Timeline for What's the intuition between formal smoothness, etaleness. and unramifiedness?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 15, 2010 at 21:29 | comment | added | Akhil Mathew | OK. Yes, the general length of EGA IV is slightly intimidating, and so far I am only familiar with the weak ZMT (basically the one in Hartshorne/EGA III that follows from the formal function business). I'll take a look at the structure theorem. | |
| Nov 15, 2010 at 5:58 | comment | added | BCnrd | Dear Akhil: Just to clarify, for your purposes it is the finitely presented ZMT that you need for the proof of the structure theorem for etale morphisms in EGA, and this structure theorem in turn is used in the EGA proof of the general ZMT. So I do not recommend at this point that you look at the general ZMT. (In Raynaud's Springer Lecture Notes book on henselian rings he gives a proof of ZMT from scratch, but I don't remember in what generality, and my recollection is that the proof in EGA is more "geometric", though much longer.) | |
| Nov 15, 2010 at 0:03 | comment | added | Akhil Mathew | OK, makes sense--thanks. I'll take a look at the general ZMT. | |
| Nov 14, 2010 at 23:59 | comment | added | BCnrd | Akhil, the 1st paragraph of their pf of flatness of smooth maps is exactly what I said: smooth is etale over affine space, so one reduces to etale case, and in etale case one stares at local structure thm for etale maps (2.3/3), which is exactly 18.4.6(ii) in EGA IV$_4$, whose proof absolutely uses ZMT (in the finitely presented case). In "Neron Models" they state general finite type case of ZMT (18.12.13 in EGA IV$_4$), but beware that the pf in this generality uses the local structure thm for etale maps. Stick to f. presented ZMT to avoid circularity. Cannot avoid ZMT. That is the engine. | |
| Nov 14, 2010 at 22:56 | comment | added | Akhil Mathew | (Unfortunately I'm not yet at the point where I can outline the argument here.) | |
| Nov 14, 2010 at 22:16 | comment | added | Akhil Mathew | OK, thanks; that makes sense. I'm still slightly confused about the last remark you make (about ZMT being necessary to prove flatness). According to N-M, p. 53, Prop. 8, an inductive argument is given (on the number of equations needed to describe $X$ locally as a subscheme of of $\mathbb{A}^n_Y$, where $X \to Y$ is a smooth morphism), that a smooth morphism which is locally of finite presentation is flat. I believe that the proof that formal smoothness for locally f.p. morphisms is equivalent to the Jacobian criterion is also mostly ZMT-free. | |
| Nov 14, 2010 at 20:39 | comment | added | BCnrd | Dear Akhil: It should also be emphasized that this "primitive element theorem" description of etale maps is the only way (that I can think of offhand) to prove that locally finitely presented and formally etale morphisms are actually flat, or even that etale maps as defined via the Jacobian criterion in "Neron Models" are flat. Of course one can cheat and impose flatness in the definition, but that just makes the carpet pop up in another corner of the room when it comes to actually verifying the definition in abstract situations. | |
| Nov 14, 2010 at 20:34 | comment | added | BCnrd | Dear Akhil: Solving formal power series by succ. approx. uses square-zero steps, so saying "Hensel's Lemma" is too fancy. Also, "Neron Models" was written by 3 people, not 1. Equivalence with "primitive element thm" formulation is deep: requires Zariski's Main Thm. Need ZMT to get to open in something finite in order to "lift" (via Nakayama) the "primitive element thm" description at a point in the fiber to a description on a Zariski-neighborhood in the total space. See EGA IV$_4$, 18.4.6(ii) (beware (i) is false in formally etale case, as I noted elsewhere on MO with counterexample). | |
| Nov 14, 2010 at 20:09 | comment | added | Akhil Mathew | Dear BCnrd, thanks! (I'd accept these comments, and many others, if you re-posted them as answers.) A couple of questions: is the means of solving formal power series equations Hensel's lemma? Also, according to Raynaud's Neron Models (Def. 3, p.36) a morphism is etale if it locally factors through an immersion into affine space such that the ideal in affine space associated to the immersion is generated by elements with linearly independent differentials and has relative dimension zero. How does get from that to open in Spec $((R[t]/(f))[1/f']))$ for one variable? | |
| Nov 14, 2010 at 19:04 | comment | added | BCnrd | In the final sentence of previous comment, "smooth case" meant "for smooth varieties". Anyway, real content is not the definitions, but rather that for locally finitely presented maps these formal criteria imply Zariski-local descriptions which look exactly like differential geometry (formally etale implies Zariski-locally looks like open in Spec(($R[t]/(f))[1/f']) \rightarrow $ Spec($R$) for some monic $f \in R[t]$ and hence flat by inspection, formally smooth implies Zariski-locally etale over an affine space, and formally unramified implies "etale-locally" a closed immersion). | |
| Nov 14, 2010 at 18:57 | comment | added | BCnrd | Try to solve finite system of formal power series eqns in several formal power series unknowns (with constant term 0, over a field) via succ. approx. That's where square-zero business comes from, and linear part matrix (= Jacobian) being injective, surjective, or isom is what controls whether can lift solns from each stage to next and whether uniquely or not. In view of submersion/immersion/inv.fn. thms in diff. geom., one arrives at Grothendieck's notions. For varieties over $k$, tangential criteria OK in smooth case at $k$-points (as then complete local ring is power series ring over $k$). | |
| Nov 14, 2010 at 18:12 | history | asked | Akhil Mathew | CC BY-SA 2.5 |