Lemma 33.35.10. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$. If $\mathcal{F}$ is $m$-regular, then $\mathcal{F}$ is $(m + 1)$-regular.

Proof. We prove this by induction on $n$. If $n = 0$ every sheaf is $m$-regular for all $m$ and there is nothing to prove. By Lemma 33.35.8 we may replace $k$ by an infinite overfield and assume $k$ is infinite. Thus we may apply Lemma 33.35.3. By Lemma 33.35.9 we know that $\mathcal{G}$ is $m$-regular. By induction on $n$ we see that $\mathcal{G}$ is $(m + 1)$-regular. Considering the long exact cohomology sequence associated to the sequence

$0 \to \mathcal{F}(m - i) \to \mathcal{F}(m + 1 - i) \to i_*\mathcal{G}(m + 1 - i) \to 0$

as in Remark 33.35.5 the reader easily deduces for $i \geq 1$ the vanishing of $H^ i(\mathbf{P}^ n_ k, \mathcal{F}(m + 1 - i))$ from the (known) vanishing of $H^ i(\mathbf{P}^ n_ k, \mathcal{F}(m - i))$ and $H^ i(\mathbf{P}^ n_ k, \mathcal{G}(m + 1 - i))$. $\square$

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