The Stacks project

Exercise 109.42.6. Let $A$ be a ring. Let $\mathbf{P}^ n_ A = \text{Proj}(A[T_0, \ldots , T_ n])$ be projective space over $A$. Let $\mathbf{A}^{n + 1}_ A = \mathop{\mathrm{Spec}}(A[T_0, \ldots , T_ n])$ and let

\[ U = \bigcup \nolimits _{i = 0, \ldots , n} D(T_ i) \subset \mathbf{A}^{n + 1}_ A \]

be the complement of the image of the closed immersion $0 : \mathop{\mathrm{Spec}}(A) \to \mathbf{A}^{n + 1}_ A$. Construct an affine surjective morphism

\[ f : U \longrightarrow \mathbf{P}^ n_ A \]

and prove that $f_*\mathcal{O}_ U = \bigoplus _{d \in \mathbf{Z}} \mathcal{O}_{\mathbf{P}^ n_ A}(d)$. More generally, show that for a graded $A[T_0, \ldots , T_ n]$-module $M$ one has

\[ f_*(\widetilde{M}|_ U) = \bigoplus \nolimits _{d \in \mathbf{Z}} \widetilde{M(d)} \]

where on the left hand side we have the quasi-coherent sheaf $\widetilde{M}$ associated to $M$ on $\mathbf{A}^{n + 1}_ A$ and on the right we have the quasi-coherent sheaves $\widetilde{M(d)}$ associated to the graded module $M(d)$.

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