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According to J. S. Milne, Étale cohomology:

A morphism $f: Y \longrightarrow X$ that is locally of finite-type is said to be unramified at $y \in Y$ if $\mathcal{O}_{Y,y}/\mathfrak{m}_{x}\mathcal{O}_{Y,y}$ is finite separable field extension of $k(x)$, where $x=f(y)$.

My question is this: Based on the definition above, if I have a finite morphism $f: Y \longrightarrow X$ of projective varieties, I could say directly that:

"A finite morphism $f: Y \longrightarrow X$ of projective varieties is said to be unramified at $y \in Y$ if $\mathcal{O}_{Y,y}/\mathfrak{m}_{x}\mathcal{O}_{Y,y}$ is finite separable field extension of $k(x)$, where $x=f(y)$" Or I need to add hypotheses about Y and X, be like normal varieties, for example?

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    $\begingroup$ As finite implies locally of finite type, your statement is correct. I have no idea why you're doing this, though. $\endgroup$ – KReiser Apr 3 at 23:11
  • $\begingroup$ First of all, thank you for answering and still putting the question. I would like to define an unramified morphism of varieties (not schemas), I would like to think about the étale morphisms of varieties to see that they are the same if the morphism is dominant and finite. $\endgroup$ – Manoel Apr 3 at 23:25
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    $\begingroup$ My advice: use schemes. Being a luddite about schemes versus varieties is a really silly thing to be doing if you're really interested in the concept of etaleness. $\endgroup$ – KReiser Apr 4 at 3:54
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    $\begingroup$ @KReiser I think that's a bit rude, and not even a well-taken point. For varieties you can just use the Jacobian criterion of smoothness to define what smooth means. Then, etale just means smooth of relative dimension zero. $\endgroup$ – Alex Youcis Apr 4 at 4:38
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    $\begingroup$ Okay, sure, smooth of relative dimension zero is a great definition for etaleness. But what is one really going to go and do with that definition in the world of varieties? Unless variety means scheme over a field (+ adjectives), what's going to come out of etaleness that you couldn't find through the analytic topology? To me, the cool stuff one finds through the concept of etaleness is the extension of the idea of a "local diffeomorphism" to places where one has no expectation that would work (+ consequences like $\pi_1^{et}$) and the connection with number theoretic stuff, all schemey! $\endgroup$ – KReiser Apr 4 at 5:08

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