Lemma 27.18.1. Let S be a scheme. Let \mathcal{A}, \mathcal{B} be two graded quasi-coherent \mathcal{O}_ S-algebras. Set p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S and q : Y = \underline{\text{Proj}}_ S(\mathcal{B}) \to S. Let \psi : \mathcal{A} \to \mathcal{B} be a homomorphism of graded \mathcal{O}_ S-algebras. There is a canonical open U(\psi ) \subset Y and a canonical morphism of schemes
over S 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 affine open W \subset S the triple
is equal to the triple associated to \psi : \mathcal{A}(W) \to \mathcal{B}(W) in Lemma 27.11.1 via the identifications p^{-1}W = \text{Proj}(\mathcal{A}(W)) and q^{-1}W = \text{Proj}(\mathcal{B}(W)) of Section 27.15.
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