27.18 Functoriality of relative Proj

This section is the analogue of Section 27.11 for the relative Proj. Let $S$ be a scheme. A graded $\mathcal{O}_ S$-algebra map $\psi : \mathcal{A} \to \mathcal{B}$ does not always give rise to a morphism of associated relative Proj. The correct result is stated as follows.

Lemma 27.18.1. Let $S$ be a scheme. Let $\mathcal{A}$, $\mathcal{B}$ be two graded quasi-coherent $\mathcal{O}_ S$-algebras. Set $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ and $q : Y = \underline{\text{Proj}}_ S(\mathcal{B}) \to S$. Let $\psi : \mathcal{A} \to \mathcal{B}$ be a homomorphism of graded $\mathcal{O}_ S$-algebras. There is a canonical open $U(\psi ) \subset Y$ and a canonical morphism of schemes

$r_\psi : U(\psi ) \longrightarrow X$

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

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

The triple $(U(\psi ), r_\psi , \theta )$ is characterized by the property that for any affine open $W \subset S$ the triple

$(U(\psi ) \cap p^{-1}W,\quad r_\psi |_{U(\psi ) \cap p^{-1}W} : U(\psi ) \cap p^{-1}W \to q^{-1}W,\quad \theta |_{U(\psi ) \cap p^{-1}W})$

is equal to the triple associated to $\psi : \mathcal{A}(W) \to \mathcal{B}(W)$ in Lemma 27.11.1 via the identifications $p^{-1}W = \text{Proj}(\mathcal{A}(W))$ and $q^{-1}W = \text{Proj}(\mathcal{B}(W))$ of Section 27.15.

Proof. This lemma proves itself by glueing the local triples. $\square$

Lemma 27.18.2. Let $S$ be a scheme. Let $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ be quasi-coherent graded $\mathcal{O}_ S$-algebras. Set $X = \underline{\text{Proj}}_ S(\mathcal{A})$, $Y = \underline{\text{Proj}}_ S(\mathcal{B})$ and $Z = \underline{\text{Proj}}_ S(\mathcal{C})$. Let $\varphi : \mathcal{A} \to \mathcal{B}$, $\psi : \mathcal{B} \to \mathcal{C}$ be graded $\mathcal{O}_ S$-algebra maps. Then we have

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

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

with obvious notation.

Proof. Omitted. $\square$

Lemma 27.18.3. With hypotheses and notation as in Lemma 27.18.1 above. Assume $\mathcal{A}_ d \to \mathcal{B}_ d$ is surjective for $d \gg 0$. Then

1. $U(\psi ) = Y$,

2. $r_\psi : Y \to X$ is a closed immersion, and

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

Proof. Follows on combining Lemma 27.18.1 with Lemma 27.11.3. $\square$

Lemma 27.18.4. With hypotheses and notation as in Lemma 27.18.1 above. Assume $\mathcal{A}_ d \to \mathcal{B}_ d$ is an isomorphism for all $d \gg 0$. Then

1. $U(\psi ) = Y$,

2. $r_\psi : Y \to X$ is an isomorphism, and

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

Proof. Follows on combining Lemma 27.18.1 with Lemma 27.11.4. $\square$

Lemma 27.18.5. With hypotheses and notation as in Lemma 27.18.1 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. $U(\psi ) = Y$,

2. $r_\psi : Y \to X$ is a closed immersion, and

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

Proof. Follows on combining Lemma 27.18.1 with Lemma 27.11.5. $\square$

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