The Stacks project

Proof. Let $\{ T_ i \to T\} _{i \in I}$ be an fpqc covering of schemes over $S$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and $\mathcal{F}_ i = \mathcal{F}_{T_ i}$. Note that $\{ X_ i \to X_ T\} _{i \in I}$ is an fpqc covering of $X_ T$ (Topologies on Spaces, Lemma 73.9.3) and that $X_{T_ i \times _ T T_{i'}} = X_ i \times _{X_ T} X_{i'}$. Suppose that $\mathcal{F}_ i \to \mathcal{Q}_ i$ is a collection of elements of $\text{Q}_{\mathcal{F}/X/B}(T_ i)$ such that $\mathcal{Q}_ i$ and $\mathcal{Q}_{i'}$ restrict to the same element of $\text{Q}_{\mathcal{F}/X/B}(T_ i \times _ T T_{i'})$. By Remark 99.7.3 we obtain a surjective map of quasi-coherent $\mathcal{O}_{X_ T}$-modules $\mathcal{F}_ T \to \mathcal{Q}$ whose restriction to $X_ i$ recovers the given quotients. By Morphisms of Spaces, Lemma 67.31.5 we see that $\mathcal{Q}$ is flat over $T$. Finally, in the case of $\text{Q}^{fp}_{\mathcal{F}/X/B}$, i.e., if $\mathcal{Q}_ i$ are of finite presentation, then Descent on Spaces, Lemma 74.6.2 guarantees that $\mathcal{Q}$ is of finite presentation as an $\mathcal{O}_{X_ T}$-module. $\square$