nLab pro-étale site

Idea

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).

Definition

Definition

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.

Properties

In terms of weakly étale maps

Definition

A morphism $f \colon X \longrightarrow Y$ of schemes is called weakly étale if

1. $f$ is a flat morphism of schemes;

2. its diagonal $X \longrightarrow X \times_Y X$ is also flat.

(Bhatt-Scholze 13, def. 4.1.1)

Definition

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)

Generation by w-contractible rings

Definition

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)

Proposition

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)

Remark

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.

Relation to the étale topos

Definition

Since every étale morphism of schemes is in particular a pro-étale morphism, there is induced a geometric morphism

$$\nu \;\colon\; Sh(X_{proet}) \longrightarrow Sh(X_{et})$$

from the pro-étale topos to the étale topos of any scheme $X$.

Proposition

$\nu$ 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)

Examples

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).

Related concepts

• étale homotopy

• étale cohomology

• condensed set

• condensed cohesion

References

The pro-étale topology was suggested by Scholze and then fully developed in

• Bhargav Bhatt, Peter Scholze, The pro-étale topology for schemes Astérisque No. 369 (2015), 99–201(arXiv:1309.1198)

Some results used in the study of weakly étale maps appeared earlier in

• Ofer Gabber and Lorenzo Ramero, Almost ring theory, volume 1800 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2003. (arXiv:math/0201175)

A textbook account is in

• The Stacks Project, Pro-étale Cohomology (pdf)

Further developments include

• Takashi Suzuki, Duality for local fields and sheaves on the category of fields (arXiv:1310.4941)