LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-08 (gpt-6-astra)

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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.

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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 $A_{k}^{n}$ .

Here $A_{k}^{n} = k^{n}$ , with closed subsets the common zero sets of families of polynomials. On any open subset its regular functions are functions locally equal to a polynomial quotient with nonvanishing denominator. Transporting these functions through the displayed charts gives compatible regular-function structures on projective space. The identifications are homeomorphisms and isomorphisms for these structures.

Proof

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

1.1givenalgebra

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

2.1step 1.1construct

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

3.1step 2.1algebra

A homogeneous polynomial restricted to the representative with $i$ th coordinate 1 becomes an ordinary polynomial in the other coordinates. Conversely, for any polynomial $h$ in those coordinates, multiply $h (x_{j} / x_{i})_{j \neq i}$ by $x_{i}^{\deg h}$ to obtain a homogeneous polynomial with the same zero set on the chart. Thus relative projective closed sets correspond exactly to affine polynomial zero sets, proving that the bijection is a homeomorphism. The affine closed-set axioms also follow directly: intersections combine equations and finite unions use pairwise products of equations.

4.1step 2.1step 3.1algebra∎

On the overlap with chart $j$ , write $u_{l} = a_{l} / a_{i}$ and $u_{i} = 1$ . The new coordinates are $v_{l} = u_{l} / u_{j}$ , with $u_{j} \neq 0$ ; the inverse has the same form with $i, j$ exchanged. Substituting these ratios into a local polynomial quotient and clearing powers of $u_{j}$ again gives a polynomial quotient with nonvanishing denominator. Hence transition maps preserve regular functions in both directions. The local quotient condition is preserved under restriction and gluing of functions, so the transported structures agree on overlaps and define the claimed chart isomorphisms, including the one-point case $n = 0$ .

Depends on

Used by

A line meets a degree-d curve in d points counted with multiplicity Corollary
Affine and projective n-space have dimension n Corollary
General hypersurfaces give smooth complete intersections Corollary
morphism to projective space homogeneous coordinates Definition
Multiplicity of a plane curve at a point Definition
Plane projective curves and their components Definition
A nondegenerate projective quadric Example
A projective cone with a smooth conic base Example
projective conic standard charts Example
projective line two affine charts Example
Smooth plane quartic has genus three Example
Intersection with a line is the order of vanishing of the restricted equation Lemma
Invariance of the local intersection multiplicity Lemma
projective closure dense affine chart Lemma
projective hypersurface affine pieces Lemma
projective regular function chart compatibility Lemma
Projective space over a classical base and homogeneous closed loci Lemma
The resultant detects finitely many common projective points Lemma
Bertini smoothness away from the base locus 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)