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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 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.

  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 étale 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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