Let $(R,m)$ be an excellent Noetherian local ring. Let $S$ be a smooth (i.e. $R \rightarrow S$ is flat and has geometrically regular fibers) Noetherian $R$-algebra. Let $T$ be the $m S$-adic completion of $S$. Then by the universality of the tensor product construction, there is a natural map $\hat{R} \otimes_R S \rightarrow T$. My question is: Does this map have to be flat?
Neil Epstein
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