Lemma 103.11.4. Let f : \mathcal{X} \to \mathcal{Y} be a quasi-compact and quasi-separated morphism of algebraic stacks. Let \mathcal{F} be a quasi-coherent sheaf on \mathcal{X}. Then there exists a spectral sequence with E_2-page
E_2^{p, q} = H^ p(\mathcal{Y}, R^ qf_{\mathit{QCoh}, *}\mathcal{F})
converging to H^{p + q}(\mathcal{X}, \mathcal{F}).
Proof.
By Cohomology on Sites, Lemma 21.14.5 the Leray spectral sequence with
E_2^{p, q} = H^ p(\mathcal{Y}, R^ qf_*\mathcal{F})
converges to H^{p + q}(\mathcal{X}, \mathcal{F}). The kernel and cokernel of the adjunction map
R^ qf_{\mathit{QCoh}, *}\mathcal{F} \longrightarrow R^ qf_*\mathcal{F}
are parasitic modules on \mathcal{Y} (Lemma 103.10.2) hence have vanishing cohomology (Lemma 103.9.3). It follows formally that H^ p(\mathcal{Y}, R^ qf_{\mathit{QCoh}, *}\mathcal{F}) = H^ p(\mathcal{Y}, R^ qf_*\mathcal{F}) and we win.
\square
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