Lemma 85.34.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. There is a commutative diagram

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/U)_{fppf, total}), \mathcal{O}_{big, total}) \ar[r]_-h \ar[d]_{a_{fppf}} & (\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}), \mathcal{O}_ U) \ar[d]^ a \\ (\mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{fppf}), \mathcal{O}_{big}) \ar[r]^-{h_{-1}} & (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) } \]

of ringed topoi where the left vertical arrow is defined in Section 85.22 and the right vertical arrow is defined in Section 85.32.

**Proof.**
For the underlying diagram of topoi we refer to the discussion in the proof of Lemma 85.33.1. The sheaf $\mathcal{O}_ U$ is the structure sheaf of the simplicial algebraic space $U$ as defined in Section 85.32. The sheaf $\mathcal{O}_ X$ is the usual structure sheaf of the algebraic space $X$. The sheaves of rings $\mathcal{O}_{big, total}$ and $\mathcal{O}_{big}$ come from the structure sheaf on $(\textit{Spaces}/S)_{fppf}$ in the manner explained in Section 85.22 which also constructs $a_{fppf}$ as a morphism of ringed topoi. The component morphisms $h_ n = a_{U_ n}$ and $h_{-1} = a_ X$ are morphisms of ringed topoi by More on Cohomology of Spaces, Section 84.7. Finally, since the continuous functor $u : U_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/U)_{fppf, total}$ used to define $h$^{1} is given by $V/U_ n \mapsto V/U_ n$ we see that $h_*\mathcal{O}_{big, total} = \mathcal{O}_ U$ which is how we endow $h$ with the structure of a morphism of ringed simplicial sites as in Remark 85.7.1. Then we obtain $h$ as a morphism of ringed topoi by Lemma 85.7.2. Please observe that the morphisms $h_ n$ indeed agree with the morphisms $a_{U_ n}$ described above. We omit the verification that the diagram is commutative (as a diagram of ringed topoi – we already know it is commutative as a diagram of topoi).
$\square$

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