The Stacks project

Lemma 107.5.3. Assume $X \to B$ is proper as well as of finite presentation and $\mathcal{F}$ quasi-coherent of finite type. Then $\mathrm{Quot}_{\mathcal{F}/X/B} \to B$ satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition 66.41.1).

Proof. Taking base change, this immediately reduces to the following problem: given a valuation ring $R$ with fraction field $K$, an algebraic space $X$ proper over $R$, a finite type quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$, and a coherent quotient $\mathcal{F}_ K \to \mathcal{Q}_ K$, show there exists a quotient $\mathcal{F} \to \mathcal{Q}$ where $\mathcal{Q}$ is a finitely presented $\mathcal{O}_ X$-module flat over $R$ whose generic fibre is $\mathcal{Q}_ K$. Observe that by Flatness on Spaces, Theorem 76.4.5 any finite type quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ flat over $R$ is of finite presentation. We first solve the existence of $\mathcal{Q}$ affine locally.

Affine locally we arrive at the following problem: let $R \to A$ be a finitely presented ring map, let $M$ be a finite $A$-module, let $\varphi : M_ K \to N_ K$ be an $A_ K$-quotient module. Then we may consider

\[ L = \{ x \in M \mid \varphi (x \otimes 1) = 0 \} \]

The $M \to M/L$ is an $A$-module quotient which is torsion free as an $R$-module. Hence it is flat as an $R$-module (More on Algebra, Lemma 15.22.10). Since $M$ is finite as an $A$-module so is $L$ and we conclude that $L$ is of finite presentation as an $A$-module (by the reference above). Clearly $M/L$ is the unqiue such quotient with $(M/L)_ K = N_ K$.

The uniqueness in the construction of the previous paragraph guarantees these quotients glue and give the desired $\mathcal{Q}$. Here is a bit more detail. Choose a surjective ├ętale morphism $U \to X$ where $U$ is an affine scheme. Use the above construction to construct a quotient $\mathcal{F}|_ U \to \mathcal{Q}_ U$ which is quasi-coherent, is flat over $R$, and recovers $\mathcal{Q}_ K|U$ on the generic fibre. Since $X$ is separated, we see that $U \times _ X U$ is an affine scheme ├ętale over $X$ as well. Then $\mathcal{F}|_{U \times _ X U} \to \text{pr}_1^*\mathcal{Q}_ U$ and $\mathcal{F}|_{U \times _ X U} \to \text{pr}_2^*\mathcal{Q}_ U$ agree as quotients by the uniquess in the construction. Hence we may descend $\mathcal{F}|_ U \to \mathcal{Q}_ U$ to a surjection $\mathcal{F} \to \mathcal{Q}$ as desired (Properties of Spaces, Proposition 65.32.1). $\square$

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