The Stacks project

27.19 Invertible sheaves and morphisms into relative Proj

It seems that we may need the following lemma somewhere. The situation is the following:

  1. Let $S$ be a scheme.

  2. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra.

  3. Denote $\pi : \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ the relative homogeneous spectrum over $S$.

  4. Let $f : X \to S$ be a morphism of schemes.

  5. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module.

  6. 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:

  1. $V \subset S$ affine open,

  2. $U = f^{-1}(V)$, and

  3. $a \in \mathcal{A}(V)_{+}$ 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 17.24.10).

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:

  1. 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.

  2. 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.

  3. 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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