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Remark 33.35.5. In the situation of Lemmas 33.35.2 and 33.35.3 we have $H \cong \mathbf{P}^{n - 1}_ k$ with Serre twists $\mathcal{O}_ H(d) = i^*\mathcal{O}_{\mathbf{P}^ n_ k}(d)$. For every $d \in \mathbf{Z}$ we have a short exact sequence

\[ 0 \to \mathcal{F}(d - 1) \to \mathcal{F}(d) \to i_*(\mathcal{G}(d)) \to 0 \]

Namely, tensoring by $\mathcal{O}_{\mathbf{P}^ n_ k}(d)$ is an exact functor and by the projection formula (Cohomology, Lemma 20.52.2) we have $i_*(\mathcal{G}(d)) = i_*\mathcal{G} \otimes \mathcal{O}_{\mathbf{P}^ n_ k}(d)$. We obtain corresponding long exact sequences

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

This follows from the above and the fact that we have $H^ i(\mathbf{P}^ n_ k, i_*\mathcal{G}(d)) = H^ i(H, \mathcal{G}(d))$ by Cohomology of Schemes, Lemma 30.2.4 (closed immersions are affine).

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