Points in integral separated schemes are determined by their local rings

Question

I am a bit stuck with the following assertion:

Let $X$ be a separated integral scheme. Then to every (schematic) point $x \in X$ we can correspond its local ring, and look at it as a subring of the field of rational functions on $X$. How do we show that different points give rise to different such subrings?

If $X$ is affine, it is clear, since we can find a function which is zero on one point and non-zero on another. If $X$ is not affine, I guess that for two points which do not lie in same affine open subscheme we should choose affine neighbourhoods and consider their intersection, which is affine by separatedness. But what to do then?

Thanks, Sasha

Answer 1

This follows from the valuative criterion for separatedness: for every valuation ring $O$ of $K(X)$ there exists at most one point $x\in X$ such that $O$ is a local ring extension of the local ring $O_{X,x}$.

Assume $O_{X,x}=O_{X,y}=:O^\prime$. By the so-called $x-\frac{1}{x}$-Lemma there exists a valuation ring $O$ of $K(X)$ such that the extension $O^\prime\subseteq O$ is local. Hence $x=y$.