## 38.42 An operator introduced by Berthelot and Ogus

Let $X$ be a scheme. Let $D \subset X$ be an effective Cartier divisor. Let $\mathcal{I} = \mathcal{I}_ D \subset \mathcal{O}_ X$ be the ideal sheaf of $D$, see Divisors, Section 31.14. Clearly we can apply the discussion in Cohomology, Section 20.53 to $X$ and $\mathcal{I}$.

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. $\square$

Lemma 38.42.2. Let $X$ be a scheme. Let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. The functor $L\eta _\mathcal {I} : D(\mathcal{O}_ X) \to D(\mathcal{O}_ X)$ of Cohomology, Lemma 20.53.7 sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into itself. Moreover, if $X = \mathop{\mathrm{Spec}}(A)$ is affine and $D = V(f)$, then the functor $L\eta _ f$ on $D(A)$ defined in More on Algebra, Lemma 15.95.4 and the functor $L\eta _\mathcal {I}$ on $D_\mathit{QCoh}(\mathcal{O}_ X)$ correspond via the equivalence of Derived Categories of Schemes, Lemma 36.3.5.

Proof. Omitted. $\square$

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