The Stacks project

Lemma 80.11.3. In Situation 80.10.6 the functor ( is fully faithful on algebraic spaces separated over $X$. More precisely, it induces a bijection

\[ \mathop{\mathrm{Mor}}\nolimits _ X(X'_1, X'_2) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Spaces}(Y \to X, Z)}(F(X'_1), F(X'_2)) \]

whenever $X'_2 \to X$ is separated.

Proof. Since $X'_2 \to X$ is separated, the graph $i : X'_1 \to X'_1 \times _ X X'_2$ of a morphism $X'_1 \to X'_2$ over $X$ is a closed immersion, see Morphisms of Spaces, Lemma 66.4.6. Moreover a closed immersion $i : T \to X'_1 \times _ X X'_2$ is the graph of a morphism if and only if $\text{pr}_1 \circ i$ is an isomorphism. The same is true for

  1. the graph of a morphism $U \times _ X X'_1 \to U \times _ X X'_2$ over $U$,

  2. the graph of a morphism $V \times _ X X'_1 \to V \times _ X X'_2$ over $V$, and

  3. the graph of a morphism $Y \times _ X X'_1 \to Y \times _ X X'_2$ over $Y$.

Moreover, if morphisms as in (1), (2), (3) fit together to form a morphism in the category $\textit{Spaces}(Y \to X, Z)$, then these graphs fit together to give an object of $\textit{Spaces}(Y \times _ X (X'_1 \times _ X X'_2) \to X'_1 \times _ X X'_2, Z \times _ X (X'_1 \times _ X X'_2))$ whose triple of morphisms are closed immersions. The proof is finished by applying Lemmas 80.11.1 and 80.11.2. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 80.11: Formal glueing of algebraic spaces

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Your email address will not be published.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 0AF6. Beware of the difference between the letter 'O' and the digit '0'.