The Stacks project

Proposition 96.20.1. Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of algebraic stacks.

  1. Let $\mathcal{F}$ be an abelian étale sheaf on $\mathcal{X}$. Assume that $f$ is representable by algebraic spaces, surjective, and smooth. Then there is a spectral sequence

    \[ E_1^{p, q} = H^ q_{\acute{e}tale}(\mathcal{U}_ p, f_ p^{-1}\mathcal{F}) \Rightarrow H^{p + q}_{\acute{e}tale}(\mathcal{X}, \mathcal{F}) \]

  2. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{X}$. Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then there is a spectral sequence

    \[ E_1^{p, q} = H^ q_{fppf}(\mathcal{U}_ p, f_ p^{-1}\mathcal{F}) \Rightarrow H^{p + q}_{fppf}(\mathcal{X}, \mathcal{F}) \]

Proof. To see this we will check the hypotheses (1) – (4) of Lemma 96.19.8. 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. $\square$

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