Suppose that $A$ is a reduced finitely generated algebra over a field and $\mathfrak{m}\subset A$ is a maximal ideal. Is it true that the localization $A\_{\mathfrak{m}}$ is analytically unramified, i.e. the completion
$$
\widehat{A\_{\mathfrak{m}}} = \lim\limits\_{\infty\leftarrow n}A\_{\mathfrak{m}}/(\mathfrak{m}A\_{\mathfrak{m}})^n
$$
is reduced?



Is a localization of a reduced finitely generated algebra analytically unramified?
Let me expand my comment as an answer. There is a notion of excellent rings, for a precise definition look here <https://stacks.math.columbia.edu/tag/07QS> (and see Chapter 13 of Matsumura's book "Commutative Algebra" for a self-contained systematic development). We will need only one important feature of excellent rings. Namely, that morphism $A\_{\mathfrak p} \to \hat A\_{\mathfrak p}$ is regular (flat + regular geometric fibers) for any prime ideal $\mathfrak p$.


Now, we use that any field $k$ is excellent and that all finite type algebras over an excellent ring are excellent (EGA IV$\_2$ 7.8.6). Applying this to your finitely generated $k$-algebra $A$ we see that it is excellent. In particular, $A\_{\mathfrak m} \to \hat A\_{\mathfrak m}$ is regular morphism of local rings. The key is Serre's criterion for reducedness (<https://stacks.math.columbia.edu/tag/031O>) and Theorem 23.9 from Matsumura's book "Commutative ring theory". Let me state them here:



> 
> **Theorem 1(Serre's criterion for reducedness):**Let A be a noetherian ring, then it is reduced iff it has properties (R\_0) and (S\_1).
> 
> 
> **Theorem 2(Theorem 23.9)** Let $A \to B$ be a local faithfully flat morphism of local noetherian rings. If $A$ and all fibers $B\otimes\_A k(\mathfrak p)$ have property (R\_i) (resp. (S\_i)), then $B$ also has property (R\_i) (resp. (S\_i)).
> 
> 
> 


Note that $A\_{\mathfrak m}$ is reduced as a localization of a reduced ring and any regular ring (such as any fiber of $A\_{\mathfrak m} \to \hat A\_{\mathfrak m}$) has properties (R\_i) and (S\_i) for all i. Thus from regularity of the homomorphism $A\_{\mathfrak m} \to \hat A\_{\mathfrak m}$ and theorems highlighted above we conclude that $\hat A\_{\mathfrak m}$ is reduced.


P.S. If you want just a reference for this fact, you can use EGA IV$\_2$ 7.8.6 and EGA IV$\_2$ 7.8.3 (vii).


