Lemma 108.5.5 (0DP2)—The Stacks project

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Table of contents
Part 8: Topics in Moduli Theory
Chapter 108: Moduli Stacks
Section 108.5: Properties of Quot
Lemma 108.5.5 (cite)

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Lemma 108.5.5. Let $B$ be an algebraic space. Let $\pi : X \to Y$ be an affine open immersion of algebraic spaces which are separated and of finite presentation over $B$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Then the morphism $\mathrm{Quot}_{\mathcal{F}/X/B} \to \mathrm{Quot}_{\pi _*\mathcal{F}/Y/B}$ of Lemma 108.5.4 is an open immersion.

Proof. Omitted. Hint: If $(\pi _*\mathcal{F})_ T \to \mathcal{Q}$ is an element of $\mathrm{Quot}_{\pi _*\mathcal{F}/Y/B}(T)$ and for $t \in T$ we have $\text{Supp}(\mathcal{Q}_ t) \subset |X_ t|$ , then the same is true for $t' \in T$ in a neighbourhood of $t$ . $\square$

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