The Stacks project

Lemma 98.11.5. Let $\mathcal{Z}$ be an algebraic stack. Assume $\mathcal{Z}$ satisfies the equivalent conditions of Lemma 98.11.2. Then there exists a unique strictly full subcategory $\mathcal{Z}' \subset \mathcal{Z}$ such that $\mathcal{Z}'$ is an algebraic stack which satisfies the equivalent conditions of Lemma 98.11.3. The inclusion morphism $\mathcal{Z}' \to \mathcal{Z}$ is a monomorphism of algebraic stacks.

Proof. The last part is immediate from the first part and Lemma 98.8.4. Pick a field $k$ and a morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$ which is surjective, flat, and locally of finite type. Set $U = \mathop{\mathrm{Spec}}(k)$ and $R = U \times _\mathcal {Z} U$. The projections $s, t : R \to U$ are locally of finite type. Since $U$ is the spectrum of a field, it follows that $s, t$ are flat and locally of finite presentation (by Morphisms of Spaces, Lemma 65.28.7). We see that $\mathcal{Z}' = [U/R]$ is an algebraic stack by Criteria for Representability, Theorem 95.17.2. By Algebraic Stacks, Lemma 92.16.1 we obtain a canonical morphism

\[ f : \mathcal{Z}' \longrightarrow \mathcal{Z} \]

which is fully faithful. Hence this morphism is representable by algebraic spaces, see Algebraic Stacks, Lemma 92.15.2 and a monomorphism, see Lemma 98.8.4. By Criteria for Representability, Lemma 95.17.1 the morphism $U \to \mathcal{Z}'$ is surjective, flat, and locally of finite presentation. Hence $\mathcal{Z}'$ is an algebraic stack which satisfies the equivalent conditions of Lemma 98.11.3. By Algebraic Stacks, Lemma 92.12.4 we may replace $\mathcal{Z}'$ by its essential image in $\mathcal{Z}$. Hence we have proved all the assertions of the lemma except for the uniqueness of $\mathcal{Z}' \subset \mathcal{Z}$. Suppose that $\mathcal{Z}'' \subset \mathcal{Z}$ is a second such algebraic stack. Then the projections

\[ \mathcal{Z}' \longleftarrow \mathcal{Z}' \times _\mathcal {Z} \mathcal{Z}'' \longrightarrow \mathcal{Z}'' \]

are monomorphisms. The algebraic stack in the middle is nonempty by Lemma 98.4.3. Hence the two projections are isomorphisms by Lemma 98.11.4 and we win. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 98.11: Residual gerbes

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Your email address will not be published.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 06MR. Beware of the difference between the letter 'O' and the digit '0'.