Remark 108.10.4 (Numerical invariants). Let $B, X, Y$ be as in the introduction to this section. Let $I$ be a set and for $i \in I$ let $E_ i \in D(\mathcal{O}_{Y \times _ B X})$ be perfect. Let $P : I \to \mathbf{Z}$ be a function. Recall that
\[ \mathit{Mor}_ B(Y, X) \subset \mathrm{Hilb}_{Y \times _ B X/B} \]
is an open subspace, see Quot, Lemma 99.12.2. Thus we can define
\[ \mathit{Mor}^ P_ B(Y, X) = \mathit{Mor}_ B(Y, X) \cap \mathrm{Hilb}^ P_{Y \times _ B X/B} \]
where $\mathrm{Hilb}^ P_{Y \times _ B X/B}$ is as in Remark 108.7.6. The morphism
\[ \mathit{Mor}^ P_ B(Y, X) \longrightarrow \mathit{Mor}_ B(Y, X) \]
is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \mathbf{Z}$, $E_ i = \mathcal{L}^{\otimes i}$ for some invertible $\mathcal{O}_{Y \times _ B X}$-module $\mathcal{L}$. In the last case we sometimes use the notation $\mathit{Mor}^{P, \mathcal{L}}_ B(Y, X)$.
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