27.19 Invertible sheaves and morphisms into relative Proj
It seems that we may need the following lemma somewhere. The situation is the following:
Let S be a scheme.
Let \mathcal{A} be a quasi-coherent graded \mathcal{O}_ S-algebra.
Denote \pi : \underline{\text{Proj}}_ S(\mathcal{A}) \to S the relative homogeneous spectrum over S.
Let f : X \to S be a morphism of schemes.
Let \mathcal{L} be an invertible \mathcal{O}_ X-module.
Let \psi : f^*\mathcal{A} \to \bigoplus _{d \geq 0} \mathcal{L}^{\otimes d} be a homomorphism of graded \mathcal{O}_ X-algebras.
Given this data set
U(\psi ) = \bigcup \nolimits _{(U, V, a)} U_{\psi (a)}
where (U, V, a) satisfies:
V \subset S affine open,
U = f^{-1}(V), and
a \in \mathcal{A}(V)_{+} is homogeneous.
Namely, then \psi (a) \in \Gamma (U, \mathcal{L}^{\otimes \deg (a)}) and U_{\psi (a)} is the corresponding open (see Modules, Lemma 17.25.10).
Lemma 27.19.1. With assumptions and notation as above. The morphism \psi induces a canonical morphism of schemes over S
r_{\mathcal{L}, \psi } : U(\psi ) \longrightarrow \underline{\text{Proj}}_ S(\mathcal{A})
together with a map of graded \mathcal{O}_{U(\psi )}-algebras
\theta : r_{\mathcal{L}, \psi }^*\left( \bigoplus \nolimits _{d \geq 0} \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(d) \right) \longrightarrow \bigoplus \nolimits _{d \geq 0} \mathcal{L}^{\otimes d}|_{U(\psi )}
characterized by the following properties:
For every open V \subset S and every d \geq 0 the diagram
\xymatrix{ \mathcal{A}_ d(V) \ar[d]_{\psi } \ar[r]_{\psi } & \Gamma (f^{-1}(V), \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma (\pi ^{-1}(V), \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(d)) \ar[r]^{\theta } & \Gamma (f^{-1}(V) \cap U(\psi ), \mathcal{L}^{\otimes d}) }
is commutative.
For any d \geq 1 and any open subscheme W \subset X such that \psi |_ W : f^*\mathcal{A}_ d|_ W \to \mathcal{L}^{\otimes d}|_ W is surjective the restriction of the morphism r_{\mathcal{L}, \psi } agrees with the morphism W \to \underline{\text{Proj}}_ S(\mathcal{A}) which exists by the construction of the relative homogeneous spectrum, see Definition 27.16.7.
For any affine open V \subset S, the restriction
(U(\psi ) \cap f^{-1}(V), r_{\mathcal{L}, \psi }|_{U(\psi ) \cap f^{-1}(V)}, \theta |_{U(\psi ) \cap f^{-1}(V)})
agrees via i_ V (see Lemma 27.15.4) with the triple (U(\psi '), r_{\mathcal{L}, \psi '}, \theta ') of Lemma 27.14.1 associated to the map \psi ' : A = \mathcal{A}(V) \to \Gamma _*(f^{-1}(V), \mathcal{L}|_{f^{-1}(V)}) induced by \psi .
Proof.
Use characterization (3) to construct the morphism r_{\mathcal{L}, \psi } and \theta locally over S. Use the uniqueness of Lemma 27.14.1 to show that the construction glues. Details omitted.
\square
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