Situation 108.4.7 (0DNC): Numerical invariants

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

\[ K_ i = Rf_{T, *}(E_{i, T} \otimes _{\mathcal{O}_{X_ T}}^\mathbf {L} \mathcal{F}) \]

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

\[ \chi _ i : |T| \longrightarrow \mathbf{Z},\quad \chi _ i(t) = \chi (X_ t, E_{i, t} \otimes _{\mathcal{O}_{X_ t}}^\mathbf {L} \mathcal{F}_ t) = \chi (K_ i \otimes _{\mathcal{O}_ T}^\mathbf {L} \kappa (t)) \]

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

\[ \mathcal{C}\! \mathit{oh}^ P_{X/B} \subset \mathcal{C}\! \mathit{oh}_{X/B} \]

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 $t \in T$.

The Stacks project

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