# The local ring of the generic point of a prime divisor

Suppose $X$ is a noetherian integral separated scheme which is regular in codimension one, i.e. every local ring $O_x$ of dimension one is regular. Let $Y$ be a prime divisor of $X$, i.e. $Y$ is a closed integral subscheme of codimension one.

The last paragraph on page 130 of Hartshorne's Algebraic Gometry states:

If $Y$ is a prime divisor on $X$, let $\eta\in Y$ be its generic point. Then the local ring $O_{\eta,X}$ is a discrete valuation ring with quotient field $K$, the function field of $X$.

1. How do I see that $O_{\eta,X}$ has dimension one from the fact that Y has codimension 1 in X (which means $Y$ is a maximal proper closed irreducible set)?
2. How do I see that the quotient field of $O_{\eta,X}$ is $K(X)$? This only seems true if $\eta$ were the generic point of $X$, but it is not.

Let $$X$$ be any scheme, then I claim $$\dim O_{x,X} = \operatorname{codim} \{x\}^-$$ for any $$x\in X$$.

First step is to reduce to the case where $$X$$ is an affine scheme. Consider an affine open $$\operatorname{Spec} A$$ containing $$x$$, for each irreducible closed set $$K$$ of $$X$$ containing $$x$$, we obtain an irreducible closed set of $$\operatorname{Spec} A$$ containing $$x$$ by $$K\mapsto K\cap \operatorname{Spec} A$$. For each irreducible closed set of $$\operatorname{Spec} A$$ containing $$x$$, we obtain an irreducible closed set of $$X$$ containing $$x$$ by $$C\mapsto C^-$$, with closure taken inside $$X$$. We show this establishes a bijection. Clearly $$C^-\cap \operatorname{Spec} A=C$$, we have one sided inverse. For the other side, we need to show that $$(K\cap \operatorname{Spec} A)^- = K$$. Observe that $$K - \operatorname{Spec} A$$ and $$(K\cap \operatorname{Spec} A)^-$$ are two closed sets of $$X$$ whose union is $$K$$, and $$(K\cap \operatorname{Spec} A)^-$$ contains $$x$$ which is non-empty, we are thus done by irreducibility of $$K$$. (In fact this bijection works with same proof for any open set of $$X$$ containing $$x$$, not just $$\operatorname{Spec} A$$.)

Now let $$P\in \operatorname{Spec} A$$, we have $$\dim O_{P} = \dim A_P = \operatorname{codim} P$$. For each irreducible closed subset $$K$$ containing $$P$$, $$K$$ has a unique generic point $$Q$$, whence $$K = \{Q\}^- = V(Q)$$. Since $$P\in V(Q)$$ we have $$P\supset Q$$. Therefore an ascending chain of irreducible closed subset containing $$P$$ corresponds to a descending chain of prime ideals (the generic points of the irreducible closed subsets) from $$P$$. We thus conclude that $$\dim O_P = \operatorname{codim} \{P\}^-$$.

This solves question $$1$$ since any regular local ring of dimension $$1$$ is a DVR. For question two, we again reduce to an affine cover $$\operatorname{Spec} A$$ of $$\eta$$. Let $$\xi$$ be the generic point of $$X$$, we have $$\xi \in \operatorname{Spec} A$$ corresponds to the $$0$$ ideal since $$A$$ is a domain. Clearly the quotient ring of $$A_\eta$$ is $$A_0 = K(O_{\xi,X}) = K(X)$$.

• First,the generic point $\eta$ correspond to zero ideal $0$,right?Then,$dim O_{\eta,X}=dim O_{\eta,U}=dim A_{0}=0$,where $U=Spec(A)$ is an open set of an open cover of $X$ for some integral domain .Why I get a differnet fact? Am I missing something? – Jiabin Du Nov 22 '17 at 9:21
• Is the isomorphism between the quotient field of $\mathcal{O}_{\eta}$ and $K(X)$ natural, i.e., independent of the affine cover chosen? – Colescu Jul 15 '20 at 14:56

As for 1. this is an algebraic theorem that a ring A is DVR iff it's Noetherian, normal, and Spec A = {0, m} where m != 0 is maximal. You can find it in Miles Reid "Undergraduate Commutative Algebra".

As for why $\mathcal{O}_{\eta,X}$ is dimension one, this is a general fact that (under Noetherian hypothesis) increasing sequences of irreducible subvarieties correspond to decreasing sequences of prime ideals, and vice versa. I cannot give any good reference, as it is covered in every textbook in "dimension" chapter.

As for 2, let $\nu$ be the generic point of $X$. Then, $\mathcal{O}_{\nu, X} = K(X)$ is localization of $\mathcal{O}_{\eta,X}$ at its maximal ideal, which is the same as quotient field.

• Your answer for $1$ is irrelevant as you are talking about the Galois correspondence between the prime ideals of a finitely generated $k$ algebra that is an integral domain and the subvarieties of the variety. Your answer for $2$ is incorrect because $O_{\nu,X}$ is the localization of $O_{\eta,X}$ at its zero ideal, not the maximal ideal. It is trivial that if you localize a local ring at its maximal ideal you get back itself, since anything outside the maximal ideal has to be a unit. See my answer. – mez Jul 16 '15 at 12:10