Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 20th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarted entries * [[weakly étale morphism of schemes]] * [[pro-étale cohomology]] Question: We have the implications [[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]] but can one say more specifically what kind of generalized finite presentability condition makes a formally étale morphism a weakly étale morphism?

   started entries

   • weakly étale morphism of schemes

   • pro-étale cohomology

   Question: We have the implications

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

   but can one say more specifically what kind of generalized finite presentability condition makes a formally étale morphism a weakly étale morphism?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeNov 20th 2013
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexWell, an $A$-algebra $B$ is finitely presentable in the classical sense if and only if $B$ is finitely presentable as an object in ${}^{A /} \mathbf{CRing}$ (qua locally finitely presentable category). But I very much doubt this generalises to the non-affine case.

   Well, an $A$-algebra $B$ is finitely presentable in the classical sense if and only if $B$ is finitely presentable as an object in ${}^{A /} \mathbf{CRing}$ (qua locally finitely presentable category). But I very much doubt this generalises to the non-affine case.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 20th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn the affine case, what is the generalization of finitely presentable that makes a formally étale morphism be weakly étale?

   In the affine case, what is the generalization of finitely presentable that makes a formally étale morphism be weakly étale?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 20th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere is nothing like actually reading an article from the beginning. ;-) I have added the following to [[weakly etale morphism of schemes]] and also created _[[pro-etale morphism of schemes]]_; *** In fact a weakly étale morphism is equivalently a [[formally étale morphism]] which is "locally [[pro-object|pro-finitely]] presentable" (dually locally of [[ind-object|ind]]-finite rank) in the following sense +-- {: .num_defn #IndEtale} ###### Definition For $A \to B$ a [[homomorphism]] of [[rings]], say that it is an **[[ind-étale morphism]]** if that $A$-[[associative algebra|algebra]] $B$ is a [[filtered colimit]] of $A$-[[étale algebras]]. =-- +-- {: .num_prop} ###### Proposition Let $f \;\colon\; A \longrightarrow B$ be a [[homomorphism]] of [[rings]]. * If $f$ is ind-étale, def. \ref{IndEtale}, then it is weakly étale, def. \ref{WeaklyEtale}. 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 [[composition|composite]] $g\circ f$ is ind-étale. =-- ([Bhatt-Scholze 13, theorem 1.3](#BhattScholze13)) +-- {: .num_cor} ###### Corollary The [[sheaf toposes]] over the [[sites]] of weak étale morphisms and of [[pro-étale morphisms of schemes]] into some base [[scheme]] are [[equivalence of categories|equivalent]], see at _[[pro-étale site]]_. =--

   There is nothing like actually reading an article from the beginning. ;-) I have added the following to weakly etale morphism of schemes and also created pro-etale morphism of schemes;

   ---

   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

   +– {: .num_defn #IndEtale}

   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.

   =–

   +– {: .num_prop}

   Proposition

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

   • If $f$ is ind-étale, def. \ref{IndEtale}, then it is weakly étale, def. \ref{WeaklyEtale}.

   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)

   +– {: .num_cor}

   Corollary

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

   =–