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.



Completion of a local ring of a curve
Let me expand and generalize my comments above. We can prove the following



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> **Proposition.** Let $X$ be a projective scheme of dimension $n$ which is defined over an algebraically closed field $k$. If $p \in X$ is a closed, regular point and $\mathcal{O}\_{X, \, p}$ is the local ring of $X$ at $p$, then there is an isomorphism $$\widehat{\mathcal{O}}\_{X, \, p}= k[[x\_1, \ldots, x\_n]].$$
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*Proof.* The ring $\mathcal{O}\_{X, \, p}$ is a Noetherian regular local ring of dimension $n$, whose residue field is $k$ since $p \in X$ is a closed point and $k$ is algebraically closed. Therefore its $\mathfrak{m}$-adic completion $\widehat{\mathcal{O}}\_{X, \,p}$ is a Noetherian complete regular ring with the same residue field and the same dimension, see 


Atiyah-MacDonald: *Introduction to Commutative Algebra*, Proposition 10.15, Theorem 10.26, Corollary 11.19, Proposition 11.24. 


By Cohen structure theorem it follows that $\widehat{\mathcal{O}}\_{X, \,p}$ is a formal power series over its residue field. Since its dimension is $n$, this implies that it is isomorphic to $k[[x\_1, \ldots, x\_n]]$ and we are done.


