I have a question about a statement in the proof of Theorem 6.9 of Hartshorne's Algebraic Geometry. The setup is this:
$Y$ is a projective curve and $C$ is an abstract nonsingular curve (although i think that it can be seen as a projecitve nonsingular curve for the matter of this question). Furthermore $\varphi \colon C \to Y$ is dominant morphism. Let $K$ be the funciton field of $C$ and $K'$ the function filed of $Y$.
Then he claims: For any $Q\in Y$ the local ring $\mathcal{O}_Q$ is dominated by some discrete valuation ring of $R$ of $K/k$, ($k$ being the algebraically closesd ground field). Take for example a localization of the integral closure of $\mathcal{O}_Q$ at a maximal ideal.
But i fail to see how this yields a discrete valuation ring, at least by means of simple arguments.
My thoughts so far: We can embed $\mathcal{O}_Q$ in $K$ with the induced field homomorphism $\varphi^* \colon K' \to K$. $\mathcal{O}_Q $ is a local noetherian domain of dimension $1$ (not necessarily integerally closed), so the integral closure in $K$ is an integrally closed domain of dimension $1$. So if we were to show that it was noetherian, its localization at a prime ideal would be a discrete valuation ring.
Is there an easy way to see that the integral closure is noetherian? I believe the Krull–Akizuki theorem tells us that the integral closure is indeed noetherian again; but this theorem is not mentioned anywhere in the book previously. Am I missing something completely? Or is there an alternative way to see that $\mathcal{O}_Q$ is dominated by a discrete valuation ring of $K/k$?
Any help is appreciated.
Best Answer
Instead of using the theorem of Krull-Akizuki one could also invoke the following result: the integral closure of an integral finitely generated algebra $A$ over a field $k$ in a finite extension of the fraction field of $A$ is a finitely generated $k$-algebra too. In particular it is noetherian. This result is mentioned although not proved in Hartshorne as Theorem 3.9 A.