Question:

Completion of a local ring of a curve

Lillian: 3 days ago

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.

Answer:
Logan: 3 days ago

My answer is most likely going to be rephrasing Francesco's answer above. Here's how I think about your question. IMHO, the Cohen Structure Theorem is too big a thermonuclear weapon to invoke, because from memory, the hardest part of the proof is the existence of a coefficient field. In your case, we get a coefficient field almost for free! What we need to prove is this:

Proposition: Let $A$ be an $n$-dimensional regular ring that is an integral domain. Furthermore, suppose that $A$ is a finite-type $k$-algebra with $k$ algebraically closed. Then for any maximal ideal $\mathfrak{m}$, we have $\widehat{A}_{\mathfrak{m}} \cong k[[t_1,\ldots,t_n ]]$.

Proof: Let $x_1,\ldots,x_n$ be a regular system of parameters of $\mathfrak{m}A_{\mathfrak{m}}$. By the universal property of the power series ring, we have a ring homomorphism $\varphi : k[[t_1,\ldots,t_n]] \to \widehat{A}_{\mathfrak{m}}$, sending $t_i$ to the image of $x_i$ in $\widehat{A}_{\mathfrak{m}}$. Since the residue field of $\widehat{A}_{\mathfrak{m}}$ is isomorphic to $k$ by Zariski's Lemma, it follows by Nakayama's Lemma and Stacks 031D (http://stacks.math.columbia.edu/tag/031D) that $\varphi$ is surjective.

The ring $\widehat{A}_{\mathfrak{m}}$ is a regular local ring, a fortiori an integral domain. Hence $\varphi$ is a surjective ring homomorphism between two integral domains of the same dimension and thus is an isomorphism. Done!

Added: The proposition above is not true if we delete the regularity hypothesis. Consider the union of two lines $X = \operatorname{Spec} k[x,y]/(xy)$. Let $p$ denote the origin of the plane $\Bbb{A}^2$. Since $x$ and $y$ are non-zero in $\mathcal{O}_{X,p}$, they are non-zero in the completion since $\mathcal{O}_{X,p} \to \widehat{\mathcal{O}}_{X,p}$ is injective by Krull's Intersection Theorem. It follows that $\widehat{\mathcal{O}}_{X,p}$ is not isomorphic to the power series ring in one variable.