Lemma 33.35.9. In the situation of Lemma 33.35.3, if $\mathcal{F}$ is $m$-regular, then $\mathcal{G}$ is $m$-regular on $H \cong \mathbf{P}^{n - 1}_ k$.

Proof. Recall that $H^ i(\mathbf{P}^ n_ k, i_*\mathcal{G}) = H^ i(H, \mathcal{G})$ by Cohomology of Schemes, Lemma 30.2.4. Hence we see that for $i \geq 1$ we get

$H^ i(\mathbf{P}^ n_ k, \mathcal{F}(m - i)) \to H^ i(H, \mathcal{G}(m - i)) \to H^{i + 1}(\mathbf{P}^ n_ k, \mathcal{F}(m - 1 - i))$

by Remark 33.35.5. The lemma follows. $\square$

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