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

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 étale 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 étale 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). $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)