The Stacks project

Remark 21.41.3. Let $\mathcal{C}$ be a site. Let $\epsilon : \mathcal{A}_\bullet \to \mathcal{O}$ be an augmentation (Simplicial, Definition 14.20.1) in the category of sheaves of rings. Assume $\epsilon $ induces a quasi-isomorphism $s(\mathcal{A}_\bullet ) \to \mathcal{O}$. In this case we obtain an exact functor of triangulated categories

Namely, for any object $K$ of $D(\mathcal{A}_\bullet )$ we have $L\pi _!(K) = L\pi _!(K \otimes _{\mathcal{A}_\bullet }^\mathbf {L} \mathcal{O})$ by Lemma 21.41.2. Thus we can define the displayed functor as the composition of $- \otimes ^\mathbf {L}_{\mathcal{A}_\bullet } \mathcal{O}$ with the functor $L\pi _! : D(\Delta \times \mathcal{C}, \pi ^{-1}\mathcal{O}) \to D(\mathcal{O})$ of Remark 21.38.6. In other words, we obtain a $\mathcal{O}$-module structure on $L\pi _!(K)$ coming from the (canonical, functorial) identification of $L\pi _!(K)$ with $L\pi _!(K \otimes _{\mathcal{A}_\bullet }^\mathbf {L} \mathcal{O})$ of the lemma.