ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Projective zero-space over an affine base

Example

Let $A$ be a commutative ring with $1$ (Commutative ring). Then $P_{A}^{0} ≅ Spec A$ (Relative projective space from standard charts, Projective space is Proj of a polynomial ring), and under this identification every twisting sheaf is trivial: $O_{P_{A}^{0}} (d) ≅ O_{Spec A} for every d \in Z$ (Twisting sheaf on Proj). Consequently $H^{0} (P_{A}^{0}, O (d)) ≅ A, H^{q} (P_{A}^{0}, O (d)) = 0 (q > 0),$ for every $d \in Z$ (Sheaf cohomology as right derived global sections). The zero ring $A = 0$ , the case $d = 0$ and negative $d$ are included. For a general base scheme $S$ the same definition gives $P_{S}^{0} ≅ S$ , with one chart and no gluing; only this identification of schemes is asserted, and no general vanishing of higher cohomology over a nonaffine base is claimed.

Facts & Assumptions

Given: A commutative ring $A$ with $1$ , an integer $d \in Z$ , the scheme $P_{A}^{0}$ with its twisting sheaf $O (d)$ , and the Axiom of Choice as inherited from the affine suppliers.

[A1]

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

[F1]

There is a canonical isomorphism $P_{A}^{0} ≅ Proj A [x_{0}]$ for the total-degree grading; the points of $Proj A [x_{0}]$ are the homogeneous primes not containing the irrelevant ideal $(x_{0})$ , so each of them omits $x_{0}$ and lies in the standard open $D_{+} (x_{0})$ ; and $D_{+} (x_{0}) = Spec ((A [x_{0}]_{x_{0}})_{0})$ is the single standard affine chart. (Projective space is Proj of a polynomial ring, Standard opens of Proj, Points of Proj of a graded ring, Standard opens are affine)

[F2]

The chart ring of the single chart is $(A [x_{0}]_{x_{0}})_{0} ≅ A$ via $a \mapsto a / 1$ ; more generally the degree-zero part of the localisation of the shifted module $A [x_{0}] (d)$ is $A [x_{0}] (d)_{(x_{0})} = {c x_{0}^{d} : c \in A} ≅ A,$ a free $A$ -module of rank one with generator $x_{0}^{d}$ , for every $d \in Z$ (including negative $d$ , where $x_{0}^{d}$ denotes the unit $x_{0}^{- ∣ d ∣}$ of the localisation). [algebra]

[F3]

The twisting sheaf is $O (d) = \tilde{A [x_{0}] (d)}$ (Twisting sheaf on Proj), its sections on the chart $D_{+} (x_{0})$ are the degree-zero localisation $A [x_{0}] (d)_{(x_{0})}$ , and the restriction of $O (d)$ to that chart is the associated sheaf of this $A$ -module; an isomorphism of $A$ -modules induces an isomorphism of the associated sheaves on $Spec A$ (Module sheaf on an affine scheme, Sections of the associated sheaf on basic opens).

[F4]

For an affine scheme $X = Spec R$ and a quasi-coherent $O_{X}$ -module $F$ one has $H^{q} (X, F) = 0$ for every $q > 0$ , and $H^{0} (Spec R, O) ≅ R$ via the canonical map; the structure sheaf of a scheme is quasi-coherent, being over an affine open the associated sheaf of its coordinate ring. (Affine acyclicity of quasi-coherent sheaves, Global functions on Spec A recover A, Quasi-coherent module on a scheme)

[F5]

Cohomology of twists on projective space includes the case $n = 0$ : $P_{R}^{0} = Spec R$ and $H^{0} (P_{R}^{0}, O (m)) ≅ R$ for every $m \in Z$ , with all higher groups zero, for every commutative ring $R$ . (Cohomology of O(d) on projective space)

[F6]

For an arbitrary base scheme $S$ the relative projective space is $P_{S}^{n} = S \times_{Spec Z} P_{Z}^{n}$ ; for $n = 0$ there is one chart $U_{0} = Spec Z$ , no gluing takes place, and $P_{Z}^{0} = Spec Z$ , so that $P_{S}^{0} ≅ S$ ; for $S = \emptyset$ one has $P_{\emptyset}^{n} = \emptyset$ . (Relative projective space from standard charts)

Verification

technique · direct: identify the single standard chart with $\operatorname{Spec}A$, compute the degree-zero localisations defining the twists, transport the resulting trivialisations into the affine vanishing theorem, and record the relative case and the degenerate boundaries

1.1F1

The single chart covers the space. By [F1] the points of $Proj A [x_{0}]$ omit $x_{0}$ , so every point lies in $D_{+} (x_{0})$ ; hence the single standard chart is the whole space, and $P_{A}^{0} = D_{+} (x_{0}) = Spec ((A [x_{0}]_{x_{0}})_{0})$ .

2.1F2step 1.1algebra

The chart ring. The map $A \to (A [x_{0}]_{x_{0}})_{0}$ , $a \mapsto a / 1$ , is an isomorphism: a degree-zero fraction has the form $a x_{0}^{m} / x_{0}^{m} = a / 1$ , and $a / 1 = b / 1$ forces $a = b$ by comparing coefficients after clearing the powers of $x_{0}$ ; for $A = 0$ both rings are zero. Hence $P_{A}^{0} ≅ Spec A$ for every commutative ring $A$ .

2.2F2F3step 1.1

The twisting sheaves are trivial. By [F3] and [F2], for every $d \in Z$ the sections of $O (d)$ on the chart are $A [x_{0}] (d)_{(x_{0})} = A \cdot x_{0}^{d}$ , a free rank-one $A$ -module, and the associated sheaf of this module on $Spec A$ is isomorphic to $\tilde{A} = O$ through the module isomorphism $c \mapsto c x_{0}^{d}$ ; since the chart is the whole space by [step 1.1], this is a global isomorphism $O_{P_{A}^{0}} (d) ≅ O_{Spec A}$ , valid for every $d \in Z$ including $d = 0$ and negative $d$ .

3.1F4F5step 2.2

The cohomology. The isomorphism of [step 2.2] identifies $H^{q} (P_{A}^{0}, O (d))$ with $H^{q} (Spec A, O)$ ; by [F4] the target is $A$ in degree zero, through the canonical isomorphism $A \to Γ (Spec A, O)$ , and vanishes for $q > 0$ because the structure sheaf is quasi-coherent on the affine scheme $Spec A$ . This gives $H^{0} (P_{A}^{0}, O (d)) ≅ A$ and $H^{q} (P_{A}^{0}, O (d)) = 0$ for $q > 0$ ; the $n = 0$ clause of [F5] states the same conclusion directly, independent of the trivialisation.

4.1A1F3F4F5F6step 2.1step 2.2step 3.1cases: zero ring and d=0 and general base∎

General base, boundaries and choice accounting. For an arbitrary base scheme $S$ , [F6] gives $P_{S}^{0} ≅ S$ with one chart and no gluing, and $P_{\emptyset}^{0} = \emptyset$ ; only this identification is asserted, because for a nonaffine base the same reasoning would reduce the question to $H^{q} (S, O_{S})$ , which is not claimed to vanish. The cases $A = 0$ , where $Spec A = \emptyset$ and the degree-zero group is the zero ring $A = 0$ in agreement with [step 2.1] and [step 3.1], and $d = 0$ , where $O (0) = O$ by [F3] and [step 2.2] reduces to the identity, are both included. The Axiom of Choice [A1] is inherited through the affine vanishing and global-sections suppliers of [F4] and the projective cohomology of [F5]; no chart, resolution or trivialisation is chosen here.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

59 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, Cohomology of Schemes, Chapter 30, Section 30.8 (Tag 01XV) (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), Section 19.1 (standard reference, not scraped)