The Stacks project

Example 85.23.1. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\mathcal{C} = X_{spaces, {\acute{e}tale}}$ be the étale site on the category of algebraic spaces étale over $X$, see Properties of Spaces, Definition 66.18.2. Denote $\mathcal{O}_\mathcal {C}$ the structure sheaf, i.e., the sheaf given by the rule $U \mapsto \Gamma (U, \mathcal{O}_ U)$. Denote $\mathcal{A}_ U$ the category of quasi-coherent $\mathcal{O}_ U$-modules. Let $\mathcal{B} = \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and for $V \in \mathcal{B}$ set $d_ V = 0$ and let $\text{Cov}_ V$ denote the coverings $\{ V_ i \to V\} $ with $V_ i$ affine for all $i$. Then the assumptions (1), (2), (3) are satisfied. See Properties of Spaces, Lemmas 66.29.2 and 66.29.7 for properties (1) and (2) and the vanishing in (3) follows from Cohomology of Schemes, Lemma 30.2.2 and the discussion in Cohomology of Spaces, Section 69.3.