nLab comparison theorem (étale cohomology)

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 $A$ be either of

• a finite field;

• the p-adic integers $\mathbb{Z}_p$ for some prime $p \geq 2$;

• the p-adic numbers $\mathbb{A}_{p}$

(but not for instance the integers themselves).

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

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

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

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

Related concepts

• analytification, GAGA

• comparison theorem

• comparison theorem (crystalline cohomology)

• Chow's theorem

References

The original reference is

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

• Alexander Grothendieck, SGA I, Exposé XII

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

• eom, Comparison theorem

• James Milne, section 21 of Lectures on Étale Cohomology

In the context of Berkovich analytic spaces see also

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