The Stacks project

Lemma 102.9.3. Let $\tau \in \{ {\acute{e}tale}, fppf\} $. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be a parasitic object of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$.

  1. $H^ i_\tau (\mathcal{X}, \mathcal{F}) = 0$ for all $i$.

  2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then $R^ if_*\mathcal{F}$ (computed in $\tau $-topology) is a parasitic object of $\textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y})$.

Proof. We first reduce (2) to (1). By Sheaves on Stacks, Lemma 95.21.2 we see that $R^ if_*\mathcal{F}$ is the sheaf associated to the presheaf

\[ y \longmapsto H^ i_\tau \Big(V \times _{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big) \]

Here $y$ is a typical object of $\mathcal{Y}$ lying over the scheme $V$. By Lemma 102.9.2 it suffices to show that these cohomology groups are zero when $y : V \to \mathcal{Y}$ is flat. Note that $\text{pr} : V \times _{y, \mathcal{Y}} \mathcal{X} \to \mathcal{X}$ is flat as a base change of $y$. Hence by Lemma 102.9.2 we see that $\text{pr}^{-1}\mathcal{F}$ is parasitic. Thus it suffices to prove (1).

To see (1) we can use the spectral sequence of Sheaves on Stacks, Proposition 95.20.1 to reduce this to the case where $\mathcal{X}$ is an algebraic stack representable by an algebraic space. Note that in the spectral sequence each $f_ p^{-1}\mathcal{F} = f_ p^*\mathcal{F}$ is a parasitic module by Lemma 102.9.2 because the morphisms $f_ p : \mathcal{U}_ p = \mathcal{U} \times _\mathcal {X} \ldots \times _\mathcal {X} \mathcal{U} \to \mathcal{X}$ are flat. Reusing this spectral sequence one more time (as in the proof of Lemma 102.5.1) we reduce to the case where the algebraic stack $\mathcal{X}$ is representable by a scheme $X$. Then $H^ i_\tau (\mathcal{X}, \mathcal{F}) = H^ i((\mathit{Sch}/X)_\tau , \mathcal{F})$. In this case the vanishing follows easily from an argument with Čech coverings, see Descent, Lemma 35.12.2. $\square$

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