103.5 Higher direct images of types of modules

The following lemma is the basis for our understanding of higher direct images of certain types of sheaves of modules. There are two versions: one for the étale topology and one for the fppf topology.

Lemma 103.5.1. Let $\mathcal{M}$ be a rule which associates to every algebraic stack $\mathcal{X}$ a subcategory $\mathcal{M}_\mathcal {X}$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ such that

1. $\mathcal{M}_\mathcal {X}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ (see Homology, Definition 12.10.1) for all algebraic stacks $\mathcal{X}$,

2. for a smooth morphism of algebraic stacks $f : \mathcal{Y} \to \mathcal{X}$ the functor $f^*$ maps $\mathcal{M}_\mathcal {X}$ into $\mathcal{M}_\mathcal {Y}$,

3. if $f_ i : \mathcal{X}_ i \to \mathcal{X}$ is a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| = \bigcup |f_ i|(|\mathcal{X}_ i|)$, then an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ is in $\mathcal{M}_\mathcal {X}$ if and only if $f_ i^*\mathcal{F}$ is in $\mathcal{M}_{\mathcal{X}_ i}$ for all $i$, and

4. if $f : \mathcal{Y} \to \mathcal{X}$ is a morphism of algebraic stacks such that $\mathcal{X}$ and $\mathcal{Y}$ are representable by affine schemes, then $R^ if_*$ maps $\mathcal{M}_\mathcal {Y}$ into $\mathcal{M}_\mathcal {X}$.

Then for any quasi-compact and quasi-separated morphism $f : \mathcal{Y} \to \mathcal{X}$ of algebraic stacks $R^ if_*$ maps $\mathcal{M}_\mathcal {Y}$ into $\mathcal{M}_\mathcal {X}$. (Higher direct images computed in étale topology.)

Proof. Let $f : \mathcal{Y} \to \mathcal{X}$ be a quasi-compact and quasi-separated morphism of algebraic stacks and let $\mathcal{F}$ be an object of $\mathcal{M}_\mathcal {Y}$. Choose a surjective smooth morphism $\mathcal{U} \to \mathcal{X}$ where $\mathcal{U}$ is representable by a scheme. By Sheaves on Stacks, Lemma 96.21.3 taking higher direct images commutes with base change. Assumption (2) shows that the pullback of $\mathcal{F}$ to $\mathcal{U} \times _\mathcal {X} \mathcal{Y}$ is in $\mathcal{M}_{\mathcal{U} \times _\mathcal {X} \mathcal{Y}}$ because the projection $\mathcal{U} \times _\mathcal {X} \mathcal{Y} \to \mathcal{Y}$ is smooth as a base change of a smooth morphism. Hence (3) shows we may replace $\mathcal{Y} \to \mathcal{X}$ by the projection $\mathcal{U} \times _\mathcal {X} \mathcal{Y} \to \mathcal{U}$. In other words, we may assume that $\mathcal{X}$ is representable by a scheme. Using (3) once more, we see that the question is Zariski local on $\mathcal{X}$, hence we may assume that $\mathcal{X}$ is representable by an affine scheme. Since $f$ is quasi-compact this implies that also $\mathcal{Y}$ is quasi-compact. Thus we may choose a surjective smooth morphism $g : \mathcal{V} \to \mathcal{Y}$ where $\mathcal{V}$ is representable by an affine scheme.

In this situation we have the spectral sequence

$E_2^{p, q} = R^ q(f \circ g_ p)_*g_ p^*\mathcal{F} \Rightarrow R^{p + q}f_*\mathcal{F}$

of Sheaves on Stacks, Proposition 96.21.1. Recall that this is a first quadrant spectral sequence hence we may use the last part of Homology, Lemma 12.25.3. Note that the morphisms

$g_ p : \mathcal{V}_ p = \mathcal{V} \times _\mathcal {Y} \ldots \times _\mathcal {Y} \mathcal{V} \longrightarrow \mathcal{Y}$

are smooth as compositions of base changes of the smooth morphism $g$. Thus the sheaves $g_ p^*\mathcal{F}$ are in $\mathcal{M}_{\mathcal{V}_ p}$ by (2). Hence it suffices to prove that the higher direct images of objects of $\mathcal{M}_{\mathcal{V}_ p}$ under the morphisms

$\mathcal{V}_ p = \mathcal{V} \times _\mathcal {Y} \ldots \times _\mathcal {Y} \mathcal{V} \longrightarrow \mathcal{X}$

are in $\mathcal{M}_\mathcal {X}$. The algebraic stacks $\mathcal{V}_ p$ are quasi-compact and quasi-separated by Morphisms of Stacks, Lemma 101.7.8. Of course each $\mathcal{V}_ p$ is representable by an algebraic space (the diagonal of the algebraic stack $\mathcal{Y}$ is representable by algebraic spaces). This reduces us to the case where $\mathcal{Y}$ is representable by an algebraic space and $\mathcal{X}$ is representable by an affine scheme.

In the situation where $\mathcal{Y}$ is representable by an algebraic space and $\mathcal{X}$ is representable by an affine scheme, we choose anew a surjective smooth morphism $\mathcal{V} \to \mathcal{Y}$ where $\mathcal{V}$ is representable by an affine scheme. Going through the argument above once again we once again reduce to the morphisms $\mathcal{V}_ p \to \mathcal{X}$. But in the current situation the algebraic stacks $\mathcal{V}_ p$ are representable by quasi-compact and quasi-separated schemes (because the diagonal of an algebraic space is representable by schemes).

Thus we may assume $\mathcal{Y}$ is representable by a scheme and $\mathcal{X}$ is representable by an affine scheme. Choose (again) a surjective smooth morphism $\mathcal{V} \to \mathcal{Y}$ where $\mathcal{V}$ is representable by an affine scheme. In this case all the algebraic stacks $\mathcal{V}_ p$ are representable by separated schemes (because the diagonal of a scheme is separated).

Thus we may assume $\mathcal{Y}$ is representable by a separated scheme and $\mathcal{X}$ is representable by an affine scheme. Choose (yet again) a surjective smooth morphism $\mathcal{V} \to \mathcal{Y}$ where $\mathcal{V}$ is representable by an affine scheme. In this case all the algebraic stacks $\mathcal{V}_ p$ are representable by affine schemes (because the diagonal of a separated scheme is a closed immersion hence affine) and this case is handled by assumption (4). This finishes the proof. $\square$

Here is the version for the fppf topology.

Lemma 103.5.2. Let $\mathcal{M}$ be a rule which associates to every algebraic stack $\mathcal{X}$ a subcategory $\mathcal{M}_\mathcal {X}$ of $\textit{Mod}(\mathcal{O}_\mathcal {X})$ such that

1. $\mathcal{O}_\mathcal {X}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_\mathcal {X})$ for all algebraic stacks $\mathcal{X}$,

2. for a smooth morphism of algebraic stacks $f : \mathcal{Y} \to \mathcal{X}$ the functor $f^*$ maps $\mathcal{M}_\mathcal {X}$ into $\mathcal{M}_\mathcal {Y}$,

3. if $f_ i : \mathcal{X}_ i \to \mathcal{X}$ is a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| = \bigcup |f_ i|(|\mathcal{X}_ i|)$, then an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is in $\mathcal{M}_\mathcal {X}$ if and only if $f_ i^*\mathcal{F}$ is in $\mathcal{M}_{\mathcal{X}_ i}$ for all $i$, and

4. if $f : \mathcal{Y} \to \mathcal{X}$ is a morphism of algebraic stacks and $\mathcal{X}$ and $\mathcal{Y}$ are representable by affine schemes, then $R^ if_*$ maps $\mathcal{M}_\mathcal {Y}$ into $\mathcal{M}_\mathcal {X}$.

Then for any quasi-compact and quasi-separated morphism $f : \mathcal{Y} \to \mathcal{X}$ of algebraic stacks $R^ if_*$ maps $\mathcal{M}_\mathcal {Y}$ into $\mathcal{M}_\mathcal {X}$. (Higher direct images computed in fppf topology.)

Proof. Identical to the proof of Lemma 103.5.1. $\square$

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