The Stacks project

Lemma 70.11.4. In Situation 70.11.1. 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 70.11.3 choose a scheme $U$ and a surjective ├ętale 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 65.32.1 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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