Originally Posted by

**topspin1617** If anyone is interested in the context of this question, I am reading through the proof of Lemma 6.5 in Hartshorne's Algebraic Geometry.

We have fields $\displaystyle k\subseteq K$ such that $\displaystyle \mathrm{tr.deg}_k(K)=1$. Also, assume $\displaystyle k$ is algebraically closed.

Let $\displaystyle y\in K\setminus k$, and let $\displaystyle B$ be the integral closure of the ring $\displaystyle k[y]$ in $\displaystyle K$. It can be shown that $\displaystyle B$ is a Dedekind domain which is finitely generated as a $\displaystyle k$-algebra (and I'm fine with that).

Now, suppose $\displaystyle R$ is a discrete valuation ring between $\displaystyle k$ and $\displaystyle K$ such that $\displaystyle y\in R$. Since $\displaystyle R$ is integrally closed (being a DVR), we must have $\displaystyle B\subseteq R$.

Here's where I get lost: they say that if $\displaystyle \mathfrak{m}_R$ is the (unique) maximal ideal of $\displaystyle R$, then $\displaystyle \mathfrak{n}=B\cap \mathfrak{m}_R$ is a maximal ideal of $\displaystyle B$, and that $\displaystyle B$ is *dominated* by $\displaystyle R$. What I don't see is (1) how we know that $\displaystyle \mathfrak{n}$ is a maximal ideal of $\displaystyle B$ (instead of just prime), and (2) why they are saying that $\displaystyle B$ is local. They seem to be claiming this, as the relation of domination is defined only for two local rings.