The Stacks project

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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