LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 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.

projective coordinate morphisms well defined

Statement

A same-degree homogeneous tuple $F_{0}, \dots, F_{m}$ having no common zero on $X$ defines a projective morphism $X \to P_{k}^{m}$ .

Proof

Given: Such a tuple, of common degree $d$ .

1.1

Rescaling $a$ by $λ$ rescales every $F_{i} (a)$ by $λ^{d}$ , so the target class is representative-independent.

givenalgebra

2.1

On $D_{+} (y_{j})$ the coordinate functions are $F_{i} / F_{j}$ , regular equal-degree fractions wherever $F_{j} \neq 0$ .

step 1.1algebra

3.1

The target standard opens cover the image and the ratios agree on overlaps, so the local maps glue to a morphism.

step 2.1∎

Depends on

Used by

morphism projective line power map Example

Dependency tree · two levels

8 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)