LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-06

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.

standard projective opens are affine spaces

Statement

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 $A_{k}^{n}$ .

Proof

Given: $[a] \in D_{+} (x_{i})$ .

1.1

Multiplication by $a_{i}^{- 1}$ gives the unique representative with $i$ th coordinate $1$ .

givenalgebra

2.1

Retaining the remaining coordinates and reinserting $1$ give mutually inverse maps.

step 1.1construct

3.1

On chart overlaps these maps are coordinate ratios with nonzero denominator, so they are affine-chart isomorphisms.

step 2.1algebra∎

Depends on

Used by

projective conic standard charts Example
projective line two affine charts Example
projective closure dense affine chart Lemma
projective hypersurface affine pieces Lemma
projective regular function chart compatibility Lemma
global regular functions projective variety Theorem

Dependency tree · two levels

3 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

J. S. Milne, Algebraic Geometry, Chapter 6 (standard reference, not scraped)
Michael Artin, Algebraic Geometry, Chapter 3 (standard reference, not scraped)