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Let $\mathscr{M}\to\mathscr{N}$ be a map of (Deligne-Mumford) stacks. Recall that it is said to be representable by affine schemes if for all affine maps $\operatorname{Spec}R\to \mathscr{N}$, the pullback $\mathscr{M}\times_\mathscr{N}\operatorname{Spec}R$ is equivalent to an affine scheme.

What is a criterion for when a map of the associated Hopf algebroids which induces a map of DM-stacks $\mathscr{M}\to\mathscr{N}$ which is representable by affine schemes? (What about the analogous question for flat maps of stacks?)

I know that if $(L,W)$ is the Hopf algebroid asociated to $\mathscr{M}$ and $\mathscr{N}$, respectively, then it suffices to check the representability of $\mathscr{M}\to\mathscr{N}(L,W)$ on the morphism $\operatorname{Spec}L\to\mathscr{N}$. Is there a similar statement for the map $\mathscr{M}\to \mathscr{N}$ if $(A,\Gamma)$ is the Hopf algebroid asociated to $\mathscr{M}$?

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    $\begingroup$ 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". $\endgroup$ – Jason Starr Dec 30 '15 at 15:51
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    $\begingroup$ @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! $\endgroup$ – user62675 Dec 30 '15 at 15:56
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    $\begingroup$ 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. $\endgroup$ – Elden Elmanto Dec 30 '15 at 16:09
  • $\begingroup$ @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? $\endgroup$ – user62675 Dec 30 '15 at 16:34
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    $\begingroup$ @SanathK.Devalapurkar maybe you could add a reference to DM stacks coming from Hopf algebroids? $\endgroup$ – Mattia Talpo Jan 1 '16 at 1:09
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Of course, you mean representable by affine schemes, on usual schemes this is called an affine map. This is an important class of maps but quite restrictive, too. The question you ask is the main concern of the paper

Powell, Geoffrey M. L. On affine morphisms of Hopf algebroids. Homology, Homotopy Appl. 10 (2008), no. 1, 53–95.

The result you ask for is Theorem 1.2. on page 55. Unfortunately it is too long to reproduce it here. I hope this answers your question.

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