## 31.30 Relative Proj

Some results on relative Proj. First some very basic results. Recall that a relative Proj is always separated over the base, see Constructions, Lemma 27.16.9.

Lemma 31.30.1. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If one of the following holds

1. $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{A}_0$-algebras,

2. $\mathcal{A}$ is generated by $\mathcal{A}_1$ as an $\mathcal{A}_0$-algebra and $\mathcal{A}_1$ is a finite type $\mathcal{A}_0$-module,

3. there exists a finite type quasi-coherent $\mathcal{A}_0$-submodule $\mathcal{F} \subset \mathcal{A}_{+}$ such that $\mathcal{A}_{+}/\mathcal{F}\mathcal{A}$ is a locally nilpotent sheaf of ideals of $\mathcal{A}/\mathcal{F}\mathcal{A}$,

then $p$ is quasi-compact.

Proof. The question is local on the base, see Schemes, Lemma 26.19.2. Thus we may assume $S$ is affine. Say $S = \mathop{\mathrm{Spec}}(R)$ and $\mathcal{A}$ corresponds to the graded $R$-algebra $A$. Then $X = \text{Proj}(A)$, see Constructions, Section 27.15. In case (1) we may after possibly localizing more assume that $A$ is generated by homogeneous elements $f_1, \ldots , f_ n \in A_{+}$ over $A_0$. Then $A_{+} = (f_1, \ldots , f_ n)$ by Algebra, Lemma 10.58.1. In case (3) we see that $\mathcal{F} = \widetilde{M}$ for some finite type $A_0$-module $M \subset A_{+}$. Say $M = \sum A_0f_ i$. Say $f_ i = \sum f_{i, j}$ is the decomposition into homogeneous pieces. The condition in (3) signifies that $A_{+} \subset \sqrt{(f_{i, j})}$. Thus in both cases we conclude that $\text{Proj}(A)$ is quasi-compact by Constructions, Lemma 27.8.9. Finally, (2) follows from (1). $\square$

Lemma 31.30.2. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{O}_ S$-algebras, then $p$ is of finite type and $\mathcal{O}_ X(d)$ is a finite type $\mathcal{O}_ X$-module.

Proof. The assumption implies that $p$ is quasi-compact, see Lemma 31.30.1. Hence it suffices to show that $p$ is locally of finite type. Thus the question is local on the base and target, see Morphisms, Lemma 29.15.2. Say $S = \mathop{\mathrm{Spec}}(R)$ and $\mathcal{A}$ corresponds to the graded $R$-algebra $A$. After further localizing on $S$ we may assume that $A$ is a finite type $R$-algebra. The scheme $X$ is constructed out of glueing the spectra of the rings $A_{(f)}$ for $f \in A_{+}$ homogeneous. Each of these is of finite type over $R$ by Algebra, Lemma 10.57.9 part (1). Thus $\text{Proj}(A)$ is of finite type over $R$. To see the statement on $\mathcal{O}_ X(d)$ use part (2) of Algebra, Lemma 10.57.9. $\square$

Lemma 31.30.3. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{O}_ S \to \mathcal{A}_0$ is an integral algebra map1 and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra, then $p$ is universally closed.

Proof. The question is local on the base. Thus we may assume that $X = \mathop{\mathrm{Spec}}(R)$ is affine. Let $\mathcal{A}$ be the quasi-coherent $\mathcal{O}_ X$-algebra associated to the graded $R$-algebra $A$. The assumption is that $R \to A_0$ is integral and $A$ is of finite type over $A_0$. Write $X \to \mathop{\mathrm{Spec}}(R)$ as the composition $X \to \mathop{\mathrm{Spec}}(A_0) \to \mathop{\mathrm{Spec}}(R)$. Since $R \to A_0$ is an integral ring map, we see that $\mathop{\mathrm{Spec}}(A_0) \to \mathop{\mathrm{Spec}}(R)$ is universally closed, see Morphisms, Lemma 29.44.7. The quasi-compact (see Constructions, Lemma 27.8.9) morphism

$X = \text{Proj}(A) \to \mathop{\mathrm{Spec}}(A_0)$

satisfies the existence part of the valuative criterion by Constructions, Lemma 27.8.11 and hence it is universally closed by Schemes, Proposition 26.20.6. Thus $X \to \mathop{\mathrm{Spec}}(R)$ is universally closed as a composition of universally closed morphisms. $\square$

Lemma 31.30.4. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. The following conditions are equivalent

1. $\mathcal{A}_0$ is a finite type $\mathcal{O}_ S$-module and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra,

2. $\mathcal{A}_0$ is a finite type $\mathcal{O}_ S$-module and $\mathcal{A}$ is of finite type as an $\mathcal{O}_ S$-algebra

If these conditions hold, then $p$ is locally projective and in particular proper.

Proof. Assume that $\mathcal{A}_0$ is a finite type $\mathcal{O}_ S$-module. Choose an affine open $U = \mathop{\mathrm{Spec}}(R) \subset X$ such that $\mathcal{A}$ corresponds to a graded $R$-algebra $A$ with $A_0$ a finite $R$-module. Condition (1) means that (after possibly localizing further on $S$) that $A$ is a finite type $A_0$-algebra and condition (2) means that (after possibly localizing further on $S$) that $A$ is a finite type $R$-algebra. Thus these conditions imply each other by Algebra, Lemma 10.6.2.

A locally projective morphism is proper, see Morphisms, Lemma 29.43.5. Thus we may now assume that $S = \mathop{\mathrm{Spec}}(R)$ and $X = \text{Proj}(A)$ and that $A_0$ is finite over $R$ and $A$ of finite type over $R$. We will show that $X = \text{Proj}(A) \to \mathop{\mathrm{Spec}}(R)$ is projective. We urge the reader to prove this for themselves, by directly constructing a closed immersion of $X$ into a projective space over $R$, instead of reading the argument we give below.

By Lemma 31.30.2 we see that $X$ is of finite type over $\mathop{\mathrm{Spec}}(R)$. Constructions, Lemma 27.10.6 tells us that $\mathcal{O}_ X(d)$ is ample on $X$ for some $d \geq 1$ (see Properties, Section 28.26). Hence $X \to \mathop{\mathrm{Spec}}(R)$ is quasi-projective (by Morphisms, Definition 29.40.1). By Morphisms, Lemma 29.43.12 we conclude that $X$ is isomorphic to an open subscheme of a scheme projective over $\mathop{\mathrm{Spec}}(R)$. Therefore, to finish the proof, it suffices to show that $X \to \mathop{\mathrm{Spec}}(R)$ is universally closed (use Morphisms, Lemma 29.41.7). This follows from Lemma 31.30.3. $\square$

Lemma 31.30.5. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{A}$ is generated by $\mathcal{A}_1$ over $\mathcal{A}_0$ and $\mathcal{A}_1$ is a finite type $\mathcal{O}_ S$-module, then $p$ is projective.

Proof. Namely, the morphism associated to the graded $\mathcal{O}_ S$-algebra map

$\text{Sym}_{\mathcal{O}_ X}^*(\mathcal{A}_1) \longrightarrow \mathcal{A}$

is a closed immersion $X \to \mathbf{P}(\mathcal{A}_1)$, see Constructions, Lemma 27.18.5. $\square$

Lemma 31.30.6. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{A}_ d$ is a flat $\mathcal{O}_ S$-module for $d \gg 0$, then $p$ is flat and $\mathcal{O}_ X(d)$ is flat over $S$.

Proof. Affine locally flatness of $X$ over $S$ reduces to the following statement: Let $R$ be a ring, let $A$ be a graded $R$-algebra with $A_ d$ flat over $R$ for $d \gg 0$, let $f \in A_ d$ for some $d > 0$, then $A_{(f)}$ is flat over $R$. Since $A_{(f)} = \mathop{\mathrm{colim}}\nolimits A_{nd}$ where the transition maps are given by multiplication by $f$, this follows from Algebra, Lemma 10.39.3. Argue similarly to get flatness of $\mathcal{O}_ X(d)$ over $S$. $\square$

Lemma 31.30.7. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{A}$ is a finitely presented $\mathcal{O}_ S$-algebra, then $p$ is of finite presentation and $\mathcal{O}_ X(d)$ is an $\mathcal{O}_ X$-module of finite presentation.

Proof. Affine locally this reduces to the following statement: Let $R$ be a ring and let $A$ be a finitely presented graded $R$-algebra. Then $\text{Proj}(A) \to \mathop{\mathrm{Spec}}(R)$ is of finite presentation and $\mathcal{O}_{\text{Proj}(A)}(d)$ is a $\mathcal{O}_{\text{Proj}(A)}$-module of finite presentation. The finite presentation condition implies we can choose a presentation

$A = R[X_1, \ldots , X_ n]/(F_1, \ldots , F_ m)$

where $R[X_1, \ldots , X_ n]$ is a polynomial ring graded by giving weights $d_ i$ to $X_ i$ and $F_1, \ldots , F_ m$ are homogeneous polynomials of degree $e_ j$. Let $R_0 \subset R$ be the subring generated by the coefficients of the polynomials $F_1, \ldots , F_ m$. Then we set $A_0 = R_0[X_1, \ldots , X_ n]/(F_1, \ldots , F_ m)$. By construction $A = A_0 \otimes _{R_0} R$. Thus by Constructions, Lemma 27.11.6 it suffices to prove the result for $X_0 = \text{Proj}(A_0)$ over $R_0$. By Lemma 31.30.2 we know $X_0$ is of finite type over $R_0$ and $\mathcal{O}_{X_0}(d)$ is a quasi-coherent $\mathcal{O}_{X_0}$-module of finite type. Since $R_0$ is Noetherian (as a finitely generated $\mathbf{Z}$-algebra) we see that $X_0$ is of finite presentation over $R_0$ (Morphisms, Lemma 29.21.9) and $\mathcal{O}_{X_0}(d)$ is of finite presentation by Cohomology of Schemes, Lemma 30.9.1. This finishes the proof. $\square$

[1] In other words, the integral closure of $\mathcal{O}_ S$ in $\mathcal{A}_0$, see Morphisms, Definition 29.53.2, equals $\mathcal{A}_0$.

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