|0354652}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck (ed.) et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer (1971) {{MR|0354651}} {{ZBL|1039.14001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
+
H  ^ {q} ( X , ) = \
 +
H  ^ {q} ( X ( \mathbf C ) , F ^ { \textrm{ an } } ) .
 +
$$
  
 +
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====
 
====Comments====
A morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545039.png" /> is locally of finite type if there exists a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545040.png" /> by affine open subschemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545041.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545042.png" /> there is an open covering by affine subschemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545045.png" /> is a finitely-generated algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545046.png" /> (with respect to the homomorphism of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545047.png" /> which defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545048.png" />). The morphism is of finite type if the coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545050.png" /> can be taken finite for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545051.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545052.png" /> is finite if there exists an affine open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545054.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545056.png" /> is affine for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545057.png" />, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545058.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545059.png" /> is a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545060.png" />-module.
+
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]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545061.png" /> is said to be proper it for each topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545062.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545063.png" /> is closed. It follows that for every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545064.png" /> the base-change mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545066.png" />, is closed, so that a proper mapping of topological spaces is the same thing as a universally closed mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545067.png" /> is locally compact, a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545068.png" /> is proper if and only if the inverse image of each compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545069.png" /> is compact. Sometimes this last property is taken as a definition.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545070.png" /> be a Noetherian ring which is complete (and separated) with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545071.png" />-adic topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545072.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545073.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545074.png" /> one defines a sheaf of topological rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545075.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545076.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545077.png" />. The ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545078.png" /> is called the formal spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545079.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545080.png" />). It is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545081.png" />. 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 $  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545082.png" /> be a (locally) Noetherian scheme and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545083.png" /> a closed subscheme defined by a sheaf of ideas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545084.png" />. The formal completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545085.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545086.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545087.png" />, is the topologically ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545088.png" />. It is a (locally) Noetherian formal scheme.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545089.png" /> be a proper morphism of locally Noetherian schemes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545090.png" /> a closed subscheme, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545091.png" /> the inverse image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545092.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545094.png" /> be the formal completions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545096.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545098.png" />, respectively. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p07545099.png" /> be the induced morphism of formal schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p075450100.png" />. Then, for every coherent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p075450101.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p075450102.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p075450103.png" />, there are canonical isomorphisms
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075450/p075450104.png" /></td> </tr></table>
+
$$
 +
( R  ^ {q} f _ {*} ( M) ) \mid  _ {Y  ^  \prime  }  \cong \
 +
R  ^ {q} {\widehat{f}  } _ {*} ( M \mid  _ {X  ^  \prime  } ) ,\ \
 +
q \geq  0 .
 +
$$
  
 
This theorem can be used to prove the Zariski connectedness theorem (cf. [[Zariski theorem|Zariski theorem]]).
 
This theorem can be used to prove the Zariski connectedness theorem (cf. [[Zariski theorem|Zariski theorem]]).

Latest revision as of 17:55, 17 July 2024


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
How to Cite This Entry:
Proper morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Proper_morphism&oldid=23940
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article