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 $\mathscr{M}$?

5more comments