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The Stacks project

Lemma 71.11.4. In Situation 71.11.1. The relative Proj comes equipped with a quasi-coherent sheaf of \mathbf{Z}-graded algebras \bigoplus _{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ X(\mathcal{A})}(n) and a canonical homomorphism of graded algebras

\psi : \pi ^*\mathcal{A} \longrightarrow \bigoplus \nolimits _{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_ X(\mathcal{A})}(n)

whose base change to any scheme over X agrees with Constructions, Lemma 27.15.5.

Proof. As in the discussion following Definition 71.11.3 choose a scheme U and a surjective étale morphism U \to X, set R = U \times _ X U with projections s, t : R \to U, \mathcal{A}_ U = \mathcal{A}|_ U, \mathcal{A}_ R = \mathcal{A}|_ R, and \pi : P = \underline{\text{Proj}}_ X(\mathcal{A}) \to X, \pi _ U : P_ U = \underline{\text{Proj}}_ U(\mathcal{A}_ U) and \pi _ R : P_ R = \underline{\text{Proj}}_ U(\mathcal{A}_ R). By the Constructions, Lemma 27.15.5 we have a quasi-coherent sheaf of \mathbf{Z}-graded \mathcal{O}_{P_ U}-algebras \bigoplus _{n \in \mathbf{Z}} \mathcal{O}_{P_ U}(n) and a canonical map \psi _ U : \pi _ U^*\mathcal{A}_ U \to \bigoplus _{n \geq 0} \mathcal{O}_{P_ U}(n) and similarly for P_ R. By Constructions, Lemma 27.16.10 the pullback of \mathcal{O}_{P_ U}(n) and \psi _ U by either projection P_ R \to P_ U is equal to \mathcal{O}_{P_ R}(n) and \psi _ R. By Properties of Spaces, Proposition 66.32.1 we obtain \mathcal{O}_{P}(n) and \psi . We omit the verification of compatibility with pullback to arbitrary schemes over X. \square


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