The Stacks project

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). $\square$

[1] This result should hold for any $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$.

Comments (2)

Comment #5971 by Dario Weißmann on

What distinguishes a "typical object" from an object? To me the lemma makes perfect sense deleting that adjective (there are several instances of this in the SP but it doesn't seem to be defined? )

Comment #6150 by on

Yes, typical is a word that can be removed. Thanks and fixed here.

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