The Stacks project

Proposition 94.20.1. 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 $\textit{Mod}(\mathcal{O}_\mathcal {Y})$.

Proof. To see this we will check the hypotheses (1) – (4) of Lemma 94.18.11 and Lemma 94.18.12. The $1$-morphism $f$ is faithful by Algebraic Stacks, Lemma 92.15.2. This proves (4). Hypothesis (3) follows from the fact that $\mathcal{U}$ is an algebraic stack, see Lemma 94.16.2. To see (2) apply Lemma 94.18.10. Condition (1) is satisfied by fiat in all four cases. $\square$


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