Situation 99.8.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation. 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 $X_ T = X \times _ S T \to X$. Given such a $T$ we set
\[ \mathrm{Quot}_{\mathcal{F}/X/B}(T) = \left\{ \begin{matrix} \text{quotients }\mathcal{F}_ T \to \mathcal{Q}\text{ where } \mathcal{Q}\text{ is a quasi-coherent }
\\ \mathcal{O}_{X_ T}\text{-module of finite presentation, flat over }T
\\ \text{with support proper over }T
\end{matrix} \right\} \]
By Derived Categories of Spaces, Lemma 75.7.8 this is a subfunctor of the functor $Q^{fp}_{\mathcal{F}/X/B}$ we discussed in Section 99.7. Thus we obtain a functor
99.8.1.1
\begin{equation} \label{quot-equation-quot} \mathrm{Quot}_{\mathcal{F}/X/B} : (\mathit{Sch}/B)^{opp} \longrightarrow \textit{Sets} \end{equation}
This is the Quot functor associated to $\mathcal{F}/X/B$.
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