The Stacks project

Lemma 101.10.2. 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 101.9.2) hence have vanishing cohomology (Lemma 101.8.3). 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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