The Stacks project

Example 108.4.10 (Coherent sheaves with fixed Hilbert polynomial). Let $f : X \to B$ be as in the introduction to this section. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $P : \mathbf{Z} \to \mathbf{Z}$ be a numerical polynomial. Then we can consider the open and closed algebraic substack

consisting of flat families of coherent sheaves with proper support whose numerical invariants agree with $P$: an object $(T \to B, \mathcal{F})$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ lies in $\mathcal{C}\! \mathit{oh}^ P_{X/B}$ if and only if

for all $n \in \mathbf{Z}$ and $t \in T$. Of course this is a special case of Situation 108.4.7 where $I = \mathbf{Z} \to D(\mathcal{O}_ X)$ is given by $n \mapsto \mathcal{L}^{\otimes n}$. It follows from Lemma 108.4.9 that this is an open and closed substack. Since the functions $n \mapsto \chi (X_ t, \mathcal{F}_ t \otimes _{\mathcal{O}_{X_ t}} \mathcal{L}_ t^{\otimes n})$ are always numerical polynomials (Spaces over Fields, Lemma 72.18.1) we conclude that

is a disjoint union decomposition.