Lemma 82.6.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then there are commutative diagrams of topoi

and

with $a_ X = \pi _ X \circ \epsilon _ X$ and $a_ Y = \pi _ X \circ \epsilon _ X$.

Lemma 82.6.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then there are commutative diagrams of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{\epsilon _ X} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{fppf}) \ar[d]^{\epsilon _ Y} \\ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \ar[rr]^{f_{big, {\acute{e}tale}}} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{\acute{e}tale}) } \]

and

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{a_ X} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{fppf}) \ar[d]^{a_ Y} \\ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \ar[rr]^{f_{small}} & & \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) } \]

with $a_ X = \pi _ X \circ \epsilon _ X$ and $a_ Y = \pi _ X \circ \epsilon _ X$.

**Proof.**
This follows immediately from working out the definitions of the morphisms involved, see Topologies on Spaces, Section 71.7 and Section 82.5.
$\square$

Your email address will not be published.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.

## Comments (0)