The Stacks project

Proposition 101.10.5. Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable morphisms of algebraic stacks. Assume that

  1. $f$ is representable by algebraic spaces, surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated, and

  2. $g$ is quasi-compact and quasi-separated.

If $\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ then there is a spectral sequence

\[ E_2^{p, q} = R^ q(g \circ f_ p)_{\mathit{QCoh}, *}f_ p^*\mathcal{F} \Rightarrow R^{p + q}g_{\mathit{QCoh}, *}\mathcal{F} \]

in $\mathit{QCoh}(\mathcal{O}_\mathcal {Y})$.

Proof. Note that each of the morphisms $f_ p : \mathcal{U} \times _\mathcal {X} \ldots \times _\mathcal {X} \mathcal{U} \to \mathcal{X}$ is quasi-compact and quasi-separated, hence $g \circ f_ p$ is quasi-compact and quasi-separated, hence the assertion makes sense (i.e., the functors $R^ q(g \circ f_ p)_{\mathit{QCoh}, *}$ are defined). There is a spectral sequence

\[ E_2^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]

by Sheaves on Stacks, Proposition 94.20.1. Applying the exact functor $Q_\mathcal {Y} : \mathcal{M}_\mathcal {Y} \to \mathit{QCoh}(\mathcal{O}_\mathcal {Y})$ gives the desired spectral sequence in $\mathit{QCoh}(\mathcal{O}_\mathcal {Y})$. $\square$

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