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Timeline for Gerbes over finite fields

Current License: CC BY-SA 4.0

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S Jul 31, 2023 at 15:24 history suggested User1234
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Jul 31, 2023 at 14:20 review Suggested edits
S Jul 31, 2023 at 15:24
Jul 31, 2023 at 13:16 history edited Daniel Loughran CC BY-SA 4.0
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Jul 31, 2023 at 13:15 vote accept Daniel Loughran
Jul 30, 2023 at 6:25 comment added Angelo In case the gerbe has affine diagonal this is Theorem 8.1 in Di Proietto, Tonini and Zhang, Frobenius fixed objects of moduli. Actually, they show this for general fpqc gerbes, without finiteness hypotheses.
Jul 29, 2023 at 17:51 comment added Daniel Loughran Yes I had also thought that but didnt want to come across as a smart arse :) In any case the important condition is anon's condition (c) and how it relates to my definition.
Jul 29, 2023 at 16:51 comment added Robert Furber @DanielLoughran When I read the question, I assumed that a connected groupoid was required to be non-empty, like a connected topological space or a connected graph, for the same reason that 1 is not a prime number. So non-emptiness of $\mathcal{G}(\overline{k})$ is not missing (but maybe Niels uses a different definition of connectedness of groupoids).
Jul 29, 2023 at 14:50 comment added Daniel Loughran @anon: In any case I think multiple answers are welcome on MO and I would be very happy for you to upgrade your comment to an answer.
Jul 29, 2023 at 14:49 comment added Daniel Loughran @anon: Modulo the missing assumption that $\mathcal{G}(\bar{k})$ is non-empty, how does my definition differ from yours? Are there algebraic stacks which satisfy one definition but not the other? Just your condition (b) seems to hold automatically as $\mathcal{G}$ was assumed to be an algebraic stack, and your condition (c) feels very similar to my condition that $\mathcal{G}(\bar{k})$ is connected (by which I naturally mean that any two $\bar{k}$-points are isomorphic). It feels like a descent argument should reduce one definition to the other.
Jul 29, 2023 at 7:08 comment added Niels @anon I know your definition is correct, my comment was about the definition given by Daniel Loughran
Jul 29, 2023 at 1:56 comment added anon The definition is Giraud's original definition.
Jul 28, 2023 at 19:59 history became hot network question
Jul 28, 2023 at 19:32 comment added Niels Your definition looks incomplete, objects should exist locally, that is $\mathcal{G}(\bar{k})$ should be non empty.
Jul 28, 2023 at 18:27 comment added anon If you assume that there exists an $x$ in $G(\bar{k})$, where $\bar{k}$ is the algebraic closure of $k$, and $Aut(x)$ is an algebraic group, then it follows from Theorem 3.5 of Springer 1966 that $G(k)$ is nonempty. (Nonabelian $H^2$ in Galois cohomology, Springer, T. A., Amer. Math. Soc., Providence, R.I., 1966, pp 164-182.)
Jul 28, 2023 at 18:00 comment added anon It might help to have the correct definition of a gerbe: A stack is a gerbe if (a) each $G_{S}$ is a groupoid (all morphisms are isomorphisms); (b) there exists an $S\neq\emptyset$ such that $G_{S}\neq\emptyset$; and (c) any two objects of $G_{S}$ are locally isomorphic.
Jul 28, 2023 at 14:10 comment added Daniel Loughran Gerbes: The Final Frontier
Jul 28, 2023 at 14:04 answer added Johan timeline score: 14
Jul 28, 2023 at 12:26 comment added LSpice Neat question! I would just fix the typo, but I kind of enjoy the idea of a theory with subtitles.
Jul 28, 2023 at 11:59 history asked Daniel Loughran CC BY-SA 4.0