Holmstrom Etale cohomology of simplicial schemes

Etale cohomology of simplicial schemes

Friedlander: Etale homotopy of simplicial schemes

Deligne: Hodge III

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Etale cohomology of simplicial schemes

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] --- ## 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.

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Etale cohomology of simplicial schemes

Friedlander also defines etale topological type. This should be interesting to understand.

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Etale cohomology of simplicial schemes

Can use hypercoverings/Cech cohomology. See Friedlander.


Etale cohomology of simplicial schemes

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Etale cohomology of simplicial schemes

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Etale cohomology of simplicial schemes

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Etale cohomology of simplicial schemes

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Etale cohomology of simplicial schemes

See stuff on simplicial schemes in my private notes. See also Motivic homotopy theory.

nLab page on Etale cohomology of simplicial schemes

Created on June 10, 2014 at 21:14:54 by Andreas Holmström