higher geometry / derived geometry
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The pro-étale site is a variant of the étale site where the finiteness conditions on the fibers of étale morphisms is relaxed to a pro-finiteness condition (pro-étale morphisms)
The sheaf topos over the pro-étale site might be called the pro-étale topos. Its abelian sheaf cohomology – pro-étale cohomology – improves on that of the original étale topos in that it genuinely contains (Bhatt-Scholze 13) the Weil cohomology theory called ℓ-adic cohomology (while over the étale site this is only given by an inverse limit of abelian sheaf cohomology).
For $X$ a scheme, its pro-étale site $X_{proet}$ is the site whose objects are pro-étale morphisms into $X$ and whose Grothendieck topology is that of the fpqc site.
A morphism $f \colon X \longrightarrow Y$ of schemes is called weakly étale if
$f$ is a flat morphism of schemes;
its diagonal $X \longrightarrow X \times_Y X$ is also flat.
(Bhatt-Scholze 13, def. 4.1.1)
For $X$ a scheme, its pro-étale site $X_{proet}$ is the site whose objects are weakly étale morphisms into $X$ (hence weakly étale $X$-schemes) and whose Grothendieck topology is that of the fpqc site.
(Bhatt-Scholze 13, def. 4.1.1)
A commutative ring $R$ is a w-contractible ring if every faithfully flat pro-étale morphism $Spec A \to Spec R$ has a section.
(Bhatt-Scholze 13, def. 2.4.1)
For every commutative ring $R$, there is a a w-contractible $A$, def. , equipped with a faithfully flat pro-étale morphism $Spec A \to Spec R$.
(Bhatt-Scholze 13, lemma 2.4.9)
So the full subcategory on the w-contractible rings forms a dense subsite of the pro-étale site, consisting of objects with pro-étale homotopy type a set.
Since every étale morphism of schemes is in particular a pro-étale morphism, there is induced a geometric morphism
from the pro-étale topos to the étale topos of any scheme $X$.
$\nu$ is a surjective geometric morphism with fully faithful inverse image. Hence the ordinary étale topos is a coreflection of the pro-étale topos.
(Bhatt-Scholze 13, lemma 5.1.2)
Given a field $k$ with separable closure $\overline{k}$, then the pro-etale site of $Spec(\overline{k})$ is equivalently the category of profinite sets. The pro-etale site of $Spec(k)$ identifies with the category of profinite continuous $Gal(\overline k/k)$-sets. (Bhatt-Scholze 13, example 4.1.10).
The pro-étale topology was suggested by Scholze and then fully developed in
Some results used in the study of weakly étale maps appeared earlier in
A textbook account is in
Further developments include
Last revised on January 25, 2023 at 17:07:18. See the history of this page for a list of all contributions to it.