Proposition 100.7.4. Summary of results on locally quasi-coherent modules having the flat base change property.

Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ which is locally quasi-coherent and has the flat base change property, then $\mathcal{F}$ is a sheaf for the fppf topology, i.e., it is an object of $\textit{Mod}(\mathcal{O}_\mathcal {X})$.

The category of modules which are locally quasi-coherent and have the flat base change property is a weak Serre subcategory $\mathcal{M}_\mathcal {X}$ of both $\textit{Mod}(\mathcal{O}_\mathcal {X})$ and $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$.

Pullback $f^*$ along any morphism of algebraic stacks $f : \mathcal{X} \to \mathcal{Y}$ induces a functor $f^* : \mathcal{M}_\mathcal {Y} \to \mathcal{M}_\mathcal {X}$.

If $f : \mathcal{X} \to \mathcal{Y}$ is a quasi-compact and quasi-separated morphism of algebraic stacks and $\mathcal{F}$ is an object of $\mathcal{M}_\mathcal {X}$, then

the derived direct image $Rf_*\mathcal{F}$ and the higher direct images $R^ if_*\mathcal{F}$ can be computed in either the étale or the fppf topology with the same result, and

each $R^ if_*\mathcal{F}$ is an object of $\mathcal{M}_\mathcal {Y}$.

The category $\mathcal{M}_\mathcal {X}$ has colimits and they agree with colimits in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ as well as in $\textit{Mod}(\mathcal{O}_\mathcal {X})$.

## Comments (2)

Comment #3800 by Pieter Belmans on

Comment #3918 by Johan on