Comparison theorem (algebraic geometry)

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

Let be a scheme of finite type over , while is a constructible torsion sheaf of Abelian groups on . Then induces a sheaf on in the classical topology, and there exist canonical isomorphisms

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

Moreover, if is simply connected, then , where and are the classical and étale homotopy types of the scheme , 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)