# Representable map of Deligne-Mumford stacks

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 $\mathscr{M}$?

• 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". – Jason Starr Dec 30 '15 at 15:51
• @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! – user62675 Dec 30 '15 at 15:56
• 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. – Elden Elmanto Dec 30 '15 at 16:09
• @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? – user62675 Dec 30 '15 at 16:34
• @SanathK.Devalapurkar maybe you could add a reference to DM stacks coming from Hopf algebroids? – Mattia Talpo Jan 1 '16 at 1:09