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Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?

Let $K$ be the fraction field of $\mathcal{O}$. Assuming $A$ and $B$ are $\mathcal{O}$-flat, then also $A \widehat{\otimes} B$ is $\mathcal{O}$-flat (exercise), so it's enough to see that $C:=(A \...
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