Lemma 38.43.13. In Situation 38.43.1. Let $b : X' \to X$, $D' \subset X'$, and $Q$ be as in Lemma 38.43.7. If there exists a locally bounded complex $\mathcal{M}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules representing $M$, then there exists a locally bounded complex $\mathcal{Q}^\bullet $ of finite locally free $\mathcal{O}_{X'}$-modules representing $Q$.
Proof. Recall that $Q = L\eta _{\mathcal{I}'}Lb^*M$ where $\mathcal{I}'$ is the ideal sheaf of the effective Cartier divisor $D'$. The locally bounded complex $(\mathcal{M}')^\bullet = b^*\mathcal{M}^\bullet $ of finite locally free $\mathcal{O}_{X'}$-modules represents $Lb^*M$. Thus the lemma follows from Lemma 38.43.4. $\square$
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