96.21 Higher direct images and algebraic stacks
Let $g : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. In the sections above we have constructed a morphism of ringed topoi $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_\tau )$ for each $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. In the chapter on cohomology of sites we have explained how to define higher direct images. Hence the total direct image $Rg_*\mathcal{F}$ is defined as $g_*\mathcal{I}^\bullet $ where $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution in $\textit{Ab}(\mathcal{X}_\tau )$. The $i$th higher direct image $R^ ig_*\mathcal{F}$ is the $i$th cohomology of the total direct image. Important: it matters which topology $\tau $ is used here!
If $\mathcal{F}$ is a presheaf of $\mathcal{O}_\mathcal {X}$-modules which is a sheaf in the $\tau $-topology, then we use injective resolutions in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ to compute total direct image and higher direct images.
So far our only tool to compute the higher direct images of $g_*$ is the result on Čech complexes proved above. This requires the choice of a “covering” $f : \mathcal{U} \to \mathcal{X}$. If $\mathcal{U}$ is an algebraic space, then $f : \mathcal{U} \to \mathcal{X}$ is representable by algebraic spaces, see Algebraic Stacks, Lemma 94.10.11. Thus the proposition applies in particular to a smooth cover of the algebraic stack $\mathcal{X}$ by a scheme.
Proposition 96.21.1. Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of algebraic stacks.
Assume that $f$ is representable by algebraic spaces, surjective and smooth.
If $\mathcal{F}$ is in $\textit{Ab}(\mathcal{X}_{\acute{e}tale})$ then there is a spectral sequence
\[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]
in $\textit{Ab}(\mathcal{Y}_{\acute{e}tale})$ with higher direct images computed in the étale topology.
If $\mathcal{F}$ is in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ then there is a spectral sequence
\[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]
in $\textit{Mod}(\mathcal{Y}_{\acute{e}tale}, \mathcal{O}_\mathcal {Y})$.
Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation.
If $\mathcal{F}$ is in $\textit{Ab}(\mathcal{X})$ then there is a spectral sequence
\[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]
in $\textit{Ab}(\mathcal{Y})$ with higher direct images computed in the fppf topology.
If $\mathcal{F}$ is in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ then there is a spectral sequence
\[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]
in $\textit{Mod}(\mathcal{O}_\mathcal {Y})$.
Proof.
To see this we will check the hypotheses (1) – (4) of Lemma 96.19.11 and Lemma 96.19.12. The $1$-morphism $f$ is faithful by Algebraic Stacks, Lemma 94.15.2. This proves (4). Hypothesis (3) follows from the fact that $\mathcal{U}$ is an algebraic stack, see Lemma 96.17.2. To see (2) apply Lemma 96.19.10. Condition (1) is satisfied by fiat in all four cases.
$\square$
Here is a description of higher direct images for a morphism of algebraic stacks.
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).
$\square$
Here is a trivial base change result.
Lemma 96.21.3. Let $S$ be a scheme. Let $\tau \in \{ Zariski,\linebreak[0] {\acute{e}tale},\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\} $. Let
\[ \xymatrix{ \mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \ar[r]_{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{Y}' \ar[r]^ g & \mathcal{Y} } \]
be a $2$-cartesian diagram of algebraic stacks over $S$. Then the base change map is an isomorphism
\[ g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F} \]
functorial for $\mathcal{F}$ in $\textit{Ab}(\mathcal{X}_\tau )$ or $\mathcal{F}$ in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$.
Proof.
The isomorphism $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$ is Lemma 96.5.1 (and it holds for arbitrary presheaves). For the total direct images, there is a base change map because the morphisms $g$ and $g'$ are flat, see Cohomology on Sites, Section 21.15. To see that this map is a quasi-isomorphism we can use that for an object $y'$ of $\mathcal{Y}'$ over a scheme $V$ there is an equivalence
\[ (\mathit{Sch}/V)_{fppf} \times _{g(y'), \mathcal{Y}} \mathcal{X} = (\mathit{Sch}/V)_{fppf} \times _{y', \mathcal{Y}'} (\mathcal{Y}' \times _\mathcal {Y} \mathcal{X}) \]
We conclude that the induced map $g^{-1}R^ if_*\mathcal{F} \to R^ if'_*(g')^{-1}\mathcal{F}$ is an isomorphism by Lemma 96.21.2.
$\square$
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