nLab
pro-étale site

Contents

Context

Geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Étale morphisms

Contents

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 cohomologypro-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 XX a scheme, its pro-étale site X proetX_{proet} is the site whose objects are pro-étale morphisms into XX and whose Grothendieck topology is that of the fpqc site.

Properties

In terms of weakly étale maps

Definition

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

  1. ff is a flat morphism of schemes;

  2. its diagonal XX× YXX \longrightarrow X \times_Y X is also flat.

(Bhatt-Scholze 13, def. 4.1.1)

Definition

For XX a scheme, its pro-étale site X proetX_{proet} is the site whose objects are weakly étale morphisms into XX (hence weakly étale XX-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 RR is a w-contractible ring if every faithfully flat pro-étale morphism SpecASpecRSpec A \to Spec R has a section.

(Bhatt-Scholze 13, def. 2.4.1)

Proposition

For every commutative ring RR, there is a a w-contractible AA, def. , equipped with a faithfully flat pro-étale morphism SpecASpecRSpec 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

ν:Sh(X proet)Sh(X et) \nu \;\colon\; Sh(X_{proet}) \longrightarrow Sh(X_{et})

from the pro-étale topos to the étale topos of any scheme XX.

Proposition

ν\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 kk with separable closure k¯\overline{k}, then the pro-etale site of Spec(k¯)Spec(\overline{k}) is equivalently the category of profinite sets. The pro-etale site of Spec(k)Spec(k) identifies with the category of profinite continuous GG-sets. (Bhatt-Scholze 13, example 4.1.10).

References

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

  • 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

Further developments include

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

Last revised on May 1, 2019 at 10:38:48. See the history of this page for a list of all contributions to it.