The Stacks project

Remark 92.27.2. In the situation above, for every object $U \to X$ of $X_{\acute{e}tale}$ let $P_{\bullet , U}$ be the standard resolution of $\mathcal{O}_ X(U)$ over $\Lambda$. Set $\mathbf{A}_{n, U} = \mathop{\mathrm{Spec}}(P_{n, U})$. Then $\mathbf{A}_{\bullet , U}$ is a cosimplicial object of the fibre category $\mathcal{C}_{\mathcal{O}_ X(U)/\Lambda }$ of $\mathcal{C}_{X/\Lambda }$ over $U$. Moreover, as discussed in Remark 92.5.5 we have that $\mathbf{A}_{\bullet , U}$ is a cosimplicial object of $\mathcal{C}_{\mathcal{O}_ X(U)/\Lambda }$ as in Cohomology on Sites, Lemma 21.39.7. Since the construction $U \mapsto \mathbf{A}_{\bullet , U}$ is functorial in $U$, given any (abelian) sheaf $\mathcal{F}$ on $\mathcal{C}_{X/\Lambda }$ we obtain a complex of presheaves

whose cohomology groups compute the homology of $\mathcal{F}$ on the fibre category. We conclude by Cohomology on Sites, Lemma 21.40.2 that the sheafification computes $L_ n\pi _!(\mathcal{F})$. In other words, the complex of sheaves whose term in degree $-n$ is the sheafification of $U \mapsto \mathcal{F}(\mathbf{A}_{n, U})$ computes $L\pi _!(\mathcal{F})$.