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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.
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:
it is smooth and unramified,
it is smooth and of relative dimension $0$,
it is flat and unramified,
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):
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
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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}
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.
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This appears for instance as (Milne, prop. 2.1).
expanded to statement of the basic closure properties as follows:
+– {: .num_prop}
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 $f_2$.
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(e.g. Milne, prop. 2.11)
+– {: .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.
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