Holmstrom Generalized etale cohomology

Generalized etale cohomology

A book by Jardine

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Generalized etale cohomology

It is not clear to me what the relation is between Jardine’s concept and the theories represented by objects in Quick’s stable homotopy category.

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Generalized etale cohomology

We follow chapter 6 of Jardine: Generalized etale cohomology theories.

A gen. etale cohomology theory is a graded group $\mathbb{H}^*(T,F)$, which is associated to a presheaf of spectra $F$ on an etale site for a scheme $T$. It arises from the homotopy category of presheaves of spectra on the underlying site for $T$. If $F$ consists of presheaves of Kan complexes, then $\mathbb{H}^*(T,F)$ is a graded group consisting of morphisms in the associated stable category in the sense that

where $\Gamma^*S$ is the constant presheaf associated to the ordinary sphere spectrum $S$.

Formal definition of $\mathbb{H}^*(T,F)$ in terms of stable homotopy groups of global sections for a globally fibrant model of $F$.

Construction of the etale cohomological descent spectral sequence:

under some hyps. (Use Postnikov resolutions)

Finite approximation technique for computing the groups $\mathbb{H}^*(T,F)$

Special case: $T$ is a field. In this the case the category of sheaves on the etale site of $T$ can be identified with the classifying topos of the absolute Galois group of $T$. More on generalized Galois cohomology theories.

Discussion of Cech cohomology.

Chapter 7: We give proof of Thomason’s descent theorem for the Bott periodic K-theory of fields and its corollaries. Outline due to Thomason, but we use homotopy theory of presheaves of spectra (smash products) and the Gabber rigidity theorem.

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Generalized etale cohomology

A generalized etale cohomology theory, according to Jardine, is a cohomology theory which is represented by a presheaf of spectra on some étale site. Examples: etale K-theory, etale cohomology.

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Generalized etale cohomology

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Generalized etale cohomology

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Generalized etale cohomology

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