Is a localization of a reduced finitely generated algebra analytically unramified?

Question

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?

Answer 1

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).