The Stacks project

Lemma 85.25.1. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. There is a commutative diagram

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{LC}_{qc}/U)_{total}) \ar[r]_-h \ar[d]_{a_{qc}} & \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \ar[d]^ a \\ \mathop{\mathit{Sh}}\nolimits (\textit{LC}_{qc}/X) \ar[r]^-{h_{-1}} & \mathop{\mathit{Sh}}\nolimits (X) } \]

where the left vertical arrow is defined in Section 85.21 and the right vertical arrow is defined in Lemma 85.2.8.

Proof. Write $\mathop{\mathit{Sh}}\nolimits (X) = \mathop{\mathit{Sh}}\nolimits (X_{Zar})$. Observe that both $(\textit{LC}_{qc}/U)_{total}$ and $U_{Zar}$ fall into case A of Situation 85.3.3. This is immediate from the construction of $U_{Zar}$ in Section 85.2 and it follows from Lemma 85.21.5 for $(\textit{LC}_{qc}/U)_{total}$. Next, consider the functors $U_{n, Zar} \to \textit{LC}_{qc}/U_ n$, $U \mapsto U/U_ n$ and $X_{Zar} \to \textit{LC}_{qc}/X$, $U \mapsto U/X$. We have seen that these define morphisms of sites in Cohomology on Sites, Section 21.31. Thus we obtain a morphism of simplicial sites compatible with augmentations as in Remark 85.5.4 and we may apply Lemma 85.5.5 to conclude. $\square$

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