Lemma 33.35.16 (08AE)—The Stacks project

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Part 2: Schemes
Chapter 33: Varieties
Section 33.35: Coherent sheaves on projective space
Lemma 33.35.16 (cite)

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Lemma 33.35.16. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$ with Hilbert polynomial $P \in \mathbf{Q}[t]$. Then

\[ P(d) = \dim _ k H^0(\mathbf{P}^ n_ k, \mathcal{F}(d)) \]

for all $d \gg 0$.

Proof. This follows from the vanishing of cohomology of high enough twists of $\mathcal{F}$. See Cohomology of Schemes, Lemma 30.14.1. $\square$

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