Relation between codimension and generic point

Question

I don't know how to solve the following exercise, which is stated in Vakil's The Rising Sea: Foundations of Algebraic Geometry:

11.1.I. EXERCISE. Show that if $Y$ is an irreducible closed subset of a scheme $X$, and $\eta$ is the generic point of $Y$, then the codimension of $Y$ is the dimension of the local ring $\mathscr{O}_{X,\eta}$.

The result is general, so I want to prove directly by definition but failed. I've also thought about proving by affine case, but I don't know the relation of codimensions if reduced to affine case.

Can anyone give me a solution?

Answer 1

We can prove this by proving that the prime ideals of $O_{x,X}$ for any point $x$ in a scheme $X$ correspond to integral closed subschemes $Z$ of $X$ containing $x$. Note I haven't assumed $x$ to be closed here.

We will use the bijection between closed subsets of $X$ containing the point $x$ and closed subsets of $X$ containing the closure $\bar{x}$, which follows from closure axioms.

Now note that if $Z$ is integral, closed in $X$, then $Z\cap U$ is integral, closed in $U$, so picking an affine $U$ containing our point $x$ we get a bijection between closed integral subschemes of $X$ containing $x$ and closed integral subschemes of $U$ containing $x$.

This reduces us to the affine case, where the claim to prove becomes $\text{dim}(R_\mathfrak{p})=\text{height}(\mathfrak{p})$, which follows from noting that the only prime ideals that survive to $R_\mathfrak{p}$ are those contained in $\mathfrak{p}$.

Both parts of this argument are really saying the same thing, which is relating how prime ideals correspond to integral subschemes, and how this behaves with localisation=restriction to open subschemes.