The Stacks project

Lemma 85.25.2. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. If $a : U \to X$ gives a proper hypercovering of $X$, then

\[ a^{-1} : \mathop{\mathit{Sh}}\nolimits (X) \to \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \quad \text{and}\quad a^{-1} : \textit{Ab}(X) \to \textit{Ab}(U_{Zar}) \]

are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X)$ is as in Lemma 85.2.8.

Proof. We will prove the statement for sheaves of sets. It will be an almost formal consequence of results already established. Consider the diagram of Lemma 85.25.1. By Cohomology on Sites, Lemma 21.31.6 the functor $(h_{-1})^{-1}$ is fully faithful with quasi-inverse $h_{-1, *}$. The same holds true for the components $h_ n$ of $h$. By the description of the functors $h^{-1}$ and $h_*$ of Lemma 85.5.2 we conclude that $h^{-1}$ is fully faithful with quasi-inverse $h_*$. Observe that $U$ is a hypercovering of $X$ in $\textit{LC}_{qc}$ (as defined in Section 85.21) by Cohomology on Sites, Lemma 21.31.4. By Lemma 85.21.1 we see that $a_{qc}^{-1}$ is fully faithful with quasi-inverse $a_{qc, *}$ and with essential image the cartesian sheaves on $(\textit{LC}_{qc}/U)_{total}$. A formal argument (chasing around the diagram) now shows that $a^{-1}$ is fully faithful.

Finally, suppose that $\mathcal{G}$ is a cartesian sheaf on $U_{Zar}$. Then $h^{-1}\mathcal{G}$ is a cartesian sheaf on $\textit{LC}_{qc}/U$. Hence $h^{-1}\mathcal{G} = a_{qc}^{-1}\mathcal{H}$ for some sheaf $\mathcal{H}$ on $\textit{LC}_{qc}/X$. We compute

\begin{align*} (h_{-1})^{-1}(a_*\mathcal{G}) & = (h_{-1})^{-1} \text{Eq}( \xymatrix{ a_{0, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ (h_{-1})^{-1}a_{0, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & (h_{-1})^{-1}a_{1, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{qc, 0, *}h_0^{-1}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{qc, 1, *}h_1^{-1}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{qc, 0, *}a_{qc, 0}^{-1}\mathcal{H} \ar@<1ex>[r] \ar@<-1ex>[r] & a_{qc, 1, *}a_{qc, 1}^{-1}\mathcal{H} } ) \\ & = a_{qc, *}a_{qc}^{-1}\mathcal{H} \\ & = \mathcal{H} \end{align*}

Here the first equality follows from Lemma 85.2.8, the second equality follows as $(h_{-1})^{-1}$ is an exact functor, the third equality follows from Cohomology on Sites, Lemma 21.31.8 (here we use that $a_0 : U_0 \to X$ and $a_1: U_1 \to X$ are proper), the fourth follows from $a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$, the fifth from Lemma 85.4.2, and the sixth we've seen above. Since $a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$ we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}$ which ends the proof by fully faithfulness of $h^{-1}$. $\square$

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