Let $X \subset \mathbb{C}^n$ be an affine Cohen-Macaulay variety. I would like to know whether for every point $p \in X$, the local ring of germs of analytic functions on $X$ at $p$ is also Cohen-Macaulay. A reference would be appreciated. Below I present my (rather long) attempt at a proof.
Let us assume that $p =0$ and write $R = \mathbb{C}[x_1,\ldots,x_n]_{(x)}$ and $S = \mathbb{C}\{x_1,\ldots,x_n\}$ (the ring of power series convergent in some polydisc around the origin). Since both are local and the embedding $R \to S$ is a map of local rings, we can apply the flatness criterion. Namely, $S$ is flat over $R$ if and only if $\operatorname{Tor}^R_1(S,R/m) = 0$, where $m$ is the maximal ideal of $R$. Now a free resolution of $R/m$ is given by the Koszul complex and tensoring with $S$ it stays exact except for the rightmost term and we are done. So $S$ is flat over $R$. Now let $I \subset R$ be the ideal generated by the polynomials that vanish on $X$. So we have a commutative diagram:
$\begin{matrix} R & \rightarrow & S \\ \downarrow & & \downarrow \\ R/I & \rightarrow & R/I \otimes_R S\end{matrix}$.
Since the upper arrow is flat, then so is the lower arrow. I believe that $R/I \otimes_R S$ is the ring of germs of analytic functions at the origin on $X$. Now choose a regular sequence that testifies to the fact that $R/I$ is CM. Tensoring the associated complex by $R/I \otimes_R S$ preserves cohomology, so the same sequence is regular in the ring of germs.