Lemma 71.12.1. 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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