Lemma 71.13.1. With assumptions and notation as above. The morphism $\psi $ induces a canonical morphism of algebraic spaces over $Y$

together with a map of graded $\mathcal{O}_{U(\psi )}$-algebras

characterized by the following properties:

For $V \to Y$ étale and $d \geq 0$ the diagram

\[ \xymatrix{ \mathcal{A}_ d(V) \ar[d]_{\psi } \ar[r]_{\psi } & \Gamma (V \times _ Y X, \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma (V \times _ Y \underline{\text{Proj}}_ Y(\mathcal{A}), \mathcal{O}_{\underline{\text{Proj}}_ Y(\mathcal{A})}(d)) \ar[r]^-\theta & \Gamma (V \times _ Y U(\psi ), \mathcal{L}^{\otimes d}) } \]is commutative.

For any $d \geq 1$ and any morphism $W \to X$ where $W$ is a scheme such that $\psi |_ W : f^*\mathcal{A}_ d|_ W \to \mathcal{L}^{\otimes d}|_ W$ is surjective we have (a) $W \to X$ factors through $U(\psi )$ and (b) composition of $W \to U(\psi )$ with $r_{\mathcal{L}, \psi }$ agrees with the morphism $W \to \underline{\text{Proj}}_ Y(\mathcal{A})$ which exists by the construction of $\underline{\text{Proj}}_ Y(\mathcal{A})$, see Definition 71.11.3.

Consider a commutative diagram

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]where $X'$ and $Y'$ are schemes, set $\mathcal{A}' = g^*\mathcal{A}$ and $\mathcal{L}' = (g')^*\mathcal{L}$ and denote $\psi ' : (f')^*\mathcal{A} \to \bigoplus _{d \geq 0} (\mathcal{L}')^{\otimes d}$ the pullback of $\psi $. Let $U(\psi ')$, $r_{\psi ', \mathcal{L}'}$, and $\theta '$ be the open, morphism, and homomorphism constructed in Constructions, Lemma 71.13.1. Then $U(\psi ') = (g')^{-1}(U(\psi ))$ and $r_{\psi ', \mathcal{L}'}$ agrees with the base change of $r_{\psi , \mathcal{L}}$ via the isomorphism $\underline{\text{Proj}}_{Y'}(\mathcal{A}') = Y' \times _ Y \underline{\text{Proj}}_ Y(\mathcal{A})$ of Lemma 71.11.5. Moreover, $\theta '$ is the pullback of $\theta $.

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