LemmaStatement: AI-generatedProof: AI-generated

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.

Free stabilization passes through global sections

Statement

Assume AC and put $X = Spec A$ . For bundles $E, F$ and an integer $N \geq 0$ , $E \oplus O_{X}^{N} ≅ F \oplus O_{X}^{N}$ if and only if $Γ (X, E) \oplus A^{N} ≅ Γ (X, F) \oplus A^{N}$ . All modules here are finite projective.

Facts & Assumptions

Given: The data and hypotheses in the statement.

Goal: Establish the stated conclusion.

[F1]

The affine equivalence preserves finite direct sums and sends $A^{N}$ to $O_{X}^{N}$ . Affine bundles and finite projective modules

[F2]

It reflects and preserves isomorphism classes. Affine bundle classes are actual projective-module classes

Proof

1.1F1F2

Apply global sections to a stabilized bundle isomorphism. Its inverse remains an inverse; direct-sum compatibility identifies its source and target with the displayed modules.

2.1F1F2step 1.1∎

Conversely sheafify a module isomorphism and use the counit to identify its ends with the displayed bundles. The equivalence also supplies finite projectivity.

Depends on

Used by · 1

Spectral K-zero equality gives an actual projective stabilization

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.