**Proof.**
The easy implication follows from Cohomology on Sites, Lemma 21.46.5. For the converse, assume that $\epsilon ^*E$ has tor amplitude in $[a, b]$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. As $\epsilon $ is a flat morphism of ringed sites (Lemma 75.4.1) we have

\[ \epsilon ^*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{F}) = \epsilon ^*E \otimes ^\mathbf {L}_{\mathcal{O}_{\acute{e}tale}} \epsilon ^*\mathcal{F} \]

Thus the (assumed) vanishing of cohomology sheaves on the right hand side implies the desired vanishing of the cohomology sheaves of $E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{F}$ via Lemma 75.4.1.
$\square$

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