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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 12th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to the list of equivalent conditions in the definition at _[[étale morphism of schemes]]_ the pair "smooth+unramified". Added a remark after the definition on how to read these pairs of conditions.

   added to the list of equivalent conditions in the definition at étale morphism of schemes the pair “smooth+unramified”. Added a remark after the definition on how to read these pairs of conditions.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeNov 12th 2013
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIt would be more readable if the list had been made more compact (e.g. to fit on a small page etc.), say as follows: A morphism of schemes is étale if the one of the following four equivalent conditions hold: 1. it is [[smooth]] and [[unramified]], 2. it is [[smooth]] and of [[relative dimension]] $0$, 3. it is [[flat]] and [[unramified]], 4. it is [[formally étale]] and [[locally of finite presentation]]. BTW, [[Mariusz Wodzicki]] was telling us at the Eilenberg conference that smoothness and etalness are better replaced with a bit modified properties which are discovered very recently in the brilliant work of Bhatt and Scholze (aimed at going into the heart of l-adic cohomology theory), though he objected to the term pro-etale there as a bit inappropriate (and suggested stably flat, what is on the other hand overloaded term as well): * Bhargav Bhatt, Peter Scholze, _The pro-étale topology for schemes_, [arxiv/1309.1198](http://arxiv.org/abs/1309.1198)

   It would be more readable if the list had been made more compact (e.g. to fit on a small page etc.), say as follows:

   A morphism of schemes is étale if the one of the following four equivalent conditions hold:

   1. it is smooth and unramified,

   2. it is smooth and of relative dimension $0$,

   3. it is flat and unramified,

   4. it is formally étale and locally of finite presentation.

   BTW, Mariusz Wodzicki was telling us at the Eilenberg conference that smoothness and etalness are better replaced with a bit modified properties which are discovered very recently in the brilliant work of Bhatt and Scholze (aimed at going into the heart of l-adic cohomology theory), though he objected to the term pro-etale there as a bit inappropriate (and suggested stably flat, what is on the other hand overloaded term as well):

   • Bhargav Bhatt, Peter Scholze, The pro-étale topology for schemes, arxiv/1309.1198
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
   • (edited Nov 22nd 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAt _[[etale morphism of schemes]]_ there used to be a complaint query box *** +-- {: .standout} This proposition seems to be wrong for 2 reasons; first, A,B,R,S are 2 many symbols. Second, the statement sounds like "etale iff standard smooth" but only the direction "etale implies standard smooth" is true (and can be found in the stacks project). Since I couldn't find the cited source, I couldn't look into what the statement is supposed to be, originally. -- Konrad =-- *** The first reason is typos (symbol permutations), granted. The second, I am possibly missing something basic, but for the time being I removed the query box and edited the prop to read: *** +-- {: .num_prop} ###### Proposition A [[ring]] homomorphism of [[affine varieties]] $Spec(A) \to Spec(B)$ for $Spec(B)$ non-singular and for $A \simeq B[x_1, \cdots, x_n]/(f_1, \cdots, f_n)$ with [[polynomials]] $f_i$ is étale precisel if the [[Jacobian]] $det(\frac{\partial f_i}{\partial x_j})$ is invertible. =-- This appears for instance as ([Milne, prop. 2.1](#Milne)).

   At etale morphism of schemes there used to be a complaint query box

   ---

   +– {: .standout}

   This proposition seems to be wrong for 2 reasons; first, A,B,R,S are 2 many symbols. Second, the statement sounds like “etale iff standard smooth” but only the direction “etale implies standard smooth” is true (and can be found in the stacks project). Since I couldn’t find the cited source, I couldn’t look into what the statement is supposed to be, originally. – Konrad

   =–

   ---

   The first reason is typos (symbol permutations), granted. The second, I am possibly missing something basic, but for the time being I removed the query box and edited the prop to read:

   ---

   +– {: .num_prop}

   Proposition

   A ring homomorphism of affine varieties $Spec(A) \to Spec(B)$ for $Spec(B)$ non-singular and for $A \simeq B[x_1, \cdots, x_n]/(f_1, \cdots, f_n)$ with polynomials $f_i$ is étale precisel if the Jacobian $det(\frac{\partial f_i}{\partial x_j})$ is invertible.

   =–

   This appears for instance as (Milne, prop. 2.1).

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexexpanded to statement of the basic closure properties as follows: *** +-- {: .num_prop} ###### Proposition * A [[composite]] of two étale morphism is itself étale. * The [[pullback]] of an étale morphism is étale. * If $f_1 \circ f_2$ is étale and $f_1$ is, then so is $f_2$. =-- (e.g. [Milne, prop. 2.11](#Milne)) +-- {: .proof} ###### Proof Use that an étale morphism is a [[formally étale morphism]] with finite fibers, and that $f \colon X \to Y$ is formally étale precisely if the [[infinitesimal shape modality]] [[unit of a monad|unit]] [[natural transformation|naturality square]] $$ \array{ X &\longrightarrow& \Pi_{inf}(X) \\ \downarrow && \downarrow \\ Y &\longrightarrow& \Pi_{inf}(Y) } $$ is a [[pullback]] square. Then the three properties to be shown are equivalently the [[pasting law]] for pullback diagrams. =--

   expanded to statement of the basic closure properties as follows:

   ---

   +– {: .num_prop}

   Proposition

   • A composite of two étale morphism is itself étale.

   • The pullback of an étale morphism is étale.

   • If $f_1 \circ f_2$ is étale and $f_1$ is, then so is $f_2$.

   =–

   (e.g. Milne, prop. 2.11)

   +– {: .proof}

   Proof

   Use that an étale morphism is a formally étale morphism with finite fibers, and that $f \colon X \to Y$ is formally étale precisely if the infinitesimal shape modality unit naturality square

   $$\array{ X &\longrightarrow& \Pi_{inf}(X) \\ \downarrow && \downarrow \\ Y &\longrightarrow& \Pi_{inf}(Y) }$$

   is a pullback square. Then the three properties to be shown are equivalently the pasting law for pullback diagrams.

   =–