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?
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2$\begingroup$ Yes, it follows from the fact that $A_{\mathfrak m}$ is excellent. $\endgroup$– gdbCommented Nov 25, 2017 at 22:24
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$\begingroup$ @gdb, can you give me a reference to this fact? I mean that an excellent ring is analytically unramified. $\endgroup$– MikhailCommented Nov 25, 2017 at 22:35
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1 Answer
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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).
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1$\begingroup$ Perhaps you can add a quick proof for Theorem 1 as follows. I'll show that R_0 + S_1 implies reduced. For any ring $A$, we know that $A \hookrightarrow \prod_{\mathfrak{p} \in \text{Ass}(A)} A_{\mathfrak{p}}$. However S_1 implies that this is a product over minimal primes, and then R_0 implies that each $A_{\mathfrak{p}}$ is a field. Therefore $A$ injects into a (finite) product of fields and so is reduced. $\endgroup$ Commented Nov 26, 2017 at 23:14