The Stacks project

Proposition 96.21.1. Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of algebraic stacks.

1. Assume that $f$ is representable by algebraic spaces, surjective and smooth.

   1. If $\mathcal{F}$ is in $\textit{Ab}(\mathcal{X}_{\acute{e}tale})$ then there is a spectral sequence

      \[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]

      in $\textit{Ab}(\mathcal{Y}_{\acute{e}tale})$ with higher direct images computed in the étale topology.

   2. If $\mathcal{F}$ is in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ then there is a spectral sequence

      \[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]

      in $\textit{Mod}(\mathcal{Y}_{\acute{e}tale}, \mathcal{O}_\mathcal {Y})$.

2. Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation.

   1. If $\mathcal{F}$ is in $\textit{Ab}(\mathcal{X})$ then there is a spectral sequence

      \[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]

      in $\textit{Ab}(\mathcal{Y})$ with higher direct images computed in the fppf topology.

   2. If $\mathcal{F}$ is in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ then there is a spectral sequence

      \[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]

      in $\textit{Mod}(\mathcal{O}_\mathcal {Y})$.