The Stacks project

Lemma 21.39.7. Notation and assumptions as in Example 21.39.1. Let $U_\bullet $ be a cosimplicial object in $\mathcal{C}$ such that for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the simplicial set $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U)$ is homotopy equivalent to the constant simplicial set on a singleton. Then

\[ L\pi _!(\mathcal{F}) = \mathcal{F}(U_\bullet ) \]

in $D(\textit{Ab})$, resp. $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp. $\textit{Mod}(\underline{B})$.

Proof. As $L\pi _!$ agrees for modules and abelian sheaves by Lemma 21.38.5 it suffices to prove this when $\mathcal{F}$ is an abelian sheaf. For $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the abelian sheaf $j_{U!}\mathbf{Z}_ U$ is a projective object of $\textit{Ab}(\mathcal{C})$ since $\mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathbf{Z}_ U, \mathcal{F}) = \mathcal{F}(U)$ and taking sections is an exact functor as the topology is chaotic. Every abelian sheaf is a quotient of a direct sum of $j_{U!}\mathbf{Z}_ U$ by Modules on Sites, Lemma 18.28.8. Thus we can compute $L\pi _!(\mathcal{F})$ by choosing a resolution

\[ \ldots \to \mathcal{G}^{-1} \to \mathcal{G}^0 \to \mathcal{F} \to 0 \]

whose terms are direct sums of sheaves of the form above and taking $L\pi _!(\mathcal{F}) = \pi _!(\mathcal{G}^\bullet )$. Consider the double complex $A^{\bullet , \bullet } = \mathcal{G}^\bullet (U_\bullet )$. The map $\mathcal{G}^0 \to \mathcal{F}$ gives a map of complexes $A^{0, \bullet } \to \mathcal{F}(U_\bullet )$. Since $\pi _!$ is computed by taking the colimit over $\mathcal{C}^{opp}$ (Lemma 21.38.8) we see that the two compositions $\mathcal{G}^ m(U_1) \to \mathcal{G}^ m(U_0) \to \pi _!\mathcal{G}^ m$ are equal. Thus we obtain a canonical map of complexes

\[ \text{Tot}(A^{\bullet , \bullet }) \longrightarrow \pi _!(\mathcal{G}^\bullet ) = L\pi _!(\mathcal{F}) \]

To prove the lemma it suffices to show that the complexes

\[ \ldots \to \mathcal{G}^ m(U_1) \to \mathcal{G}^ m(U_0) \to \pi _!\mathcal{G}^ m \to 0 \]

are exact, see Homology, Lemma 12.25.4. Since the sheaves $\mathcal{G}^ m$ are direct sums of the sheaves $j_{U!}\mathbf{Z}_ U$ we reduce to $\mathcal{G} = j_{U!}\mathbf{Z}_ U$. The complex $j_{U!}\mathbf{Z}_ U(U_\bullet )$ is the complex of abelian groups associated to the free $\mathbf{Z}$-module on the simplicial set $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U)$ which we assumed to be homotopy equivalent to a singleton. We conclude that

\[ j_{U!}\mathbf{Z}_ U(U_\bullet ) \to \mathbf{Z} \]

is a homotopy equivalence of abelian groups hence a quasi-isomorphism (Simplicial, Remark 14.26.4 and Lemma 14.27.1). This finishes the proof since $\pi _!j_{U!}\mathbf{Z}_ U = \mathbf{Z}$ as was shown in the proof of Lemma 21.38.5. $\square$

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