The Stacks project

Lemma 44.2.2. Let $X \to S$ be a morphism of schemes. If $X \to S$ is of finite presentation, then the functor $\mathrm{Hilb}^ d_{X/S}$ is limit preserving (Limits, Remark 32.6.2).

Proof. Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a limit of affine schemes over $S$. We have to show that $\mathrm{Hilb}^ d_{X/S}(T) = \mathop{\mathrm{colim}}\nolimits \mathrm{Hilb}^ d_{X/S}(T_ i)$. Observe that if $Z \to X_ T$ is an element of $\mathrm{Hilb}^ d_{X/S}(T)$, then $Z \to T$ is of finite presentation. Hence by Limits, Lemma 32.10.1 there exists an $i$, a scheme $Z_ i$ of finite presentation over $T_ i$, and a morphism $Z_ i \to X_{T_ i}$ over $T_ i$ whose base change to $T$ gives $Z \to X_ T$. We apply Limits, Lemma 32.8.5 to see that we may assume $Z_ i \to X_{T_ i}$ is a closed immersion after increasing $i$. We apply Limits, Lemma 32.8.8 to see that $Z_ i \to T_ i$ is finite locally free of degree $d$ after possibly increasing $i$. Then $Z_ i \in \mathrm{Hilb}^ d_{X/S}(T_ i)$ as desired. $\square$

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