Let $X$ be a noetherian integral (separated) scheme which is regular in codimension one. Let $Y$ be a prime divisor and let $\eta$ be the generic point of $Y.$ It seems I am missing something easy but why $\mathcal{O}_{X, \eta}$ is a DVR with the quotient field the function field of $X?$
And when it is said, $X$ is regular (non-singular) of codimension one, does it follow from the definition that the local ring of a codimension one closed subscheme is regular in general? (otherwise, the terminology doesn't make sense to me!)