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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