The Stacks project

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

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

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

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

The triple $(U(\psi ), r_\psi , \theta )$ is characterized by the property that for any affine open $W \subset S$ the triple

\[ (U(\psi ) \cap p^{-1}W,\quad r_\psi |_{U(\psi ) \cap p^{-1}W} : U(\psi ) \cap p^{-1}W \to q^{-1}W,\quad \theta |_{U(\psi ) \cap p^{-1}W}) \]

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.

Proof. This lemma proves itself by glueing the local triples. $\square$

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