The Stacks project

Proof. By Lemma 99.8.2 we have that $\mathrm{Quot}_{\mathcal{F}/X/B}$ is a sheaf in the fppf topology. Let $\textit{Quot}_{\mathcal{F}/X/B}$ be the stack in groupoids corresponding to $\mathrm{Quot}_{\mathcal{F}/X/S}$, see Algebraic Stacks, Section 94.7. By Algebraic Stacks, Proposition 94.13.3 it suffices to show that $\textit{Quot}_{\mathcal{F}/X/B}$ is an algebraic stack. Consider the $1$-morphism of stacks in groupoids

on $(\mathit{Sch}/S)_{fppf}$ which associates to the quotient $\mathcal{F}_ T \to \mathcal{Q}$ the module $\mathcal{Q}$. By Theorem 99.6.1 we know that $\mathcal{C}\! \mathit{oh}_{X/B}$ is an algebraic stack. By Algebraic Stacks, Lemma 94.15.4 it suffices to show that this $1$-morphism is representable by algebraic spaces.

Let $T$ be a scheme over $S$ and let the object $(h, \mathcal{G})$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ over $T$ correspond to a $1$-morphism $\xi : (\mathit{Sch}/T)_{fppf} \to \mathcal{C}\! \mathit{oh}_{X/B}$. The $2$-fibre product

is a stack in setoids, see Stacks, Lemma 8.6.7. The corresponding sheaf of sets (i.e., functor, see Stacks, Lemmas 8.6.7 and 8.6.2) assigns to a scheme $T'/T$ the set of surjections $u : \mathcal{F}_{T'} \to \mathcal{G}_{T'}$ of quasi-coherent modules on $X_{T'}$. Thus we see that $\mathcal{Z}$ is representable by an open subspace (by Flatness on Spaces, Lemma 77.9.3) of the algebraic space $\mathit{Hom}(\mathcal{F}_ T, \mathcal{G})$ from Proposition 99.3.10. $\square$