Proposition 103.11.7. Let f : \mathcal{U} \to \mathcal{X} and g : \mathcal{X} \to \mathcal{Y} be composable morphisms of algebraic stacks. Assume that
f is representable by algebraic spaces, surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated, and
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 96.21.1. Applying the exact functor Q_\mathcal {Y} : \textit{LQCoh}^{fbc}(\mathcal{O}_\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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