nLab
weakly étale morphism of schemes

Context

Geometry

Cohomology

cohomology

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Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A variant of étale morphism of schemes where the finiteness conditions on étale morphisms are relaxed.

Used in the definition of pro-étale site and pro-étale cohomology.

Definition

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)

Properties

(Gabber-Ramero 03, theorem 2.5.36, prop. 3.2.16 Bhatt-Scholze 13, prop. 2.3.3. (2))

Remark

As discussed there, an étale morphism is a formally étale morphism which is locally of finite presentation.

In fact a weakly étale morphism is equivalently a formally étale morphism which is “locally pro-finitely presentable” (dually locally of ind-finite rank) in the following sense

Definition

For ABA \to B a homomorphism of rings, say that it is an ind-étale morphism if that AA-algebra BB is a filtered colimit of AA-étale algebras.

Proposition

Let f:ABf \;\colon\; A \longrightarrow B be a homomorphism of rings.

  • If ff is ind-étale, def. , then it is weakly étale, def. .

Almost conversely

(Bhatt-Scholze 13, theorem 1.3)

Corollary

The sheaf toposes over the sites of weak étale morphisms and of pro-étale morphisms of schemes into some base scheme are equivalent, both define the pro-étale topos over the pro-étale site.

étale morphism\Rightarrow pro-étale morphism \Rightarrow weakly étale morphism \Rightarrow formally étale morphism

References

Last revised on June 30, 2016 at 08:05:06. See the history of this page for a list of all contributions to it.