Situation 111.45.1. Let $k$ be a field. Let $X = \mathbf{P}^ n_ k$ be $n$-dimensional projective space. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Recall that

Recall that the *Hilbert polynomial* of $\mathcal{F}$ is the function

We also recall that $\mathcal{F}(t) = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(t)$ where $\mathcal{O}_ X(t)$ is the $t$th twist of the structure sheaf as in Constructions, Definition 27.10.1. In Varieties, Subsection 33.35.13 we have proved the Hilbert polynomial is a polynomial in $t$.

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