The Stacks project

Lemma 44.2.1. Let $X \to S$ be a morphism of schemes. The functor $\mathrm{Hilb}^ d_{X/S}$ satisfies the sheaf property for the fpqc topology (Topologies, Definition 34.9.12).

Proof. Let $\{ T_ i \to T\} _{i \in I}$ be an fpqc covering of schemes over $S$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$. Note that $\{ X_ i \to X_ T\} _{i \in I}$ is an fpqc covering of $X_ T$ (Topologies, Lemma 34.9.7) and that $X_{T_ i \times _ T T_{i'}} = X_ i \times _{X_ T} X_{i'}$. Suppose that $Z_ i \in \mathrm{Hilb}^ d_{X/S}(T_ i)$ is a collection of elements such that $Z_ i$ and $Z_{i'}$ map to the same element of $\mathrm{Hilb}^ d_{X/S}(T_ i \times _ T T_{i'})$. By effective descent for closed immersions (Descent, Lemma 35.34.2) there is a closed immersion $Z \to X_ T$ whose base change by $X_ i \to X_ T$ is equal to $Z_ i \to X_ i$. The morphism $Z \to T$ then has the property that its base change to $T_ i$ is the morphism $Z_ i \to T_ i$. Hence $Z \to T$ is finite locally free of degree $d$ by Descent, Lemma 35.20.30. $\square$


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