TITLE: Completion of a local ring of a curve
QUESTION [16 upvotes]: Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ the local ring of rational functions which are regular at $p$. Then, is it true that the completion of $\mathcal{O}_p(X)$ (with respect to its maximal ideal) is isomorphic to the ring $\mathbb{K}[[x]]$ of formal power series in one variable ? I think that it should follow from Cohen theorem, but I cannot find a reference for this. Most of the results on this are in commutative ring theory books (for example Matsumura) but not really in the language of algebraic geometry. Can someone please give me a reference? I need this result in a research article. Thanks in advance.

REPLY [3 votes]: One possible reference is Mumford, The red book of varieties and schemes, chap. III, § 6.