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 $\mathscr{M}$?



Representable map of Deligne-Mumford stacks
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.


