The Stacks project

Remark 108.7.6 (Numerical invariants). Let $f : X \to B$ be as in the introduction to this section. Let $I$ be a set and for $i \in I$ let $E_ i \in D(\mathcal{O}_ X)$ be perfect. Let $P : I \to \mathbf{Z}$ be a function. Recall that $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$, see Quot, Lemma 99.9.2. Thus we can define

\[ \mathrm{Hilb}^ P_{X/B} = \mathrm{Quot}^ P_{\mathcal{O}_ X/X/B} \]

where $\mathrm{Quot}^ P_{\mathcal{O}_ X/X/B}$ is as in Remark 108.5.9. The morphism

\[ \mathrm{Hilb}^ P_{X/B} \longrightarrow \mathrm{Hilb}_{X/B} \]

is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \mathbf{Z}$ and $E_ i = \mathcal{L}^{\otimes i}$ for some invertible $\mathcal{O}_ X$-module $\mathcal{L}$. In the last case we sometimes use the notation $\mathrm{Hilb}^{P, \mathcal{L}}_{X/B}$.

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