The Stacks project

Proposition 98.8.4. 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 sheaf on $X$. If $f$ is of finite presentation and separated, then $\mathrm{Quot}_{\mathcal{F}/X/B}$ is an algebraic space. If $\mathcal{F}$ is of finite presentation, then $\mathrm{Quot}_{\mathcal{F}/X/B} \to B$ is locally of finite presentation.

Proof. By Lemma 98.8.2 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 93.7. By Algebraic Stacks, Proposition 93.13.3 it suffices to show that $\textit{Quot}_{\mathcal{F}/X/B}$ is an algebraic stack. Consider the $1$-morphism of stacks in groupoids

\[ \textit{Quot}_{\mathcal{F}/X/S} \longrightarrow \mathcal{C}\! \mathit{oh}_{X/B} \]

on $(\mathit{Sch}/S)_{fppf}$ which associates to the quotient $\mathcal{F}_ T \to \mathcal{Q}$ the module $\mathcal{Q}$. By Theorem 98.6.1 we know that $\mathcal{C}\! \mathit{oh}_{X/B}$ is an algebraic stack. By Algebraic Stacks, Lemma 93.15.4 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

\[ \mathcal{Z} = (\mathit{Sch}/T)_{fppf} \times _{\xi , \mathcal{C}\! \mathit{oh}_{X/B}} \textit{Quot}_{\mathcal{F}/X/S} \]

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 76.9.3) of the algebraic space $\mathit{Hom}(\mathcal{F}_ T, \mathcal{G})$ from Proposition 98.3.10. $\square$

Comments (2)

Comment #7792 by Laurent Moret-Bailly on

In the proof, the "coherent sheaf " is only quasi-coherent of finite presentation.

Comment #8030 by on

Thanks. I lazily fixed this by just saying module. The reader has to check that it is indeed something that gives an object of the stack Coh. See this.

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