Lemma 96.21.2. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks^{1} 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