Does every stack with a connection admit an atlas with a connection? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:56:38Z http://mathoverflow.net/feeds/question/92224 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92224/does-every-stack-with-a-connection-admit-an-atlas-with-a-connection Does every stack with a connection admit an atlas with a connection? Thanos D. Papaïoannou 2012-03-26T04:02:49Z 2012-03-26T12:41:40Z <p>Dear all,</p> <p>Let $S$ be a scheme in characteristic $0$, and let $\mathscr{X}/S$ mean a <em>crystal in Artin stacks over $S$</em> in the sense of <a href="http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19%28Crystals%29.pdf" rel="nofollow">this handout</a>, page 4, Definition 0.5, where we replace the scheme $X$ by a stack $\mathscr{X}$.</p> <p>Does $\mathscr{X}$ always admit an atlas $Y$ over $S$ which can be enhanced to a crystal of schemes over $S$? In other words, is there a crystal $Y$ in schemes over $S$ together with a map of algebraic stacks $Y\to \mathscr{X}$ which is smooth and surjective?</p>