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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c0236501.png" /> in classical and étale topologies.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c0236502.png" /> be a scheme of finite type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c0236503.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c0236504.png" /> is a constructible torsion sheaf of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c0236505.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c0236506.png" /> induces a sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c0236507.png" /> in the classical topology, and there exist canonical isomorphisms
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<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/c/c023/c023650/c0236508.png" /></td> </tr></table>
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A theorem on the relations between homotopy invariants of schemes of finite type over the field $\mathbf C$
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in classical and étale topologies.
  
On the other hand, a finite topological covering of a smooth scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c0236509.png" /> of finite type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c02365010.png" /> has a unique algebraic structure (Riemann's existence theorem). The fundamental étale group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c02365011.png" /> [[#References|[1]]] is therefore the pro-finite completion of the ordinary group of classes of homotopically equivalent loops:
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Let  $X$
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be a scheme of finite type over $  \mathbf C $,
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while  $  F $
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is a constructible torsion sheaf of Abelian groups on  $  X _ {\textrm{ et } }  $.  
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Then  $  F $
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induces a sheaf on  $  X $
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in the classical topology, and there exist 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/c/c023/c023650/c02365012.png" /></td> </tr></table>
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$$
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H  ^ {q} ( X _ {\textrm{ et } }  , F)  \cong \
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H  ^ {q} ( X _ {\textrm{ class } }  , F).
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$$
  
Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c02365013.png" /> is simply connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c02365014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c02365015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c02365016.png" /> are the classical and étale homotopy types of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023650/c02365017.png" />, respectively (see [[#References|[1]]], [[#References|[2]]]).
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On the other hand, a finite topological covering of a smooth scheme  $  X $
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of finite type over  $  \mathbf C $
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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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$$
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\pi _ {1} ( X _ {\textrm{ et } }  )  = \
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[ \pi _ {1} ( X _ {\textrm{ class } }  )] \widehat{ {}}  .
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$$
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Moreover, if $  X _ {\textrm{ class } }  $
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is simply connected, then $  X _ {\textrm{ et } }  = \widehat{X}  _ { \mathop{\rm cl}  } $,  
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where $  X _ { \mathop{\rm cl}  } $
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and $  X _ {\textrm{ et } }  $
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are the classical and étale homotopy types of the scheme $  X $,  
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respectively (see {{Cite|1}}, {{Cite|2}}).
  
 
====References====
 
====References====
<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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Sullivan,  "Geometric topology" , M.I.T.  (1971)  (Notes)</TD></TR></table>
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<table>
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<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>
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<TR><TD valign="top">{{Ref|2}}</TD> <TD valign="top">  D. Sullivan,  "Geometric topology" , M.I.T.  (1971)  (Notes)</TD></TR>
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</table>

Latest 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
[2] 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=13795
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article