CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Affine and projective n-space have dimension n

Statement

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ .

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

If $X$ is an irreducible classical variety, then $\dim X = {trdeg}_{k} k (X) < \infty$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dimension equals transcendence degree).

[F2]

For every open cover $T = ⋃_{i \in I} U_{i}$ of a Noetherian space, $\dim T = \sup_{i} \dim U_{i}$ , with empty supremum $- \infty$ . (Dimension can be computed on an open cover).

[F3]

For every $i$ , normalization of the $i$ th coordinate identifies $D_{+} (x_{i})$ with $A_{k}^{n}$ . In particular $[a_{0} : \dots : a_{n}] \mapsto (a_{1} / a_{0}, \dots, a_{n} / a_{0})$ identifies $D_{+} (x_{0})$ with $A_{k}^{n}$ . (standard projective opens are affine spaces).

Proof

1.1

The polynomial coordinates are algebraically independent and generate the fraction field $k (x_{1}, \dots, x_{n})$ of affine space. Thus its transcendence degree, and hence its geometric dimension, is $n$ . When $n = 0$ the field is $k$ and affine space is one point.

2.1

The $n + 1$ standard projective opens are affine $n$ -spaces. Their open cover computes projective dimension as the supremum of their dimensions, namely $n$ . For $n = 0$ this is the single chart of the one-point projective space.

F2F3step 1.1∎

Depends on

Used by

Dependency tree · two levels

9 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

Arapura Example 4.1.1 (standard reference, not scraped)