The Stacks project

70.12 Functoriality of relative proj

This section is the analogue of Constructions, Section 27.18.

Lemma 70.12.1. 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 70.11.3. Details omitted. $\square$

Lemma 70.12.2. 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 70.12.3. With hypotheses and notation as in Lemma 70.12.1 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 70.12.4. With hypotheses and notation as in Lemma 70.12.1 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 70.12.5. With hypotheses and notation as in Lemma 70.12.1 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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