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
\[ \chi (X, \mathcal{F}) = \sum \nolimits _{i = 0}^ n (-1)^ i \dim _ k H^ i(X, \mathcal{F}) \]
Recall that the Hilbert polynomial of $\mathcal{F}$ is the function
\[ t \longmapsto \chi (X, \mathcal{F}(t)) \]
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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