The Stacks project

Remark 21.39.13. Let $\mathcal{C}$ and $B$ be as in Example 21.39.1. Assume there exists a cosimplicial object as in Lemma 21.39.7. Let $\mathcal{O} \to \underline{B}$ be a map sheaf of rings on $\mathcal{C}$ which induces an isomorphism $L\pi _!\mathcal{O} \to L\pi _!\underline{B}$. In this case we obtain an exact functor of triangulated categories

Namely, for any object $K$ of $D(\mathcal{O})$ we have $L\pi ^{\textit{Ab}}_!(K) = L\pi ^{\textit{Ab}}_!(K \otimes _{\mathcal{O}}^\mathbf {L} \underline{B})$ by Lemma 21.39.12. Thus we can define the displayed functor as the composition of $- \otimes ^\mathbf {L}_\mathcal {O} \underline{B}$ with the functor $L\pi _! : D(\underline{B}) \to D(B)$. In other words, we obtain a $B$-module structure on $L\pi _!(K)$ coming from the (canonical, functorial) identification of $L\pi _!(K)$ with $L\pi _!(K \otimes _\mathcal {O}^\mathbf {L} \underline{B})$ of the lemma.