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Difference between revisions of "Comparison theorem (algebraic geometry)"

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A theorem on the relations between homotopy invariants of schemes of finite type over the field $ \mathbf C $
+
A theorem on the relations between homotopy invariants of schemes of finite type over the field $\mathbf C$
 
in classical and étale topologies.
 
in classical and étale topologies.
  
Let  $ X $
+
Let  $X$
 
be a scheme of finite type over  $  \mathbf C $,  
 
be a scheme of finite type over  $  \mathbf C $,  
 
while  $  F $
 
while  $  F $
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On the other hand, a finite topological covering of a smooth scheme  $  X $
 
On the other hand, a finite topological covering of a smooth scheme  $  X $
 
of finite type over  $  \mathbf C $
 
of finite type over  $  \mathbf C $
has a unique algebraic structure (Riemann's existence theorem). The fundamental étale group of  $  X _ {\textrm{ et } }  $[[#References|[1]]] is therefore the pro-finite completion of the ordinary group of classes of homotopically equivalent loops:
+
has a unique algebraic structure (Riemann's existence theorem). The fundamental étale group of  $  X _ {\textrm{ et } }  $ {{Cite|1}} is therefore the pro-finite completion of the ordinary group of classes of homotopically equivalent loops:
  
 
$$  
 
$$  
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and  $  X _ {\textrm{ et } }  $
 
and  $  X _ {\textrm{ et } }  $
 
are the classical and étale homotopy types of the scheme  $  X $,  
 
are the classical and étale homotopy types of the scheme  $  X $,  
respectively (see [[#References|[1]]], [[#References|[2]]]).
+
respectively (see {{Cite|1}}, {{Cite|2}}).
  
 
====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  M. Artin,  "The étale topology of schemes" , ''Proc. Internat. Congress Mathematicians (Moscow, 1966)'' , Mir  (1968)  pp. 44–56 {{ZBL|0199.24603}}</TD></TR>
+
<TR><TD valign="top">{{Ref|1}}</TD> <TD valign="top">  M. Artin,  "The étale topology of schemes" , ''Proc. Internat. Congress Mathematicians (Moscow, 1966)'' , Mir  (1968)  pp. 44–56 {{ZBL|0199.24603}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  D. Sullivan,  "Geometric topology" , M.I.T.  (1971)  (Notes)</TD></TR>
+
<TR><TD valign="top">{{Ref|@}}</TD> <TD valign="top">  D. Sullivan,  "Geometric topology" , M.I.T.  (1971)  (Notes)</TD></TR>
 
</table>
 
</table>

Revision as of 13:16, 6 April 2023


A theorem on the relations between homotopy invariants of schemes of finite type over the field $\mathbf C$ in classical and étale topologies.

Let $X$ be a scheme of finite type over $ \mathbf C $, while $ F $ is a constructible torsion sheaf of Abelian groups on $ X _ {\textrm{ et } } $. Then $ F $ induces a sheaf on $ X $ in the classical topology, and there exist canonical isomorphisms

$$ H ^ {q} ( X _ {\textrm{ et } } , F) \cong \ H ^ {q} ( X _ {\textrm{ class } } , F). $$

On the other hand, a finite topological covering of a smooth scheme $ X $ of finite type over $ \mathbf C $ has a unique algebraic structure (Riemann's existence theorem). The fundamental étale group of $ X _ {\textrm{ et } } $ [1] is therefore the pro-finite completion of the ordinary group of classes of homotopically equivalent loops:

$$ \pi _ {1} ( X _ {\textrm{ et } } ) = \ [ \pi _ {1} ( X _ {\textrm{ class } } )] \widehat{ {}} . $$

Moreover, if $ X _ {\textrm{ class } } $ is simply connected, then $ X _ {\textrm{ et } } = \widehat{X} _ { \mathop{\rm cl} } $, where $ X _ { \mathop{\rm cl} } $ and $ X _ {\textrm{ et } } $ are the classical and étale homotopy types of the scheme $ X $, respectively (see [1], [2]).

References

[1] M. Artin, "The étale topology of schemes" , Proc. Internat. Congress Mathematicians (Moscow, 1966) , Mir (1968) pp. 44–56 Zbl 0199.24603
[@] D. Sullivan, "Geometric topology" , M.I.T. (1971) (Notes)
How to Cite This Entry:
Comparison theorem (algebraic geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Comparison_theorem_(algebraic_geometry)&oldid=53598
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article