Completion (as a module) of the integral closure of a Cohen-Macaulay local ring whose completion is reduced Let $(R,\mathfrak m)$ be a Cohen-Macaulay local ring. Let $\overline R$ be the integral closure of $R$ in the total ring of fractions of $R$.
Let $\widehat R$ be the $\mathfrak m$-adic completion of $R$, and assume it is reduced (https://en.m.wikipedia.org/wiki/Analytically_unramified_ring), hence $\overline R$ is a module finite extension of $R$.
Now $\widehat R \otimes_R \overline R$ has a natural $R$-algebra structure.
My question is: Is $\widehat R \otimes_R \overline R$ a normal ring ?
[ Note that since $R$ is reduced, so $\overline R$ is a normal ring by https://stacks.math.columbia.edu/tag/030C .
(Does Serre's normality criteria help in answering my question https://en.m.wikipedia.org/wiki/Serre%27s_criterion_for_normality ?) ]
 A: I am pretty sure $\widehat{R} \otimes_R \overline{R}$ is not necessarily normal because of the following consequence of a theorem of Heitmann (1993):
Claim. For every dimension $d \ge 2$, there exists a local Cohen–Macaulay UFD $(R,\mathfrak{m})$ of dimension $d$ such that $\hat{R}$ is reduced but not normal.
Note that in this case, $R$ is normal, hence $R = \overline{R}$. There may be earlier counterexamples to your question, too.
To prove the claim, let $T$ be a complete local $\mathbf{Q}$-algebra of dimension $d$ that is Cohen–Macaulay and reduced, but not normal (for example, when $d = 2$, let $T$ be the completion of the local ring of the Whitney umbrella at the origin: $\mathbf{C}[[x,y,z]]/(x^2-y^2z)$). By Theorem 8 in [Heitmann 1993], there exists a local UFD $R$ such that $\hat{R} \cong T$.
On the other hand, we have the following affirmative answer in dimension 1:
Proposition. Let $(R,\mathfrak{m})$ be an analytically unramified noetherian local ring, and denote by $\overline{R}$ the integral closure of $R$ in the total ring of fractions of $R$. Then, $\hat{R} \otimes_R \overline{R}$ is normal.
I don't have a reference for this, so here is a proof.
Proof. Consider the cocartesian diagram
$$\require{AMScd}\begin{CD}
R @>>> \overline{R}\\
@VVV @VVV\\
\hat{R} @>>> \hat{R} \otimes_R \overline{R}
\end{CD}$$
where by base change, the horizontal arrows are module-finite, and the vertical arrows are faithfully flat.
To show that $\hat{R} \otimes_R \overline{R}$ is normal, it suffices to show that every localization at a maximal ideal $\mathfrak{n} \subseteq \hat{R} \otimes_R \overline{R}$ is normal. Moreover, since $\overline{R}$ is normal, by [Matsumura 1989, Cor. to Thm. 23.9], it suffices to show that for every such maximal ideal, the induced homomorphism
$$\varphi_{\mathfrak{n}}\colon \overline{R}_{\mathfrak{n} \cap \overline{R}} \longrightarrow (\hat{R} \otimes_R \overline{R})_{\mathfrak{n}}$$
has normal fibers. Note that $\mathfrak{n}$ contracts to the maximal ideal in $\hat{R}$ by the integrality of the bottom horizontal arrow, and hence $\mathfrak{n} \cap R = \mathfrak{m}$.
For the closed fiber, we tensor the cocartesian diagram above by $R/\mathfrak{m}$ to obtain the cocartesian diagram
$$\begin{CD}
R/\mathfrak{m} @>>> R/\mathfrak{m} \otimes_R \overline{R}\\
@V=VV @VV=V\\
\hat{R}/\mathfrak{m}\hat{R} @>>> \hat{R}/\mathfrak{m}\hat{R} \otimes_R \overline{R}
\end{CD}$$
The closed fiber of $\varphi_{\mathfrak{n}}$ is a closed fiber of the right vertical arrow, and is therefore a finite field extension of $R/\mathfrak{m}$, which is normal.
To show that the generic fiber of $\varphi_{\mathfrak{n}}$ is normal, let $\mathfrak{p} = (0) \cap R$ be the contraction of the zero ideal in $\overline{R}_{\mathfrak{n} \cap \overline{R}}$. Since $R$ is reduced, the localization $R_\mathfrak{p}$ is a field $K$, and localizing the cocartesian diagram above at $\mathfrak{p}$, we obtain the cocartesian diagram
$$\begin{CD}
K @>=>> K\\
@VVV @VVV\\
\hat{R} \otimes_R K @>=>> \hat{R} \otimes_R K
\end{CD}$$
It therefore suffices to show that $\hat{R} \otimes_R K$ is normal. By [Matsumura 1987, p. 261], the ring $\hat{R} \otimes_R K$ is a zero-dimensional localization of $\hat{R}$. Since $\hat{R}$ is reduced, we therefore see that $\hat{R} \otimes_R K$ is regular, hence normal. $\blacksquare$
