On the local structure of stacks - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:15:33Z http://mathoverflow.net/feeds/question/120699 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120699/on-the-local-structure-of-stacks On the local structure of stacks Nullstellensatz 2013-02-03T20:09:18Z 2013-03-06T09:46:33Z <p>1) Is it true that any Deligne-Mumford stack is locally a quotient stack $[X/G]$ with a finite group $G$?</p> <p>2) Is it true that any Deligne-Mumford stack can be globally presented as a quotient stack $[X/G]$ with a non necessarily finite group $G$? For example, Geigle and Lenzing give such a presentation for stacky projective lines <a href="http://www.math.washington.edu/~smith/WPLSeminar/GL87.pdf" rel="nofollow">here</a>.</p> <p>3) What about Artin stacks?</p> http://mathoverflow.net/questions/120699/on-the-local-structure-of-stacks/120702#120702 Answer by Jacob Bell for On the local structure of stacks Jacob Bell 2013-02-03T20:37:14Z 2013-03-06T09:46:33Z <p>the one I know without having to look in the literature is 1)</p> <p>Lemma 2.2.3 of <a href="http://arxiv.org/pdf/math/9908167v2.pdf" rel="nofollow">http://arxiv.org/pdf/math/9908167v2.pdf</a></p> <p>I think 2) is true as well (maybe you need to add the adjective tameness appropriately?) and for 3) there should be a result of Kresch saying that your stack can be stratified by quotient stacks. But I'd have to look this stuff up.</p> <p>EDIT:</p> <p>for 1) I should say etale topology.</p> <p>for 2) I was thinking about this result (Theorem 4.4 and Proposition 5.1) by Kresch <a href="http://www.math.uzh.ch/fileadmin/user/kresch/publikation/geodm.pdf" rel="nofollow">http://www.math.uzh.ch/fileadmin/user/kresch/publikation/geodm.pdf</a></p> <p>for 3), the result I was misremembering was Proposition 3.5 of <a href="http://arxiv.org/pdf/1002.4372.pdf" rel="nofollow">http://arxiv.org/pdf/1002.4372.pdf</a>, and the first paragraph of the proof. (it's for stacks with affine stabilisers)</p> <p>SECOND EDIT:</p> <p>There is a paper by Edidin-Hassett-Kresch-Vistoli where the investigate when an Artin stack is a quotient stack. It turns out that this is closely related to the pushforward of the structure sheaf of a smooth atlas to admit a surjection from a vector bundle. Here is the review by Vezzosi. <a href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&amp;s1=611835&amp;vfpref=html&amp;r=34&amp;mx-pid=1844577" rel="nofollow">http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&amp;s1=611835&amp;vfpref=html&amp;r=34&amp;mx-pid=1844577</a></p> http://mathoverflow.net/questions/120699/on-the-local-structure-of-stacks/120711#120711 Answer by David Carchedi for On the local structure of stacks David Carchedi 2013-02-03T22:08:36Z 2013-02-03T22:08:36Z <p>I am not an expert in the algebraic category, however, I know that 2) is an open problem in the differentiable category; it is not known if every smooth orbifold is a global quotient stack. It <em>is</em> true in the differentiable category when you ask for the orbifold to be effective ("reduced" in algebrogeometric lingo). I have good (topos-theoretic) reasons to suspect that if $\mathscr{X}$ is a Deligne-Mumford stack such that for each scheme $T,$ the subgroupoid of $\mathscr{X}\left(T\right)$ on those maps $T \to \mathscr{X}$ which are etale is equivalent to a set, then $\mathscr{X}$ is a global quotient. (This condition is basically means that the isotropy groups act faithfully). I would be very surprised if 3) were true.</p>