Talk:Étale morphism

message: I removed the reference to monic polynomials in the section
concening the localization of k[x_1,...,x_n]/<f_1,...,f_n>
since (a) I don't know what a monic polynomial in several variables
is and (b) I don't believe such a condition is necessary.
RJChapman 18:57, 27 November 2006 (UTC)
replies: text: I changed the condition "flat+unramified" to "flat+G-unramified". The difference is that "unramified" requires locally of finite type whereas "G-unramified" requires locally of finite presentation. Milne works under the general hypothesis of all schemes being locally Noetherian where the notions coincide, but in general one should make the distinction. 84.59.181.40 (talk) 22:46, 16 August 2017 (UTC)

message: Could somebody add an IPA pronunciation of the term étale?
Baccala@freesoft.org 08:09, 29 September 2007 (UTC)
replies: text:  Added the correct French pronunciation of the word étale (), and reverted a recent change to . The latter likely reflects local usage (somewhere in the US?); however, also in all English-language discussions I've had where étale morphisms have been discussed in the UK and the US, people have quite invariably used  essentially the French pronunciation. Thus, I'd stick to the French version. Stca74 (talk) 09:05, 10 August 2008 (UTC)
replies: text: Then please link it as French rather than English, and make it obvious to the reader that it isn't English. kwami (talk) 11:22, 2 March 2010 (UTC)
thread_title:  Pronunciation of étale 

message: It is not true that X ->  Y étale implies dim X = dim Y. Juste take Y = Spec (DVR) and X = generic point. I will remove change this statement. Liu (talk) 19:50, 6 January 2011 (UTC)
thread_title:  Comparison of dimensions 

message: An anonymous user recently tried to correct the definition of standard étale.  However, the definition formerly in the article and the definition given by the anonymous user are equivalent.  I wanted to record the equivalence of the two for anyone else who came along:
We have  a homomorphism of commutative rings, and we know f and g in R[x] such that f is monic and .  Suppose first that the derivative f′ is a unit in R[x]g.  Then it's obvious that f′ is also a unit in S: If we choose h to be the inverse of f′ in R[x]g, then h is still the inverse of f′ in S.  For the other direction, assume that f′ is a unit in S.  This means that there exists some h in R[x]g such that , where k is some other element of R[x]g.  The trick is that we replace g by .  Of course, localizing R[x] at g isn't the same as localizing it at g1; but that doesn't matter, because S localized at  is isomorphic to S: The fk term vanishes in S, so all the localization does is make 1 a unit, which was already true.  So we make the replacement, and then f′ is a unit in R[x]_{g_1}.
Geometrically, R[x] is a one-dimensional affine space over R (I'm leaving out the application of Spec here).  The space R[x]g is what you get by leaving out a hypersurface; you do this to remove all the bad points (where the morphism isn't étale).  Localizing at g1 simply leaves out more.  The space S is defined in R[x] by looking at the solutions of  and leaving out the bad points using g.  What we showed above is that leaving out bad points using g1 produces the same effect; in other words, the hypersurface defined by  doesn't meet S.  Ozob (talk) 14:11, 4 December 2012 (UTC)
thread_title:  Definition of standard étale