group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
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
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.
The original reference is
A textbook account is in
Reviews and lecture notes include
James Milne, section 21 of Lectures on Étale Cohomology
In the context of Berkovich analytic spaces see also
Last revised on April 4, 2023 at 22:24:31. See the history of this page for a list of all contributions to it.