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.
$\square$
Comments (0)