The Stacks project

Definition 99.11.8. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.

  1. We say the residual gerbe of $\mathcal{X}$ at $x$ exists if the equivalent conditions (1), (2), and (3) of Lemma 99.11.7 hold.

  2. If the residual gerbe of $\mathcal{X}$ at $x$ exists, then the residual gerbe of $\mathcal{X}$ at $x$1 is the strictly full subcategory $\mathcal{Z}_ x \subset \mathcal{X}$ constructed in Lemma 99.11.7.

[1] This clashes with [LM-B] in spirit, but not in fact. Namely, in Chapter 11 they associate to any point on any quasi-separated algebraic stack a gerbe (not necessarily algebraic) which they call the residual gerbe. We will see in Morphisms of Stacks, Lemma 100.31.1 that on a quasi-separated algebraic stack every point has a residual gerbe in our sense which is then equivalent to theirs. For more information on this topic see [Appendix B, rydh_etale_devissage].

Comments (3)

Comment #964 by Niels Borne on

"We will see in Morphisms of Stacks, Lemma 77.21.1 "

The link seems broken (points to http://tag/06RD instead of http://stacks.math.columbia.edu/tag/06RD).

Probably the reason is that it is generated dynamically, but the algorithm fails within the footnote.

Comment #966 by on

Yup, you are absolutely right in your guess. I've filed it as an issue on GitHub (see https://github.com/stacks/stacks-website/issues/86) and it'll be fixed whenever I do another round of bug fixing. Thanks!

Comment #999 by on

OK, this is now fixed. Thanks for reporting.

There are also:

  • 2 comment(s) on Section 99.11: 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Your email address will not be published.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 06MU. Beware of the difference between the letter 'O' and the digit '0'.