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 such that . Also, assume is algebraically closed.
Let , and let be the integral closure of the ring in . It can be shown that is a Dedekind domain which is finitely generated as a -algebra (and I'm fine with that).
Now, suppose is a discrete valuation ring between and such that . Since is integrally closed (being a DVR), we must have .
Here's where I get lost: they say that if is the (unique) maximal ideal of , then is a maximal ideal of , and that is dominated by . What I don't see is (1) how we know that is a maximal ideal of (instead of just prime), and (2) why they are saying that is local. They seem to be claiming this, as the relation of domination is defined only for two local rings.