The Stacks project

Situation 99.7.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. For any scheme $T$ over $B$ we will denote $X_ T$ the base change of $X$ to $T$ and $\mathcal{F}_ T$ the pullback of $\mathcal{F}$ via the projection morphism $X_ T = X \times _ B T \to X$. Given such a $T$ we set

We identify quotients if they have the same kernel. Suppose that $T' \to T$ is a morphism of schemes over $B$ and $\mathcal{F}_ T \to \mathcal{Q}$ is an element of $\text{Q}_{\mathcal{F}/X/B}(T)$. Then the pullback $\mathcal{Q}' = (X_{T'} \to X_ T)^*\mathcal{Q}$ is a quasi-coherent $\mathcal{O}_{X_{T'}}$-module flat over $T'$ by Morphisms of Spaces, Lemma 67.31.3. Thus we obtain a functor

This is the functor of quotients of $\mathcal{F}/X/B$. We define a subfunctor

which assigns to $T$ the subset of $\text{Q}_{\mathcal{F}/X/B}(T)$ consisting of those quotients $\mathcal{F}_ T \to \mathcal{Q}$ such that $\mathcal{Q}$ is of finite presentation as an $\mathcal{O}_{X_ T}$-module. This is a subfunctor by Properties of Spaces, Section 66.30.