7:51 AM
2 hours later…
9:43 AM
@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.
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.
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Jan '1811
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♦ SGA Over 9000: Séminaire d'Geometric …
I want to draw a commutative diagram in the room description s...