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

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

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