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For many times, I find people working on schemes over DVRs, and prove theorems on such setting. For example, my latest experience is the "semi-stable reduction theorem" by Kempf, Knudsen, Mumford and Saint-Donat: they proved semi-stable reduction theorem first over $\mathbb{C}$, then turned to the situation over DVR.

It is "natural" for me to work over $\mathbb{C}$, or over character $p$, or even over arbitrary schemes. But why people like to choose DVRs in particular? This question would break to two parts:

(1) Is there any technical advantage to work over DVR?

(2) What is the importance to consider this special case? Does this come from the interests of number theory? I have no clue of that.

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If X is a smooth curve then the local ring at a point is a DVR. I think of Spec of a DVR like the geometric object which encodes the zariski local properties of a smooth curve around some point. Schemes over DVRs naturally come up when you pull back to this object –  Daniel Barter Aug 16 at 13:28

2 Answers 2

up vote 6 down vote accepted

Well, varieties and schemes typically come in families. If you accept that, then it's natural to consider the simplest case where the base is one dimensional and regular. This includes geometers smooth curves, and things like $Spec\mathbb{Z}$ or $Spec \mathbb{Z}_p$ of interest to number theorists. If you want to concentrate your attention to near a specific fibre, then you may as well localize to work over a DVR, which is a bit like working over a small disk... Not sure what else to say.

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Technically, as you know, the crucial thing is that for any noetherian scheme $B$ and distinct points $s, \eta \in B$ with $s \in \overline{\{\eta\}}$, there exists a map ${\rm{Spec}}(R) \rightarrow B$ for a dvr $R$ such that the closed point goes to $s$ and the generic point goes to $\eta$. Pullback over $R$ encodes these two fibers at the cost of a huge extension of their residue fields (especially at $s$). So for issues about geometric fibers, base as dvr captures a lot. And of course flatness is easy to control over a dvr (such as for schematic closures from the generic fiber, etc.). –  user54268 Aug 16 at 14:39

(1) What are the technical advantages of working over a DVR?

They are many. DVR are very simple rings, they are both local and principal. The principality in particular means that the flatness (which is a fundament and not sue say to grasp property in algebraic geometry) of a module or algebra over that sort of ring becomes something much simpler to understand, to check and to use, namely the propriety of being torsion-free. Also, a DVR is integrally closed, and that's a property that is very useful for certain type of reasoning in algebraic geometry.

(2) The interest of working with a DVR in algebraic geometry does not come primarily from number theory. For example, you may have seen the valuative criterion of properness: to check if a morphism $f: X \rightarrow Y$ is proper, under certain mild hypothesis, you only have to check that the pull-back of this morphism over any spectrum of DVR $Y'$ is. This reduces the verification of an important relative property over an arbitrary base to the case of much simpler base, see (1), and is a crucial tool in algebraic geometry.

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Your comment #2 is closely related to my comment to Arapura's answer, if one thinks in terms of Chow's Lemma and taking $s$ in the boundary of the projective closure of a non-closed quasi-projective scheme. –  user54268 Aug 17 at 5:19

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