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The Stacks project

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

r_\psi : U(\psi ) \longrightarrow P

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

\theta = \theta _\psi : r_\psi ^*\left( \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_ P(d) \right) \longrightarrow \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{U(\psi )}(d).

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

(U(\psi ) \times _ X W,\quad r_\psi |_{U(\psi ) \times _ X W} : U(\psi ) \times _ X W \to P \times _ X W,\quad \theta |_{U(\psi ) \times _ X W})

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

Proof. This lemma follows from étale localization and the case of schemes, see discussion following Definition 71.11.3. Details omitted. \square


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