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Dear all,

Let $S$ be a scheme in characteristic $0$, and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme $X$ by a stack $\mathscr{X}$.

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?

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  • $\begingroup$ Shouldn't this have an algebraic-geometry tag? I'd put it in myself, but my phone browser is not cooperating. $\endgroup$
    – David Roberts ♦
    Mar 26, 2012 at 8:43
  • $\begingroup$ Do you demand that the cocycle condition hold strictly, or up to 2-isomorphism (together with an additional compatibility)? $\endgroup$
    – S. Carnahan ♦
    Mar 26, 2012 at 10:39
  • $\begingroup$ @David: Thank you, I've added it myself! $\endgroup$
    – Thanos D. Papaïoannou
    Mar 26, 2012 at 13:10
  • $\begingroup$ @Scott: Thank you for the comment, Scott, I demand the cocycle condition hold only up to isomorphism, which itself must satisfy the natural cocycle condition. $\endgroup$
    – Thanos D. Papaïoannou
    Mar 26, 2012 at 13:12

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