Lemma 20.46.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\mathcal{O}_ Y)$. 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 20.46.3 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, Lemma 17.20.2 the modules $f^*\mathcal{E}^ i$ are flat. Thus by Lemma 20.46.3 we conclude that $Lf^*E$ has tor amplitude in $[a, b]$.
$\square$

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