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

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