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I have been reading the paper "Good Reduction of Abelian Varieties" of Serre-Tate and in particular the part where they show that the Tate module being unramified implies good reduction.

As in the paper, let $\mathcal O_v$ be a DVR (discrete valuation ring), $K_v$ it's field of fractions, $k_v$ it's residue field and $A/K_v$ an abelian variety. Also, let $A_v$ be the Neron model of $A$ over $\mathcal O_v$.

In the first part of the proof, they show that the reduction of $A_v$ (= $A_v\times_{\mathcal O_v}k_v$) is proper over $k_v$ and need to show that $A_v$ itself is proper over $\mathcal O_v$. They do it by reducing to the complete case and then applying some results from EGA (that I don't understand very well).

I wanted to try proving it myself using the valuative criterion for properness. To this end, let $R$ be a DVR over $\mathcal O_v$ with fraction field $K$ and suppose we are given maps $\operatorname{Spec} K \to A_v$ and $\operatorname{Spec} R \to \operatorname{Spec} \mathcal O_v$ that form a commuting diagram. We need to show that there is exactly one lift from $\operatorname{Spec} R \to A_v$ that fills in the diagram.

The proposed proof now splits into three cases. Either, $\operatorname{Spec} R \to \operatorname{Spec} \mathcal O_v$ factors through the generic point of $\operatorname{Spec}\mathcal O_v$ or through the closed point or neither happens (that is, the map is surjective).

In the first case, we can replace $A_v$ by $A/K_v$ and since this is proper, we are done. In the second case, we can replace $A_v$ by $A_v\otimes_{\mathcal O_v}k_v$ and this has also been shown to be proper and we are done.

In the final case, I would like to argue as follows: Since $A_v$ is the Neron model, $A_v(\operatorname{Spec} R) = A(\operatorname{Spec} K)$ and so there is a unique lift. Unfortunately, this argument will only work if $\operatorname{Spec} R$ is smooth over $\operatorname{Spec} O_v$ but of course this is not always true.

Is there a way to fix this argument? I can see one way to fix it if the following were true:

If $X/\mathcal O_v$ is a smooth, separated scheme, is it true that $X/\mathcal O_v$ is proper if and only if the valuative criterion is satisfied for all smooth DVRS $R/\mathcal O_v$? Or even for all smooth DVR's and all DVR's that do not map surjectively onto $\mathcal O_v$?

Is this true?

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    $\begingroup$ 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. $\endgroup$
    – nfdc23
    Commented Sep 25, 2017 at 1:13
  • $\begingroup$ I see... Do you know if there is another way to make the argument work using the valuative criterion? $\endgroup$
    – Asvin
    Commented Sep 25, 2017 at 1:22
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    $\begingroup$ 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. $\endgroup$
    – nfdc23
    Commented Sep 25, 2017 at 1:59
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    $\begingroup$ 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 $\endgroup$
    – nfdc23
    Commented Sep 25, 2017 at 4:10

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