Friedlander: Etale homotopy of simplicial schemes
Deligne: Hodge III
There exists a first quadrant spectral sequence
E^{s,t}_1 = H^t(X_s, F_s) \implies H^{s+t}(X_{\bullet} , F)
category: Spectral Sequences [private]
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## Etale cohomology of simplicial schemes
Def. Etale site of a simplicial scheme. Let $X_{\bullet}$ be a simplicial scheme. An object in $Et(X_{\bullet})$ is an etale map $U \to X_n$ for some $n$. Maps are commutative squares of the obvious form, where the bottom map $X_n \to X_m$ is a specified structure map of $X_{\bullet}$. A covering of $U \to X_n$ is a family $\{U_i \to U \}$ of etale morphisms of $X_n$-schemes such that the images of the $U_i$ cover $U$.
(Many more interesting details in Friedlander)
A sheaf is a contravariant functor on $Et(X)$ satisfying the sheaf condition. This is equivalent to a collection of sheaves $F_n$ on $X_n$ satisfying some compatibility criteria.
Now etale cohomology is defined in the usual way, as the right derived functors of the "global sections functor". The "global sections functor" here is the map sending a sheaf of abelian groups to the kernel of the map $d_0^* - d_1^* : F(X_0) \to F(X_1)$. Equivalently,
$$ H^i(X_{\bullet} , F) = Ext^i_{AbShv(X_{\bullet})}(\mathbb{Z}, F)
Can also define etale cohomology of bisimplicial schemes.
Friedlander also defines etale topological type. This should be interesting to understand.
Can use hypercoverings/Cech cohomology. See Friedlander.
arXiv: Experimental full text search
AG (Algebraic geometry)
Mixed
Article by Quick?
See stuff on simplicial schemes in my private notes. See also Motivic homotopy theory.
nLab page on Etale cohomology of simplicial schemes