The Stacks project

Situation 108.4.7 (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. Given an object $(T \to B, \mathcal{F})$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ denote $E_{i, T}$ the derived pullback of $E_ i$ to $X_ T$. The object

of $D(\mathcal{O}_ T)$ is perfect and its formation commutes with base change, see Derived Categories of Spaces, Lemma 75.25.1. Thus the function

is locally constant by Derived Categories of Spaces, Lemma 75.26.3. Let $P : I \to \mathbf{Z}$ be a map. Consider the substack

consisting of flat families of coherent sheaves with proper support whose numerical invariants agree with $P$. More precisely, an object $(T \to B, \mathcal{F})$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ is in $\mathcal{C}\! \mathit{oh}^ P_{X/B}$ if and only if $\chi _ i(t) = P(i)$ for all $i \in I$ and $t \in T$.