Confused about generic points of closed integral subschemes in Hartshorne

Question

On page 130 of Hartshorne's Algebraic Geometry, a prime divisor on $X$ is defined as a closed integral subscheme $Y$ of codimension one. It is then claimed that the if $\eta \in Y$ is the generic point of $Y$, then the local ring $\mathcal{O}_{\eta, X}$ is a DVR with quotient field being equal to the function field of $X$. I was of the understanding that the local ring at the generic point of any integral scheme was always a field and that this field was \emph{defined} to be the function field. What am I missing here?

Answer 1

Here Hartshorne is talking about the local ring of $\eta$ inside $X$, not $Y$. And $\eta$ is not the generic point of $X$.