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Let $B$ be an abelian variety over a DVR with good reduction, and let $A$ be a subquotient of $B$. Then $A$ has good reduction.

I know a proof of this statement using Neron-Ogg-Shafarevich. Is there a more direct argument (e.g. not mentioning Galois action)?

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    $\begingroup$ Serre and Tate write that this corollary was known to Koizumi and Shimura. I have been unable to obtain the article, but it is indexed by MathSciNet: MR116017. (The review strongly suggests it is a direct argument, and there certainly does not appear to be mention of Galois actions.) $\endgroup$ – R. van Dobben de Bruyn Aug 2 at 20:22
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    $\begingroup$ The Koizumi/Shimura paper can be found in Shimura's Collected Works. They show that if there is a surjective homomorphism from A to B and A has good reduction, then so does B. It was written in 1959 before most people knew of schemes and such questions were difficult. It looks geometric and direct, but makes unpleasant reading today. A modern version must be possible. $\endgroup$ – anon Aug 3 at 0:10
  • $\begingroup$ I see an easy argument in equicharacteristic zero, but I’m blanking in mixed characteristic. (In equichar zero: suffices to prove the statements for quotients and subs separately, but then enough to prove for subs by duality. For subs, take the closure of the generic fiber of your subscheme; this is a gp scheme hence smooth by the char zero assumption.) $\endgroup$ – Daniel Litt Aug 3 at 1:27
  • $\begingroup$ @DanielLitt: The normalisation of the closure of the generic fibre probably works in general. $\endgroup$ – ulrich Aug 3 at 7:54
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    $\begingroup$ @user126532: I don't think this question or this user is of the same nature as those that you are referencing. $\endgroup$ – Daniel Litt Aug 3 at 22:15

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