The Stacks project

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

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

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 $W$ ├ętale 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 ├ętale localization and the case of schemes, see discussion following Definition 70.11.3. Details omitted. $\square$


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