The Stacks project

Lemma 105.3.8. Let $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to (\mathcal{Y} \subset \mathcal{Y}')$ be a morphism of thickenings of algebraic stacks. Let $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ and $\Delta ' : \mathcal{X}' \to \mathcal{X}' \times _{\mathcal{Y}'} \mathcal{X}'$ be the corresponding diagonal morphisms. Then each property from the following list is satisfied by $\Delta $ if and only if it is satisfied by $\Delta '$: (a) representable by schemes, (b) affine, (c) surjective, (d) quasi-compact, (e) universally closed, (f) integral, (g) quasi-separated, (h) separated, (i) universally injective, (j) universally open, (k) locally quasi-finite, (l) finite, (m) unramified, (n) monomorphism, (o) immersion, (p) closed immersion, and (q) proper.

Proof. Observe that

\[ (\Delta , \Delta ') : (\mathcal{X} \subset \mathcal{X}') \longrightarrow (\mathcal{X} \times _\mathcal {Y} \mathcal{X} \subset \mathcal{X}' \times _{\mathcal{Y}'} \mathcal{X}') \]

is a morphism of thickenings (Lemma 105.3.7). Moreover $\Delta $ and $\Delta '$ are representable by algebraic spaces by Morphisms of Stacks, Lemma 100.3.3. Hence, via the discussion in Properties of Stacks, Section 99.3 the lemma follows for cases (a), (b), (c), (d), (e), (f), (g), (h), (i), and (j) by using More on Morphisms of Spaces, Lemma 75.10.1.

Lemma 105.3.7 tells us that $\mathcal{X} = (\mathcal{X} \times _\mathcal {Y} \mathcal{X}) \times _{(\mathcal{X}' \times _{\mathcal{Y}'} \mathcal{X}')} \mathcal{X}'$. Moreover, $\Delta $ and $\Delta '$ are locally of finite type by the aforementioned Morphisms of Stacks, Lemma 100.3.3. Hence the result for cases (k), (l), (m), (n), (o), (p), and (q) by using More on Morphisms of Spaces, Lemma 75.10.3. $\square$

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Your email address will not be published.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