## 101.5 The key lemma

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 101.5.1. 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 94.20.3 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 94.20.1. 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 99.7.8. 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 101.5.2. 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. $\mathcal{M}_\mathcal {X}$ 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 101.5.1. $\square$

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