Lemma 38.42.1. Let $X$ be a scheme. Let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_ X$-modules such that $\mathcal{F}^ i$ is $\mathcal{I}$-torsion free for all $i$. Then $\eta _\mathcal {I}\mathcal{F}^\bullet$ is a complex of quasi-coherent $\mathcal{O}_ X$-modules. Moreover, if $U = \mathop{\mathrm{Spec}}(A) \subset X$ is affine open and $D \cap U = V(f)$, then $\eta _ f(\mathcal{F}^\bullet (U))$ is canonically isomorphic to $(\eta _\mathcal {I}\mathcal{F}^\bullet )(U)$.

Proof. Omitted. $\square$

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