Proposition 103.8.1. 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 in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, 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 $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ is a weak Serre subcategory 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 $f^* : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

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 $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, then

the total 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 $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})$.

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

Given $\mathcal{F}$ and $\mathcal{G}$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then the tensor product $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

Given $\mathcal{F}$ of finite presentation and $\mathcal{G}$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

## Comments (2)

Comment #3800 by Pieter Belmans on

Comment #3918 by Johan on