## 99.8 The Quot functor

In this section we prove the Quot functor is an algebraic space.

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 $\mathcal{F}/X/B$.

In Situation 99.8.1 we sometimes think of $\mathrm{Quot}_{\mathcal{F}/X/B}$ as a functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\mathrm{Quot}_{\mathcal{F}/X/B} \to B$. Namely, if $T$ is a scheme over $S$, then an element of $\mathrm{Quot}_{\mathcal{F}/X/B}(T)$ is a pair $(h, \mathcal{Q})$ where $h$ is a morphism $h : T \to B$ and $Q$ is a finitely presented, $T$-flat quotient $\mathcal{F}_ T \to \mathcal{Q}$ on $X_ T = X \times _{B, h} T$ with support proper over $T$. In particular, when we say that $\mathrm{Quot}_{\mathcal{F}/X/B}$ is an algebraic space, we mean that the corresponding functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$ is an algebraic space.

Lemma 99.8.2. In Situation 99.8.1. The functor $\mathrm{Quot}_{\mathcal{F}/X/B}$ satisfies the sheaf property for the fpqc topology.

**Proof.**
In Lemma 99.7.4 we have seen that the functor $\text{Q}^{fp}_{\mathcal{F}/X/S}$ is a sheaf. Recall that for a scheme $T$ over $S$ the subset $\mathrm{Quot}_{\mathcal{F}/X/S}(T) \subset \text{Q}_{\mathcal{F}/X/S}(T)$ picks out those quotients whose support is proper over $T$. This defines a subsheaf by the result of Descent on Spaces, Lemma 74.11.19 combined with Morphisms of Spaces, Lemma 67.30.10 which shows that taking scheme theoretic support commutes with flat base change.
$\square$

Sanity check: $\mathrm{Quot}_{\mathcal{F}/X/B}$ plays the same role among algebraic spaces over $S$.

Lemma 99.8.3. In Situation 99.8.1. Let $T$ be an algebraic space over $S$. We have

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})}(T, \mathrm{Quot}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_ T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}
\\ \mathcal{Q}\text{ is of finite presentation and}
\\ \text{flat over }T\text{ with support proper over }T
\end{matrix} \right\} \]

where $\mathcal{F}_ T$ denotes the pullback of $\mathcal{F}$ to the algebraic space $X \times _{B, h} T$.

**Proof.**
Observe that the left and right hand side of the equality are subsets of the left and right hand side of the second equality in Lemma 99.7.5. To see that these subsets correspond under the identification given in the proof of that lemma it suffices to show: given $h : T \to B$, a surjective étale morphism $U \to T$, a finite type quasi-coherent $\mathcal{O}_{X_ T}$-module $\mathcal{Q}$ the following are equivalent

the scheme theoretic support of $\mathcal{Q}$ is proper over $T$, and

the scheme theoretic support of $(X_ U \to X_ T)^*\mathcal{Q}$ is proper over $U$.

This follows from Descent on Spaces, Lemma 74.11.19 combined with Morphisms of Spaces, Lemma 67.30.10 which shows that taking scheme theoretic support commutes with flat base change.
$\square$

Proposition 99.8.4. 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 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

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

on $(\mathit{Sch}/S)_{fppf}$ which associates to the quotient $\mathcal{F}_ T \to \mathcal{Q}$ the module $\mathcal{Q}$. By Theorem 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

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

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 77.9.3) of the algebraic space $\mathit{Hom}(\mathcal{F}_ T, \mathcal{G})$ from Proposition 99.3.10.
$\square$

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