ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

Opposite weights on the affine plane and its coordinate ring

Example

For $T = C^{*}$ acting on $X = A^{2}$ by $t (x, y) = (t x, t^{- 1} y)$ , the coordinate action gives degrees $\deg x = - 1$ , $\deg y = 1$ . Thus $x^{a} y^{b}$ has degree $b - a$ , and $A_{0} = C [x y]$ . The generating submodule $W = C x + C y$ gives the identity embedding into its dual plane, whose weights are $(1, - 1)$ . Assume AC for the cited A-page affine dictionary.

Facts & Assumptions

Given: The displayed action and AC (The Axiom of Choice).

[F1]

Function and point weights have opposite signs (Torus rational modules and affine actions are lattice gradings).

[F2]

A generating rational submodule gives the dual evaluation embedding (Every complex affine algebraic action has a finite-dimensional equivariant closed embedding).

Verification

1.1F1givenalgebra

Multiplication by $t$ and $t^{- 1}$ is regular on $T \times X$ and the group identities hold coordinatewise, so this is an action in Classical complex affine algebraic actions and rational modules. Inverse pullback sends $x$ to $t^{- 1} x$ and $y$ to $t y$ ; multiplication gives $t^{b - a} x^{a} y^{b}$ . Laurent-monomial independence gives the direct weight decomposition, and degree zero monomials are exactly $(x y)^{a}$ , so $A_{0} = C [x y]$ .

2.1F2step 1.1∎

The two coordinate functions span a stable rational submodule generating $A$ . Applying F2, evaluation $e v (x, y)$ has coordinates $(x, y)$ in the dual basis and dual weights $(1, - 1)$ . It is the identity isomorphism of affine planes, making the general embedding construction explicit.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Michel Brion, Introduction to actions of algebraic groups (2010) (standard reference, not scraped)
Philippe Gille, Introduction to reductive group schemes over rings, full notes retrieved 2026-10-02 (standard reference, not scraped)
J. S. Milne, Algebraic Groups (2022) (standard reference, not scraped)