Applications of valuation rings

If you're familiar with complex analysis, the collection of meromorphic functions on an open subset $U \subseteq \mathbb{C}$ can be endowed with many discrete valuations, one for each point of $U$. Given a meromorphic function $f$, for each $a \in U$ we can write $f(z) = (z - a)^v g(z)$ for some $v \in \mathbb{Z}$, where $g$ is a function holomorphic at $a$ with $g(a) \neq 0$. We define the order (of vanishing) of $f$ at $a$, denoted $\operatorname{ord}_a(f)$, to be this $v$. One can show that $\operatorname{ord}_a$ is a discrete valuation.

Similarly, in algebraic geometry DVRs can be used to measure the order of vanishing of a function at a point. Given a curve $C$ and a point $P \in C$, then the local ring at the point $P$ is a DVR iff $C$ is nonsingular at $P$. Basically, a function $f$ regular at $P$ has order of vanishing $v$ if $f \in \mathfrak{m}^v$ and $v$ is the smallest such positive integer, where $\mathfrak{m}$ is the maximal ideal corresponding to $P$.

This also allows us to detect the singularities of a curve algebraically. For instance, consider the cuspidal cubic $C: y^2 = x^3$. The origin is a singular point of $C$, as is clear from looking at a plot of the curve, or by computing partial derivatives, and this is reflected by the fact that the local ring $\left(\frac{k[x,y]}{(y^2 - x^3)}\right)_{(x,y)}$ is not a DVR.

A very natural set of examples come from number theory(which is apparently where the name divisor comes from!), namely number fields (i.e. finite extensions of the field of rational numbers $\mathbf{Q}$). The simplest case is $\mathbf{Q}$ itself considered as such an extension. One fixes a prime number $p$, and given any rational number $\alpha = \frac{a}{b}$ write $\alpha$ as $p^n\frac{a'}{b'}$ where the integers $a'$ and $b'$ are relatively prime and are not divisible by the fixed prime number $p$. It is easy to see (but needs proof) that such an $n \in \mathbf{Z}$ is uniquely determined(this is simply because $\mathbf{Z}$ is a unique factorization domain). Then, one gets a function on $\mathbf{Q}$ by defining $\nu_p(\frac{a}{b}):=n$, as found above. It is then an exercise to prove that $\nu_p$ is a valuation on $\mathbf{Q}$. It turns out that (Theorem of Ostrowski) apart from the usual absolute value, these are all valuations on $\mathbf{Q}$.

A parallel line of construction can be carried out by replacing $\mathbf{Q}$ with the field of rational functions over an algebraically closed field $k$, namely $k(X)$. In this case for any element $\alpha \in k$, there is a unique integer $n$ so that: $$f(X) = \frac{p(X)}{q(X)} = (X-\alpha)^n\frac{p'(X)}{q'(X)};$$ where $p'$ and $q'$ are relatively prime polynomials in $k[X]$. As was the case before, $k[X]$ is a UFD, hence $n$ is uniquely determined. Once again, we define $\nu_\alpha(f(X)) := n$. One must check that this defines a valuation on $k(X)$. An analogue for Ostrowski's theorem is still valid here with the only remaining function on $k(X)$ defining a valuation being the $\mathrm{deg}$ function: which is defined as the difference of the degrees of $p$ and $q$.

It must be stressed that the construction for $\mathbf{Q}$ can be carried out for any number field, and the construction for $k(X)$ can be carried out for any algebraic curve.

There is, of course, the tropical geometry world where valuations play a central role. this tool is used frequently to solve, at least, certain enumerative problems in algebraic geometry.