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71.12 Functoriality of relative proj

This section is the analogue of Constructions, Section 27.18.

Lemma 71.12.1. Let S be a scheme. Let X be an algebraic space over S. Let \psi : \mathcal{A} \to \mathcal{B} be a map of quasi-coherent graded \mathcal{O}_ X-algebras. Set P = \underline{\text{Proj}}_ X(\mathcal{A}) \to X and Q = \underline{\text{Proj}}_ X(\mathcal{B}) \to X. There is a canonical open subspace U(\psi ) \subset Q and a canonical morphism of algebraic spaces

r_\psi : U(\psi ) \longrightarrow P

over X and a map of \mathbf{Z}-graded \mathcal{O}_{U(\psi )}-algebras

\theta = \theta _\psi : r_\psi ^*\left( \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_ P(d) \right) \longrightarrow \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{U(\psi )}(d).

The triple (U(\psi ), r_\psi , \theta ) is characterized by the property that for any scheme W étale over X the triple

(U(\psi ) \times _ X W,\quad r_\psi |_{U(\psi ) \times _ X W} : U(\psi ) \times _ X W \to P \times _ X W,\quad \theta |_{U(\psi ) \times _ X W})

is equal to the triple associated to \psi : \mathcal{A}|_ W \to \mathcal{B}|_ W of Constructions, Lemma 27.18.1.

Proof. This lemma follows from étale localization and the case of schemes, see discussion following Definition 71.11.3. Details omitted. \square

Lemma 71.12.2. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{A}, \mathcal{B}, and \mathcal{C} be quasi-coherent graded \mathcal{O}_ X-algebras. Set P = \underline{\text{Proj}}_ X(\mathcal{A}), Q = \underline{\text{Proj}}_ X(\mathcal{B}) and R = \underline{\text{Proj}}_ X(\mathcal{C}). Let \varphi : \mathcal{A} \to \mathcal{B}, \psi : \mathcal{B} \to \mathcal{C} be graded \mathcal{O}_ X-algebra maps. Then we have

U(\psi \circ \varphi ) = r_\varphi ^{-1}(U(\psi )) \quad \text{and} \quad r_{\psi \circ \varphi } = r_\varphi \circ r_\psi |_{U(\psi \circ \varphi )}.

In addition we have

\theta _\psi \circ r_\psi ^*\theta _\varphi = \theta _{\psi \circ \varphi }

with obvious notation.

Proof. Omitted. \square

Lemma 71.12.3. With hypotheses and notation as in Lemma 71.12.1 above. Assume \mathcal{A}_ d \to \mathcal{B}_ d is surjective for d \gg 0. Then

  1. U(\psi ) = Q,

  2. r_\psi : Q \to R is a closed immersion, and

  3. the maps \theta : r_\psi ^*\mathcal{O}_ P(n) \to \mathcal{O}_ Q(n) are surjective but not isomorphisms in general (even if \mathcal{A} \to \mathcal{B} is surjective).

Proof. Follows from the case of schemes (Constructions, Lemma 27.18.3) by étale localization. \square

Lemma 71.12.4. With hypotheses and notation as in Lemma 71.12.1 above. Assume \mathcal{A}_ d \to \mathcal{B}_ d is an isomorphism for all d \gg 0. Then

  1. U(\psi ) = Q,

  2. r_\psi : Q \to P is an isomorphism, and

  3. the maps \theta : r_\psi ^*\mathcal{O}_ P(n) \to \mathcal{O}_ Q(n) are isomorphisms.

Proof. Follows from the case of schemes (Constructions, Lemma 27.18.4) by étale localization. \square

Lemma 71.12.5. With hypotheses and notation as in Lemma 71.12.1 above. Assume \mathcal{A}_ d \to \mathcal{B}_ d is surjective for d \gg 0 and that \mathcal{A} is generated by \mathcal{A}_1 over \mathcal{A}_0. Then

  1. U(\psi ) = Q,

  2. r_\psi : Q \to P is a closed immersion, and

  3. the maps \theta : r_\psi ^*\mathcal{O}_ P(n) \to \mathcal{O}_ Q(n) are isomorphisms.

Proof. Follows from the case of schemes (Constructions, Lemma 27.18.5) by étale localization. \square


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