Exercise 111.42.7. Let $A$ be a ring and let $\mathbf{P}^ n_ A = \text{Proj}(A[T_0, \ldots , T_ n])$ be projective space over $A$. Carefully compute the cohomology of the Serre twists $\mathcal{O}_{\mathbf{P}^ n_ A}(d)$ of the structure sheaf on $\mathbf{P}^ n_ A$. Feel free to use Čech cohomology and the agreement of Čech cohomology with cohomology for quasi-coherent sheaves and affine open coverings of separated schemes.
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