The Stacks project

Remark 36.6.6. Let $X$, $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$, and $\mathcal{F}$ be as in Remark 36.6.4. Denote $\mathcal{F}^\bullet $ the complex (36.6.4.1). By Lemma 36.6.5 the cohomology sheaves of $\mathcal{F}^\bullet $ are supported on $Z$ hence $\mathcal{F}^\bullet $ is an object of $D_ Z(\mathcal{O}_ X)$. On the other hand, the equality $\mathcal{F}^0 = \mathcal{F}$ determines a canonical map $\mathcal{F}^\bullet \to \mathcal{F}$ in $D(\mathcal{O}_ X)$. As $i_* \circ R\mathcal{H}_ Z$ is a right adjoint to the inclusion functor $D_ Z(\mathcal{O}_ X) \to D(\mathcal{O}_ X)$, see Cohomology, Lemma 20.34.2, we obtain a canonical commutative diagram

in $D(\mathcal{O}_ X)$ functorial in the $\mathcal{O}_ X$-module $\mathcal{F}$.