Let $X$ be a smooth projective variety, and $Y$ a subvariety of codimension one (both are irreducible). I want to show that the local ring $\mathcal{O}_{Y,X}$ at the subvariety $Y$ (which is nothing but the local ring $\mathcal{O}_{\eta,X}$ at the generic point $\eta$ of $Y$) is a discrete valuation ring.
I know the following:
- The local ring $\mathcal{O}_{P,X}$ at any point $P \in X$ (closed or not) is regular. (If $P$ is closed, this is the definition of "smooth", and any local ring at a non-closed point is a localisation of a local ring at a closed point, whence regular).
- Any Noetherian regular local ring of (Krull) dimension one is a discrete valuation ring.
It seems I need to prove that $\dim \mathcal{O}_{Y,X} = 1$, and since $Y$ is of codimension one in $X$, this would follow from the more general fact that for any subvariety $Z \subset X$, $\dim \mathcal{O}_{Z,X} = \text{codim}_{X} Z$.
My question is: Is this reasoning correct, and if yes, how do I prove it.
Thanks for your help.