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

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Standard opens are affine

Statement

Assume the Axiom of Choice as inherited from the affine scheme construction (The Axiom of Choice). Let $S = ⨁_{e \geq 0} S_{e}$ be a commutative nonnegatively graded ring, let $X = Proj S$ be the scheme of Proj carries a scheme structure, and let $f \in S_{+}$ be homogeneous of positive degree with $D_{+} (f) = {p \in Proj S : f \notin p}$ (Standard opens of Proj). Then the canonical chart map

$φ_{f} : D_{+} (f) ⟶ Spec S_{(f)}, S_{(f)} = (S [f^{- 1}])_{0},$

is an isomorphism of schemes (The underlying space of an affine spectrum). This includes the case of an empty chart: if $f$ is nilpotent then $D_{+} (f) = \emptyset$ and $S_{(f)} = 0$ , and $Spec 0 = \emptyset$ .

Facts & Assumptions

Given: The Axiom of Choice; a commutative nonnegatively graded ring $S$ ; a homogeneous element $f \in S_{+}$ of positive degree.

[A1]

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

[F1]

There is a scheme $Proj S$ with open subscheme identifications $φ_{f} : D_{+} (f) \to Spec S_{(f)}$ for every homogeneous $f \in S_{+}$ of positive degree, whose charts $D_{+} (f)$ form an affine open cover, such that $φ_{f}$ carries the structure sheaf of $Proj S$ to that of $Spec S_{(f)}$ and such that on overlaps $D_{+} (f g)$ the identifications agree; if $f$ is nilpotent then $D_{+} (f) = \emptyset$ and $S_{(f)} = 0$ , and if $S = 0$ then $Proj S = \emptyset$ . (Proj carries a scheme structure)

[F2]

$D_{+} (f) = {p \in Proj S : f \notin p}$ is an open subset; if $f$ is nilpotent then $f$ lies in every prime ideal, so $D_{+} (f) = \emptyset$ . (Standard opens of Proj)

[F3]

For the zero ring there are no prime ideals, so $Spec 0 = \emptyset$ . (The underlying space of an affine spectrum)

Proof

technique · direct: read the chart map off the gluing theorem and dispose of the empty chart

1.1F1given

By [F1] the scheme $X = Proj S$ is equipped with the open subscheme identifications $φ_{f} : D_{+} (f) \to Spec S_{(f)}$ , one for each homogeneous $f$ of positive degree, agreeing on the overlaps $D_{+} (f g)$ ; these are the canonical chart maps of the construction, being the maps used in the gluing data.

2.1F1step 1.1

The map $φ_{f}$ is an isomorphism of schemes onto the affine scheme $Spec S_{(f)}$ , with inverse the corresponding chart identification; this is precisely the first clause of [F1], and it identifies the structure sheaves, so $D_{+} (f)$ is an affine open subscheme with coordinate ring $S_{(f)}$ .

2.2F1F2F3step 1.1

Empty charts. If $f$ is nilpotent then $D_{+} (f) = \emptyset$ by [F2] and $S_{(f)} = 0$ by [F1], while $Spec 0 = \emptyset$ by [F3]; the canonical chart map is therefore the unique map $\emptyset \to \emptyset$ , which is an isomorphism. If $S = 0$ then $Proj S = \emptyset$ by [F1], every homogeneous $f$ is nilpotent, and the same conclusion holds.

3.1

The Axiom of Choice [A1] is inherited only through the construction of $Proj S$ in [F1] (prime existence and affine gluing); no choice is made in the present argument. Steps 2.1 and 2.2 prove the claim for every homogeneous $f$ of positive degree, including the empty chart. [A1, step 2.1, step 2.2] \qed

Depends on

Used by

An upper jump of h0 in a flat projective family Example
Generator cocycle for H1 of O(-2) Example
Polynomial Proj charts Example
Projective zero-space over an affine base Example
Chow lemma for proper Noetherian schemes Lemma
Finite twisted locally free resolutions on projective space Lemma
Hypersurface cohomology sequence Lemma
Laurent-monomial decomposition of the projective Cech complex Lemma
Regular hyperplane step for coherent support induction Lemma
Sections of a graded-module sheaf on a standard open Lemma
Cohomology of O(d) on projective space Theorem
Projective space is Proj of a polynomial ring Theorem

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Constructions of Schemes, Section 27.8 (Tag 01M3) (standard reference, not scraped)
Ravi Vakil, The Rising Sea, August 2022 draft, Section 4.5 (standard reference, not scraped)