nLab
comparison theorem (étale cohomology)

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Étale morphisms

Contents

Idea

In the context of étale cohomology of schemes, the Artin comparison theorem (Artin 66) says that under some conditions on a scheme and a sheaf of coefficients, the étale cohomology of the scheme coincides with the ordinary cohomology (e.g. singular cohomology) of its underlying complex analytic topology.

Historically this kind of statement was a central motivation for the development of étale cohomology in the first place.

Specifically, let AA be either of

(but not for instance the integers themselves).

Then for XX a variety over the complex numbers and X anX^{an} its analytification to the topological space of complex points X()X(\mathbb{C}) with its complex analytic topology, then there is an isomorphism

H (X et,A)H (X an,A) H^\bullet(X_{et}, A) \simeq H^\bullet(X^{an}, A)

between the étale cohomology of XX and the ordinary cohomology of X anX^{an}.

Notice that on the other hand for instance if instead X=Spec(k)X = Spec(k) is the spectrum of a field, then its étale cohomology coincides with the Galois cohomology of kk. In this way étale cohomology interpolates between “topological” and “number theoretic” notions of cohomology.

References

The original reference is

  • Michael Artin, The étale topology of schemes , Proc. Internat. Congress Mathematicians (Moscow, 1966) , Mir (1968) pp. 44–56

A textbook account is in

  • Vladimir Danilov, chapter 3 of Cohomology of algebraic varieties, in I. Shafarevich (ed.), Algebraic Geometry II, volume 35 of Encyclopedia of mathematical sciences, Springer 1991 (GoogleBooks))

Reviews and lecture notes include

In the context of Berkovich analytic spaces see also

  • Vladimir Berkovich, On the comparison theorem for étale cohomology of non-archimedean analytic spaces (pdf)

Last revised on October 28, 2015 at 16:01:52. See the history of this page for a list of all contributions to it.