LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

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A projective morphism has a relative Proj presentation

Statement

Assume the Axiom of Choice as inherited from the relative-Proj and closed subscheme constructions (The Axiom of Choice). Let $f : X \to S$ be an H-projective morphism (Projective morphisms before Proj). Then:

There is an integer $n \geq 0$ and the graded quasi-coherent $O_{S}$ -algebra $A = {Sym}_{O_{S}} (O_{S}^{n + 1})$ (Symmetric algebra of a quasi-coherent module) with ${Proj}_{S} A = P_{S}^{n}$ (Relative Proj of a graded quasi-coherent algebra), such that $X$ admits a closed $S$ -immersion $X ↪ {Proj}_{S} A;$ equivalently, H-projectivity is the case of a closed subscheme of a relative Proj of a free graded algebra on $n + 1$ generators in degree one.
If $S = Spec R$ is affine, then every such closed immersion has image $Proj (R [x_{0}, \dots, x_{n}] / J)$ for a unique $b$ -saturated homogeneous ideal $J \subseteq R [x_{0}, \dots, x_{n}]$ , $b = (x_{0}, \dots, x_{n})$ (Closed subschemes of projective space and saturated ideals).
For arbitrary $S$ , every such closed immersion with image $Z$ has a global quotient presentation $Z ≅ {Proj}_{S} (A / I),$ where $I \subseteq A = O_{S} [x_{0}, \dots, x_{n}]$ is a quasi-coherent homogeneous ideal sheaf. It can be chosen canonically so that on every affine open of $S$ its homogeneous ideal is the saturated ideal in (2). No extra hypothesis on $S$ or global generation of the ideal sheaf is required; the quotient here is a sheaf of graded algebras on $S$ .

Facts & Assumptions

Given: An H-projective morphism $f : X \to S$ , an integer $n \geq 0$ with a closed $S$ -immersion $X \to P_{S}^{n}$ , and the Axiom of Choice as inherited.

[A1]

The Axiom of Choice states that every family of nonempty sets has a choice function. (The Axiom of Choice)

[F1]

$f$ is H-projective if for some $n \geq 0$ it factors as a closed immersion $X \to P_{S}^{n}$ followed by the structure morphism $P_{S}^{n} \to S$ . (Projective morphisms before Proj)

[F2]

${Sym}_{O_{S}} (O_{S}^{n + 1})$ is a quasi-coherent graded $O_{S}$ -algebra which on an affine open $U = Spec R \subseteq S$ restricts to the polynomial algebra $R [x_{0}, \dots, x_{n}]$ with its total-degree grading, the generators corresponding to the standard basis of $R^{n + 1}$ . (Symmetric algebra of a quasi-coherent module)

[F3]

${Proj}_{S} A$ is constructed by gluing the absolute Proj schemes $Proj Γ (U, A)$ , and for the polynomial algebra over an affine base this is $P_{R}^{n}$ : $Proj R [x_{0}, \dots, x_{n}] ≅ P_{R}^{n}$ , compatibly with restriction to affine opens. (Relative Proj of a graded quasi-coherent algebra, Projective space is Proj of a polynomial ring)

[F4]

For $S = Spec R$ the closed subschemes of $P_{R}^{n}$ are exactly the $V_{+} (J)$ for homogeneous ideals $J \subseteq R [x_{0}, \dots, x_{n}]$ , and $V_{+} (J) = V_{+} (J^{'})$ if and only if $J^{s a t} = J^{' s a t}$ ; every closed subscheme arises from a unique saturated homogeneous ideal. (Closed subschemes of projective space and saturated ideals)

[F5]

For a homogeneous $h$ of degree $d$ , membership in a saturated ideal $J \subset R [x_{0}, \dots, x_{n}]$ is equivalent to $h / x_{i}^{d} \in J_{(x_{i})}$ for all $i = 0, \dots, n$ . (Saturation detected on projective charts)

[F6]

The chart frames of $O (1)$ satisfy $e_{i} = (x_{i} / x_{j}) e_{j}$ , and the coordinate sections generate it; homogeneous polynomials of degree $d \geq 0$ therefore give sections of $O (d)$ , with coefficient $h / x_{i}^{d}$ on the $i$ th chart. (Relative very ampleness in the finite projective-space convention)

[F7]

On $Spec R$ an associated module sheaf has sections $M_{r}$ on every distinguished open $D (r)$ . Thus a sheaf with these compatible sections on the distinguished-open basis is the associated sheaf of $M$ , and is quasi-coherent. (Sections of the associated sheaf on basic opens, Quasi-coherent module on a scheme)

Proof

technique · direct: identify the relative projective space with the Proj of the symmetric algebra chart by chart, then invoke the defining closed immersion and, over an affine base, the saturated-ideal correspondence

1.1F2algebra

Charts of the symmetric algebra. Let $A = {Sym}_{O_{S}} (O_{S}^{n + 1})$ and let $U = Spec R \subseteq S$ be affine. By [F2] the restriction $A ∣_{U}$ is the polynomial algebra $Γ (U, A) = R [x_{0}, \dots, x_{n}]$ with the standard generators $x_{i}$ in degree one, so $Proj Γ (U, A) = Proj R [x_{0}, \dots, x_{n}]$ .

1.2F4F5F6construct

Define the global ideal intrinsically. Put $Z = i (X)$ for the fixed closed immersion. For each $d \geq 0$ and open $T \subseteq S$ , let $I_{d} (T)$ consist of sections of $A_{d} (T)$ whose polynomial section of $O (d)$ vanishes on $Z \times_{S} T$ , using [F6]. This is a sheaf: vanishing can be checked on an open cover, and the polynomial-section maps commute with restriction. The direct sum $I = ⨁_{d \geq 0} I_{d}$ is a homogeneous ideal subsheaf of $A$ , since multiplying a vanishing polynomial section by any polynomial section still vanishes. Over an affine $U = Spec R$ , write $B = R [x_{0}, \dots, x_{n}]$ and let $K_{i} \subset B_{(x_{i})}$ be the chart ideals of $Z_{U}$ . Then $I_{d} (U) = {h \in B_{d} : h / x_{i}^{d} \in K_{i} for every i}$ , which is $(J_{U})_{d}$ for the saturated ideal $J_{U}$ supplied by [F4], by [F5].

2.1F3step 1.1

Identification of the relative Proj. Step 1.1 identifies the affine-local pieces of ${Proj}_{S} A$ with the corresponding absolute Proj schemes, and the gluing isomorphisms of [F3] restrict the coefficients and preserve the polynomial variables; by [F3] this gives a canonical isomorphism ${Proj}_{S} {Sym}_{O_{S}} (O_{S}^{n + 1}) ≅ P_{S}^{n}$ over $S$ , agreeing with the standard charts on each affine piece.

2.2F5F7step 1.2algebra

Quasi-coherence on the base. Fix $r \in R$ . On the base open $D (r)$ the chart ideals are $(K_{i})_{r}$ : restricting the affine quotient $B_{(x_{i})} / K_{i}$ localises its ring at $r$ . A homogeneous polynomial over $R_{r}$ can be written $h / r^{k}$ with $h \in B_{d}$ . It belongs to $I_{d} (D (r))$ exactly when $h / x_{i}^{d}$ belongs to $(K_{i})_{r}$ for every $i$ . For each of the finitely many charts this is equivalent to $r^{a_{i}} h / x_{i}^{d} \in K_{i}$ for some $a_{i} \geq 0$ . Taking $a = \max_{i} a_{i}$ gives $r^{a} h \in (J_{U})_{d}$ by step 1.2. Thus $I_{d} (D (r)) = ((J_{U})_{d})_{r}$ , with the usual restriction maps. This argument also works if the base or a chart is empty. By [F7], $I_{d} ∣_{U} = \tilde{(J_{U})_{d}}$ . Hence $I$ is a quasi-coherent graded ideal sheaf, and $A / I$ is quasi-coherent since on each affine $U$ its degree pieces are the associated modules of $(B / J_{U})_{d}$ ; the quotient identification follows on the distinguished-open basis by localisation of module quotients.

3.1F1step 2.1

The closed immersion. By [F1] there are $n \geq 0$ and a closed $S$ -immersion $X \to P_{S}^{n}$ ; composing with the isomorphism of step 2.1 exhibits $X$ as a closed subscheme of the relative Proj of the symmetric algebra on $n + 1$ degree-one generators. Conversely, a closed $S$ -immersion into that relative Proj becomes a closed $S$ -immersion into $P_{S}^{n}$ under step 2.1, so [F1] makes its source H-projective. This proves the equivalence in claim (1).

3.2F4step 2.1

The affine base case. If $S = Spec R$ , then by step 2.1 the closed immersion is a closed subscheme $Z ↪ Proj R [x_{0}, \dots, x_{n}] = P_{R}^{n}$ , and by [F4] there is a unique saturated homogeneous ideal $J \subseteq R [x_{0}, \dots, x_{n}]$ with $Z = Proj (R [x_{0}, \dots, x_{n}] / J)$ ; equivalently $Z = V_{+} (J)$ . This is claim (2).

3.3F3F4step 2.2

Recover the closed subscheme. On every affine $U$ of the base, [F4] identifies $Z_{U}$ with $Proj (B / J_{U})$ as a closed subscheme of $P_{U}^{n}$ . Step 2.2 identifies the quotient algebra sheaf on $U$ with the algebra associated to $B / J_{U}$ . The relative Proj gluing [F3] therefore gives ${Proj}_{S} (A / I) = Z$ : the local identifications agree on overlaps because on each standard chart they are the same quotient map defining the given closed subscheme. This proves (3).

4.1

Conclusion. Steps 1.1 and 2.1 identify ${Proj}_{S} Sym (O_{S}^{n + 1})$ with $P_{S}^{n}$ , step 3.1 records the defining closed immersion of the H-projective morphism, and step 3.2 gives the saturated homogeneous ideal description over an affine base, while steps 1.2, 2.2 and 3.3 construct the global quasi-coherent homogeneous ideal and its quotient presentation over an arbitrary base. The Axiom of Choice [A1] is inherited from the relative-Proj and closed-subscheme constructions; no choice is made here. [A1, step 2.1, step 3.1, step 3.2, step 1.2, step 2.2, step 3.3] \qed

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Sources

The Stacks Project, Constructions of Schemes, Sections 27.8-27.21 (standard reference, not scraped)
The Stacks Project, Lemma 31.32.1 (Tag 0801) (standard reference, not scraped)
The Stacks Project, Morphisms of Schemes, Sections 29.38, 29.40 (standard reference, not scraped)