Lemma 70.12.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\psi : \mathcal{A} \to \mathcal{B}$ be a map of quasi-coherent graded $\mathcal{O}_ X$-algebras. Set $P = \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ and $Q = \underline{\text{Proj}}_ X(\mathcal{B}) \to X$. There is a canonical open subspace $U(\psi ) \subset Q$ and a canonical morphism of algebraic spaces

over $X$ and a map of $\mathbf{Z}$-graded $\mathcal{O}_{U(\psi )}$-algebras

The triple $(U(\psi ), r_\psi , \theta )$ is characterized by the property that for any scheme $W$ étale over $X$ the triple

is equal to the triple associated to $\psi : \mathcal{A}|_ W \to \mathcal{B}|_ W$ of Constructions, Lemma 27.18.1.

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