The Stacks project

Lemma 85.36.1. 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

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

where the left vertical arrow is defined in Section 85.21 and the right vertical arrow is defined in Section 85.32.

Proof. The notation $(\textit{Spaces}/U)_{ph, total}$ indicates that we are using the construction of Section 85.21 for the site $(\textit{Spaces}/S)_{ph}$ and the simplicial object $U$ of this site1. We will use the sites $X_{spaces, {\acute{e}tale}}$ and $U_{spaces, {\acute{e}tale}}$ for the topoi on the right hand side; this is permissible see discussion in Section 85.32.

Observe that both $(\textit{Spaces}/U)_{ph, total}$ and $U_{spaces, {\acute{e}tale}}$ fall into case A of Situation 85.3.3. This is immediate from the construction of $U_{\acute{e}tale}$ in Section 85.32 and it follows from Lemma 85.21.5 for $(\textit{Spaces}/U)_{ph, total}$. Next, consider the functors $U_{n, spaces, {\acute{e}tale}} \to (\textit{Spaces}/U_ n)_{ph}$, $U \mapsto U/U_ n$ and $X_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{ph}$, $U \mapsto U/X$. We have seen that these define morphisms of sites in More on Cohomology of Spaces, Section 84.8 where these were denoted $a_{U_ n} = \epsilon _{U_ n} \circ \pi _{u_ n}$ and $a_ X = \epsilon _ X \circ \pi _ X$. 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. $\square$

[1] To distinguish from $(\textit{Spaces}/U)_{fppf, total}$ defined using the fppf topology in Section 85.33.

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