I read about valuations yesterday, fun stuff. I know a few different places I can apply them that are fun. Does anyone want to tell me why they care about them? @BalarkaSen perhaps?

@Narcissus I can tell you how I motivate them. I don't really remember the intricacies of the definition but the details shouldn't be hard to fill in if I missed it: If you take any complex algebraic curve $X$, for any point $x$ on it you could look at the germ of holomorphic functions/stalk of the structure sheaf $\mathscr{O}_{X, x}$ at $x$.

If you take a holomorphic chart $(U, \varphi)$ at $x$, you could set up a canonical isomorphism $\mathbb C\{z\} \to \mathscr{O}_{X, x}$ from the ring of convergent (single variable) Taylor series at $0$.

The fraction field $K = \text{Frac} \, \Bbb C\{z\}$ is just the ring of convergent Laurent series at $0$. Now that has a natural valuation on it, $\nu : K \to \Bbb Z \cup \{\infty\}$ such that if $f = \sum_{i = n}^\infty a_i z^i$, $\nu(f)$ is just the number $n$ (possibly negative).

Under this, the integral domain $\Bbb C\{z\}$ becomes a discrete valuation ring (given precisely by $\nu \geq 0$)

So the way to think of this is as if you choose a local parameterization of $X$ at $x$, given by the variable $z : U \subset X \to \Bbb C$, then $\mathscr{O}_{X, x}$ becomes a discrete valuation ring with uniformizing parameter $z$. I.e., every element of $\mathscr{O}_{X, x}$ can be written as $u \cdot z^k$ where $u$ is a unit and $k$ is some natural number.

Alternately, $(z)$ is the unique maximal ideal that makes $\mathscr{O}_{X, x}$ a local PID

Oh, I should have said $x$ is a smooth point of $X$

Further, you could say $\mathscr{O}_{X, x}$ is a local ring of Krull dimension 1 (pretty much literally because it's coming from the stalk of a 1-dimensional algebraic curve at a smooth point).

This pretty much sums up the various abstract definitions of a DVR: 1) Z-valued valuation ring 2) local PID 3) integrally closed Noetherian 1-dimensional local ring 4) Local ring with a uniformizing parameter ...

There is a construction in the last section of Hartshorne chapter I which is basically the following: If $A$ is an integral domain with a field $K \supset A$, look at the set of DVR's containing $A$ with fraction field $K$, and give it the topology generated by open sets of the form $U(\alpha_1, \cdots, \alpha_n)$ consisting of DVR's with fraction field $K$ and containing $A[\alpha_1, \cdots, \alpha_n]$.

This is the so-called "Zariski-Riemann space", and is one algebraic model for the notion of a "Riemann surface over $A$", I think. But the intuition is that; locally the DVR which is the stalk of the curve at that point determines the local structure of the curve, so you replace each point by the DVR.

That's pretty much what my intuition is. Unlikely to be very illuminating because we fundamentally think differently, but there you go.