Timeline for The final step in the proof of Neron-Ogg-Shafarevich as in the paper of Serre-Tate
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| Sep 25, 2017 at 4:10 | comment | added | nfdc23 | I slightly misspoke above; the argument in Neron Models appeals to a later result in EGA (but one can use the result cited by Serre and Tate instead after passing to complete $O_v$, as you can do since properness descends due to Chow's Lemma; a merit of the presentation in Neron Models is that it organizes the reasoning to apply without assuming perfectness of the residue field). You seek a short proof of EGA III$_1$ 5.5.1 with $Z_0=X_0$ in the quasi-projective case: apply the Theorem on Formal Functions to an idempotent on the special fiber of a projective closure. QED | |
| Sep 25, 2017 at 1:59 | comment | added | nfdc23 | By the way, the class of counterexamples I suggested exists in abundance provided that $k(v)$ is not perfect. But that is fine, since obviously you want Neron models to work over dvr's without parasitic perfectness hypotheses on the residue field since for any mixed-characteristic dvr $O$ (e.g., $\mathbf{Z}_{(p)}$) and smooth $O$-scheme $X$ with $X \to {\rm{Spec}}(O)$ having fibers of dimension $d>0$, if $\eta$ is a generic point of the special fiber $X_0$ then $O_{X,\eta}$ is a dvr with imperfect residue field; such dvrs show up all over in the study of abelian and semi-abelian schemes. | |
| Sep 25, 2017 at 1:22 | comment | added | Asvin | I see... Do you know if there is another way to make the argument work using the valuative criterion? | |
| Sep 25, 2017 at 1:13 | comment | added | nfdc23 | That result from EGA is very powerful and worth understanding; in 7.4/5 of Neron Models the same argument appears. Your use of the word "smooth" is inaccurate: you mean "ind-smooth", since in practice the relevant maps $O_v \to R$ are essentially never (locally) of finite type; you intend for $R$ to be a local ring on a smooth $O_v$-scheme, so it's a direct limit of smooth $O_v$-algebras. Your question has a negative answer, and it is much better to justify a counterexample yourself once you have a hint: remove a non-etale closed point from the special fiber of a proper $O_v$-scheme. | |
| Sep 24, 2017 at 23:19 | history | asked | Asvin | CC BY-SA 3.0 |