Timeline for Stacks as local quotients or via atlases
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Nov 16, 2019 at 14:59 | comment | added | witt-voodoo | One reason is Artin's axioms for a groupoid valued functor $X$ on schemes to be an algebraic stack. The proof involves a step-by-step construction of the local charts $U \to X$. To the best of my knowledge, this is the main technique used to prove moduli stacks are algebraic. In practice, these axioms can often be checked easily by constructing a suitable deformation theory for the objects being parametrized by $X$. It would be substantially harder (and seemingly irrelevant) to prove $X$ was locally $U/G$ in applications. | |
| Nov 16, 2019 at 8:29 | comment | added | Praphulla Koushik | So, atleast in differentiable stacks, there are interesting Lie groupoids which are not representable locally as Lie group actions on manifolds. May be even for proper etale Lie groupoids, there are some constructions which does not give proper etale Lie groupoids. So, restricting to proper etale Lie groupoids might not be sufficient. So, restricting to stacks that are represented locally as Lie group actions on manifolds (Algebriac group actions on schemes). | |
| Nov 16, 2019 at 8:29 | comment | added | Praphulla Koushik | I don't have enough experience to say anything serious. In the case of differentiable stacks, not every stack can be representable locally as quotient. For any geometric/differentiable stack $\mathcal{D}$, one can associate a Lie groupoid $\mathcal{G}$, with $\mathcal{D}\cong B\mathcal{G}$. A theorem of Moerdijk and Pronk says that only proper and etale Lie groupoids are representable as quotients of Lie group actions on manifolds. | |
| Nov 16, 2019 at 3:48 | answer | added | R. van Dobben de Bruyn | timeline score: 6 | |
| Nov 16, 2019 at 1:31 | history | asked | John Pardon | CC BY-SA 4.0 |