The Stacks project

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). $\square$

[1] This happened in the proof of Lemma 85.33.1 via an application of Lemma 85.5.5.

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