nLab weakly étale morphism of schemes

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 $f \colon X \longrightarrow Y$ 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)

Properties

Proposition

Every weakly étale morphism is a formally étale morphism.

(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.

Corollary

A weakly étale morphism which is locally of finite presentation is an étale morphism.

étale morphism$\Rightarrow$ weakly étale morphism $\Rightarrow$ formally étale morphism

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 $A \to B$ a homomorphism of rings, say that it is an ind-étale morphism if that $A$-algebra $B$ is a filtered colimit of $A$-étale algebras.

Proposition

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

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

Almost conversely

• If $f$ is weakly étale, then there is a faithfully flat morphism $g \colon B \to C$ which is ind-étale such that the composite $g\circ f$ is ind-étale.

(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.

Related concepts

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

References

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

• Bhargav Bhatt, Peter Scholze, The pro-étale topology for schemes (arXiv:1309.1198)