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.
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