Timeline for Representable map of Deligne-Mumford stacks
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 9, 2016 at 2:22 | vote | accept | CommunityBot | moved from User.Id=62675 by developer User.Id=36770 | |
| Apr 21, 2016 at 17:07 | answer | added | Leo Alonso | timeline score: 4 | |
| Jan 1, 2016 at 22:38 | comment | added | user62675 | (Also, Naumann's preprint arXiv:math/0503308 [math.AT] is a good reference.) | |
| Jan 1, 2016 at 22:16 | comment | added | user62675 | @MattiaTalpo My apologies. I learnt about Hopf algebroids from Appendix A.1 of Ravenel's Green book, available here: math.rochester.edu/people/faculty/doug/mybooks/ravenelA1.pdf. A short and brief summary is in Appendix B.3 of Ravenel's Orange book; I believe he has a PDF of the book on his webpage. | |
| Jan 1, 2016 at 21:47 | comment | added | Mattia Talpo | @SanathK.Devalapurkar thanks. And what's a Hopf algebroid exactly? (I actually meant a reference as in 'somewhere to look'...) | |
| Jan 1, 2016 at 3:03 | comment | added | user62675 | @MattiaTalpo Sure. If $(A,\Gamma)$ is a Hopf algebroid, we can take Spec to get $(\mathrm{Spec}(A),\mathrm{Spec}(\Gamma))$. This is a groupoid scheme. The stack associated to the groupoid scheme $[\mathrm{Spec}(\Gamma)\rightrightarrows\mathrm{Spec}(A)]$ is the DM stack coming from the Hopf algebroid $(A,\Gamma)$. | |
| Jan 1, 2016 at 1:09 | comment | added | Mattia Talpo | @SanathK.Devalapurkar maybe you could add a reference to DM stacks coming from Hopf algebroids? | |
| Dec 30, 2015 at 17:57 | comment | added | user62675 | Re the first question my comment above: it's possible to express $\mathscr{M}\times_\mathscr{N}\operatorname{Spec}\mathbf{Z}$ as a DM-stack associated to a Hopf algebroid via tensor products of Hopf algebroids. The Serre affineness criterion translates into asking that Ext of the tensor product of the Hopf algebroids should be acyclic. | |
| Dec 30, 2015 at 16:34 | comment | added | user62675 | @EldenElmanto I'm not too familiar with the algebro-geometric stuff, but is it possible to express $\mathscr{M}\times_\mathscr{N}\operatorname{Spec}\mathbf{Z}$ as a DM-stack associated to a Hopf algebroid? (Because quasicoherent sheaves over a DM-stack $\mathscr{M}(A,\Gamma)$ are equivalent to $(A,\Gamma)$-modules this will be a "criterion on the Hopf algebroid".) Also, is there a Hopf algebroid-al analogue of Artin representability? | |
| Dec 30, 2015 at 16:09 | comment | added | Elden Elmanto | one can apply Artin representability (see, for example, the intro math.harvard.edu/~lurie/papers/DAG-XIV.pdf) to $M \times_N Spec\,\mathbb{Z}$ to see if it is representable by an algebraic space. Then you are asking when is an algebraic space an affine scheme in which case you have Serre's affineness criterion: stacks.math.columbia.edu/tag/07V6. | |
| Dec 30, 2015 at 15:56 | history | edited | user62675 | CC BY-SA 3.0 |
added 36 characters in body
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| Dec 30, 2015 at 15:56 | comment | added | user62675 | @JasonStarr Ah, I didn't know that. I'll edit the question accordingly; I'm interested in the case when the map of stacks is representable by affine schemes, although I'd be interested in learning about the general case as well! | |
| Dec 30, 2015 at 15:51 | comment | added | Jason Starr | I believe that the usual definition of "representable" is that $\mathcal{M}\times_{\mathcal{N}}\text{Spec}\ R$ is a scheme (some authors allow an algebraic space). What you wrote is usually called "representable by affine schemes" or "representable by affine morphisms". | |
| Dec 30, 2015 at 15:21 | history | asked | user62675 | CC BY-SA 3.0 |