The Stacks project

Remark 99.7.3. Let $S$ be a scheme, $X$ an algebraic space over $S$, and $\mathcal{F}$ a quasi-coherent $\mathcal{O}_ X$-module. Suppose that $\{ f_ i : X_ i \to X\} _{i \in I}$ is an fpqc covering and for each $i, j \in I$ we are given an fpqc covering $\{ X_{ijk} \to X_ i \times _ X X_ j\}$. In this situation we have a bijection

Namely, let $(f_ i^*\mathcal{F} \to \mathcal{Q}_ i)_{i \in I}$ be an element of the right hand side. Then since $\{ X_{ijk} \to X_ i \times _ X X_ j\}$ is an fpqc covering we see that the pullbacks of $\mathcal{Q}_ i$ and $\mathcal{Q}_ j$ restrict to the same quotient of the pullback of $\mathcal{F}$ to $X_ i \times _ X X_ j$ (by fully faithfulness in Descent on Spaces, Proposition 74.4.1). Hence we obtain a descent datum for quasi-coherent modules with respect to $\{ X_ i \to X\} _{i \in I}$. By Descent on Spaces, Proposition 74.4.1 we find a map of quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F} \to \mathcal{Q}$ whose restriction to $X_ i$ recovers the given maps $f_ i^*\mathcal{F} \to \mathcal{Q}_ i$. Since the family of morphisms $\{ X_ i \to X\}$ is jointly surjective and flat, for every point $x \in |X|$ there exists an $i$ and a point $x_ i \in |X_ i|$ mapping to $x$. Note that the induced map on local rings $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X_ i, \overline{x_ i}}$ is faithfully flat, see Morphisms of Spaces, Section 67.30. Thus we see that $\mathcal{F} \to \mathcal{Q}$ is surjective.