Lemma 100.10.2. 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 100.9.2) hence have vanishing cohomology (Lemma 100.8.3). 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).