higher geometry / derived geometry
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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.
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)
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))
As discussed there, an étale morphism is a formally étale morphism which is locally of finite presentation.
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
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.
Let $f \;\colon\; A \longrightarrow B$ be a homomorphism of rings.
Almost conversely
(Bhatt-Scholze 13, theorem 1.3)
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
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)
Last revised on June 30, 2016 at 12:05:06. See the history of this page for a list of all contributions to it.