Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 66.28.5. Let $X$, $Y$ be algebraic spaces over $\mathbf{Z}$. If

\[ (g, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y) \]

is an isomorphism of ringed topoi, then there exists a unique morphism $f : X \to Y$ of algebraic spaces such that $(g, g^\sharp )$ is isomorphic to $(f_{small}, f^\sharp )$ and moreover $f$ is an isomorphism of algebraic spaces.

Proof. By Theorem 66.28.4 it suffices to show that $(g, g^\sharp )$ is a morphism of locally ringed topoi. By Modules on Sites, Lemma 18.40.8 (and since the site $X_{\acute{e}tale}$ has enough points) it suffices to check that the map $\mathcal{O}_{Y, q} \to \mathcal{O}_{X, p}$ induced by $g^\sharp $ is a local ring map where $q = f \circ p$ and $p$ is any point of $X_{\acute{e}tale}$. As it is an isomorphism this is clear. $\square$


Comments (0)


Your email address will not be published.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.