Lemma 30.8.3. Let $S$ be a scheme. Let $n \geq 0$ be an integer. Consider the structure morphism

$f : \mathbf{P}^ n_ S \longrightarrow S.$

We have

$R^ qf_*(\mathcal{O}_{\mathbf{P}^ n_ S}(d)) = \left\{ \begin{matrix} (\mathcal{O}_ S[T_0, \ldots , T_ n])_ d & \text{if} & q = 0 \\ 0 & \text{if} & q \not= 0, n \\ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}( (\mathcal{O}_ S[T_0, \ldots , T_ n])_{- n - 1 - d}, \mathcal{O}_ S) & \text{if} & q = n \end{matrix} \right.$

Proof. Omitted. Hint: This follows since the identifications in (30.8.1.1) are compatible with affine base change by Lemma 30.8.2. $\square$

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