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For which commutative rings k is the following true:

A k-algebra $A$ that is flat over $k$ and derived equivalent to a $k$-algebra $B$ implies that also $B$ is flat over $k$.

The motivation is this: Many results on derived equivalences start with two k-algebras A and B that are assumed to be flat and derived equivalent. Maybe it is often enough to just assume one is flat?

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    $\begingroup$ Not a complete answer, but one necessary condition, which rules out many popular commutative rings, is that $k$ should have no finitely generated module $M$ with flat dimension equal to one. $\endgroup$
    – Jeremy Rickard
    Commented Aug 21, 2019 at 8:52

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