The Stacks project

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 20.55.7 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.