The Stacks project

Proof. If (2) holds, then we may compute $E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F} = \mathcal{E}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{F}$ and it is clear that (1) holds.

Assume that (1) holds. We may represent $E$ by a bounded above complex of flat $\mathcal{O}_ X$-modules $\mathcal{K}^\bullet $, see Section 20.26. Let $n$ be the largest integer such that $\mathcal{K}^ n \not= 0$. If $n > b$, then $\mathcal{K}^{n - 1} \to \mathcal{K}^ n$ is surjective as $H^ n(\mathcal{K}^\bullet ) = 0$. As $\mathcal{K}^ n$ is flat we see that $\mathop{\mathrm{Ker}}(\mathcal{K}^{n - 1} \to \mathcal{K}^ n)$ is flat (Modules, Lemma 17.17.8). Hence we may replace $\mathcal{K}^\bullet $ by $\tau _{\leq n - 1}\mathcal{K}^\bullet $. Thus, by induction on $n$, we reduce to the case that $K^\bullet $ is a complex of flat $\mathcal{O}_ X$-modules with $\mathcal{K}^ i = 0$ for $i > b$.

Set $\mathcal{E}^\bullet = \tau _{\geq a}\mathcal{K}^\bullet $. Everything is clear except that $\mathcal{E}^ a$ is flat which follows immediately from Lemma 20.48.2 and the definitions. $\square$