Let $k$ be a field, let $R$ be a $k$-algebra, let $\{ S_i \}_{i \in I}$ be an inverse system of $k$-algebras, and let $R \to S_i$ be a $k$-algebra homomorphism making $S_i$ into a flat $R$-module. Is it true that $R \to \varprojlim_i S_i$ is also flat?
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Aren't inverse limits left exact? Ah, but one is worried about commuting with $\otimes$. See http://math.stackexchange.com/questions/181004/inverse-limit-of-modules-and-tensor-product – Hoot Jun 17 '16 at 23:49