Lemma 21.44.5. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\mathcal{O}_\mathcal {D})$. If $E$ has tor amplitude in $[a, b]$, then $Lf^*E$ has tor amplitude in $[a, b]$.

**Proof.**
Assume $E$ has tor amplitude in $[a, b]$. By Lemma 21.44.3 we can represent $E$ by a complex of $\mathcal{E}^\bullet $ of flat $\mathcal{O}$-modules with $\mathcal{E}^ i = 0$ for $i \not\in [a, b]$. Then $Lf^*E$ is represented by $f^*\mathcal{E}^\bullet $. By Modules on Sites, Lemma 18.38.1 the module $f^*\mathcal{E}^ i$ are flat. Thus by Lemma 21.44.3 we conclude that $Lf^*E$ has tor amplitude in $[a, b]$.
$\square$

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