I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an $R$-$S$-bimodule complex $X$ s.t. the derived tensor product $- \otimes^{\mathbb{L}}_R X \colon \mathcal{D}(R) \to \mathcal{D}(S)$ is a triangulated equivalence (cf. 3.1 and 3.2 here, Theorem 3.13 here).

Is there an example of two (non-flat) algebras which are derived equivalent, but for which no such bimodule complex exists? Or is there a proof of this kind of "Morita theorem" without the flatness assumption?



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