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 $ | |
| + | 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 | ||
| − | + | $$ | |
| + | H ^ {q} ( X _ {\textrm{ et } } , F) \cong \ | ||
| + | H ^ {q} ( X _ {\textrm{ class } } , F). | ||
| + | $$ | ||
| − | Moreover, if | + | 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 } } $[[#References|[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 [[#References|[1]]], [[#References|[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> | <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> | ||
Revision as of 17:45, 4 June 2020
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 |
| [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
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