Timeline for Stack associated to Groupoid object in category $\text{Sch}/S$
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Apr 18, 2020 at 3:17 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Sep 7, 2019 at 13:29 | vote | accept | Praphulla Koushik | ||
| Sep 6, 2019 at 5:01 | comment | added | Praphulla Koushik | I saw that notes just now. It seems to be giving some answer to my question. I will read that and respond... | |
| Sep 6, 2019 at 2:46 | comment | added | David Roberts♦ | As a general reference, these notes seem to me to be good, especially Appendix C: arxiv.org/abs/1708.08124 | |
| Sep 5, 2019 at 23:31 | comment | added | Praphulla Koushik | @Qfwfq I do not see anything about (being internal) in that in Wikipedia page... nlab says it is internal ncatlab.org/nlab/show/Lie+groupoid... I see the comment, “Note that Diff does not have all pullbacks, but by suitable conditions on the source and target map we can ensure that the requisite pullbacks do exist.” So, what you mentioned is correct in some sense, groupoid object (with appropriate changes) is Lie groupoid and this is what I had in mind when I said internal groupoid (with appropriate changes). | |
| Sep 5, 2019 at 20:31 | comment | added | Qfwfq | See en.wikipedia.org/wiki/Lie_groupoid -- As for the second aspect, the answer by S. Carnahan below shows you don't actually need algebraic spaces (for this). | |
| Sep 5, 2019 at 15:39 | answer | added | S. Carnahan♦ | timeline score: 2 | |
| Sep 5, 2019 at 14:19 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Sep 5, 2019 at 14:06 | comment | added | Praphulla Koushik | @Qfwfq "a mere groupoid internal to manifolds will not be a Lie groupoid in general".. Is that so :O I think it is a Lie groupoid, some time back I have written details, not sure where it is now... I am nore sure how to clarify the question, I tried to write down as clearly as I can.. I agree there is some confusion... I did not get "schemes vs alg spaces thing is orthogonal to these matters".. Can you please clarify | |
| Sep 5, 2019 at 13:39 | comment | added | Qfwfq | As for "how far" you can get without making these requirements, that's a question you should clarify better. E.g. are you looking for an explicit example of a grpd internal to schemes (or alg sp) whose quot stack is not algebraic? (BTW, I think the schemes vs alg spaces thing is orthogonal to these matters) | |
| Sep 5, 2019 at 13:35 | comment | added | Qfwfq | Well, also a mere groupoid internal to manifolds will not be a Lie groupoid in general: you need the source and target maps be submersions (and maybe some further topological condition?). For the algebraic case you have conditions such as source and target being smooth (in the sense of alg geom) and other stuff. | |
| Sep 4, 2019 at 23:41 | comment | added | Praphulla Koushik | Sorry for the confusion, I just copied the statement from the question, which intended to ask, if I take groupoid internal to schemes, to what extent the corresponding stack is away from being an algebraic stack.. I have no trouble understanding (believing) algebraic spaces and algebraic stacks correspondence as in your comment, it was not my question, sorry to make you type that much for no reason. | |
| Sep 4, 2019 at 22:54 | comment | added | Qfwfq | I'm not sure I understand your last comment. Unless I'm disregarding some subtleties (e.g. conditions on the diagonal etc), if you start with an alg stack $X$, which by definition has an atlas $U\to X$ with $U$ an alg space, you can form a groupoid $U\times_X U\rightrightarrows U$, internal to algebraic spaces, whose associated stack (a.k.a. quotient stack) is isomorphic to $X$. Viceversa, given a groupoid $X_1\rightrightarrows X_0$ in alg sp, satisfying certain conditions, the quotient stack $[X_0/X_1]$ will have an atlas $X_0\to [X_1/X_0]$ making it (if the conditions are ok) an alg stack. | |
| Sep 4, 2019 at 18:48 | comment | added | Praphulla Koushik | It does say a way to associate a Algebraic stack for a groupoid over Algebriac spaces and also says how to associate a algebraic space for a Algebriac stack... I do not see if it is saying something about “Do they cover “most” of Algebriac stacks over the stack 𝑆 with what ever topology on $\text{Sch}/S$“ | |
| Sep 4, 2019 at 18:36 | comment | added | Qfwfq | Probably all you need is buried somewhere in sections 16 and 17 here: stacks.math.columbia.edu/download/algebraic.pdf#nameddest=04T0. Also: stacks.math.columbia.edu/tag/06PI | |
| Sep 4, 2019 at 18:33 | comment | added | Praphulla Koushik | @Qfwfq I knew no offence was intended :) :) I saw that chapter of groupoids on Algebriac spaces just now.. it looks good, I will see what I can understand from that... I will also read about quotient stacks as well. | |
| Sep 4, 2019 at 18:26 | comment | added | Qfwfq | "I am taking [...]" No offence intended! - You could have a look at the Stacks Project; looks like there's a whole chapter on groupoids in algebraic spaces (stacks.math.columbia.edu/tag/0437). The association groupoid in alg sp $\mapsto$ corresponding stack is (I think) stacks.math.columbia.edu/tag/044O. For the reverse association, I think you just take the groupoid $U\times_{\mathscr{X}}U\to U$ of an atlas $U\to\mathscr{X}$. | |
| Sep 4, 2019 at 18:09 | comment | added | Praphulla Koushik | @Qfwfq I am taking “Just after reading the title I knew it was a question by you!” as a nice comment, not sure if this was supposed to be nice :D.. Can you please give some English reference... Does it follow as straight forward as in the case of Lie groupoids or it is more serious (I do not think there is no definite answer for this, just asking)... | |
| Sep 4, 2019 at 18:09 | comment | added | Qfwfq | Just after reading the title I knew it was a question by you! :-) If you allow your "geometric" groupoids to be internal to algebraic spaces instead of just schemes, and consider the lisse (smooth) topology, then I think all the answers become "yes" by Laumon--Moret-Bailly Champs algébriques, (4.3) and Proposition (4.3.1). | |
| Sep 4, 2019 at 17:49 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |