Comparison theorem (algebraic geometry)

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)