The Stacks project

Lemma 84.33.1. 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

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

where the left vertical arrow is defined in Section 84.21 and the right vertical arrow is defined in Section 84.32.

Proof. The notation $(\textit{Spaces}/U)_{fppf, total}$ indicates that we are using the construction of Section 84.21 for the site $(\textit{Spaces}/S)_{fppf}$ 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 84.32.

Observe that both $(\textit{Spaces}/U)_{fppf, total}$ and $U_{spaces, {\acute{e}tale}}$ fall into case A of Situation 84.3.3. This is immediate from the construction of $U_{\acute{e}tale}$ in Section 84.32 and it follows from Lemma 84.21.5 for $(\textit{Spaces}/U)_{fppf, total}$. Next, consider the functors $U_{n, spaces, {\acute{e}tale}} \to (\textit{Spaces}/U_ n)_{fppf}$, $U \mapsto U/U_ n$ and $X_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{fppf}$, $U \mapsto U/X$. We have seen that these define morphisms of sites in More on Cohomology of Spaces, Section 83.6 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 84.5.4 and we may apply Lemma 84.5.5 to conclude. $\square$

[1] We could also use the ├ętale topology and this would be denoted $(\textit{Spaces}/U)_{{\acute{e}tale}, total}$.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DH5. Beware of the difference between the letter 'O' and the digit '0'.