The Stacks project

71.13 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 and $Y$ an algebraic space over $S$.

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

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

  4. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  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 let $U(\psi ) \subset X$ be the open subspace with

\[ |U(\psi )| = \bigcup \nolimits _{d \geq 1} \{ \text{locus where }f^*\mathcal{A}_ d \to \mathcal{L}^{\otimes d} \text{ is surjective}\} \]

Formation of $U(\psi ) \subset X$ commutes with pullback by any morphism $X' \to X$.

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

\[ r_{\mathcal{L}, \psi } : U(\psi ) \longrightarrow \underline{\text{Proj}}_ Y(\mathcal{A}) \]

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

\[ \theta : r_{\mathcal{L}, \psi }^*\left( \bigoplus \nolimits _{d \geq 0} \mathcal{O}_{\underline{\text{Proj}}_ Y(\mathcal{A})}(d) \right) \longrightarrow \bigoplus \nolimits _{d \geq 0} \mathcal{L}^{\otimes d}|_{U(\psi )} \]

characterized by the following properties:

  1. For $V \to Y$ ├ętale 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.

  2. 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.

  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 $.

Proof. Omitted. Hints: First we observe that for a quasi-compact scheme $W$ over $X$ the following are equivalent

  1. $W \to X$ factors through $U(\psi )$, and

  2. there exists a $d$ such that $\psi |_ W : f^*\mathcal{A}_ d|_ W \to \mathcal{L}^{\otimes d}|_ W$ is surjective.

This gives a description of $U(\psi )$ as a subfunctor of $X$ on our base category $(\mathit{Sch}/S)_{fppf}$. For such a $W$ and $d$ we consider the quadruple $(d, W \to Y, \mathcal{L}|_ W, \psi ^{(d)}|_ W)$. By definition of $\underline{\text{Proj}}_ Y(\mathcal{A})$ we obtain a morphism $W \to \underline{\text{Proj}}_ Y(\mathcal{A})$. By our notion of equivalence of quadruples one sees that this morphism is independent of the choice of $d$. This clearly defines a transformation of functors $r_{\psi , \mathcal{L}} : U(\psi ) \to \underline{\text{Proj}}_ Y(\mathcal{A})$, i.e., a morphism of algebraic spaces. By construction this morphism satisfies (2). Since the morphism constructed in Constructions, Lemma 27.19.1 satisfies the same property, we see that (3) is true.

To construct $\theta $ and check the compatibility (1) of the lemma, work ├ętale locally on $Y$ and $X$, arguing as in the discussion following Definition 71.11.3. $\square$

Comments (2)

Comment #7844 by Shang Li on

In (5), L should be an invertible O_X module.

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