Lemma 96.21.2. Let S be a scheme. Let f : \mathcal{X} \to \mathcal{Y} be a 1-morphism of algebraic stacks1 over S. Let \tau \in \{ Zariski,\linebreak[0] {\acute{e}tale},\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\} . Let \mathcal{F} be an object of \textit{Ab}(\mathcal{X}_\tau ) or \textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X}). Then the sheaf R^ if_*\mathcal{F} is the sheaf associated to the presheaf
y \longmapsto H^ i_\tau \Big((\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big)
Here y is an object of \mathcal{Y} lying over the scheme V.
Proof.
Choose an injective resolution \mathcal{F}[0] \to \mathcal{I}^\bullet . By the formula for pushforward (96.5.0.1) we see that R^ if_*\mathcal{F} is the sheaf associated to the presheaf which associates to y the cohomology of the complex
\begin{matrix} \Gamma \Big((\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{I}^{i - 1}\Big)
\\ \downarrow
\\ \Gamma \Big((\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{I}^ i\Big)
\\ \downarrow
\\ \Gamma \Big((\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{I}^{i + 1}\Big)
\end{matrix}
Since \text{pr}^{-1} is exact, it suffices to show that \text{pr}^{-1} preserves injectives. This follows from Lemmas 96.17.5 and 96.17.6 as well as the fact that \text{pr} is a representable morphism of algebraic stacks (so that \text{pr} is faithful by Algebraic Stacks, Lemma 94.15.2 and that (\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X} has equalizers by Lemma 96.17.2).
\square
Comments (2)
Comment #5971 by Dario Weißmann on
Comment #6150 by Johan on