The Stacks project

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. $\square$

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