The Stacks project

Definition 100.11.8. 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 100.11.7 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 100.11.7.

[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 101.31.1 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 100.11: Residual gerbes

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View Definition 100.11.8 as pdf