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.

Local application interface for a critical-rank cancellation theorem

Statement

Assume AC and the suitable universes used for the article's perfect K-theory construction. Let $X = Spec A$ be noetherian affine and $r \geq 0$ . Suppose every constant-rank- $r$ finite projective $A$ -module is cancellative in the free-stabilization sense. Then two rank- $r$ bundles $E, F$ on $X$ are isomorphic exactly when $[E] = [F]$ in the article's spectral $K_{0} (X)$ .

Facts & Assumptions

Given: The data and hypotheses in the statement.

Goal: Establish the stated conclusion.

[F1]

Rank and isomorphism classes pass through the affine equivalence. Affine bundle classes are actual projective-module classes

[F2]

Under AC and the stated universes, spectral K-zero equality for vector bundles on a noetherian affine scheme is equivalent to a common finite free module stabilization. Spectral K-zero equality gives an actual projective stabilization

[F3]

The meaning of cancellative is fixed by Common projective complements and cancellation by free modules.

Proof

1.1F1F2F3given

If the K-zero classes agree, let $P, Q$ be the modules of sections. They have constant rank $r$ , and [F2] gives $P \oplus A^{N} ≅ Q \oplus A^{N}$ . By the assumed cancellativity, $P ≅ Q$ . By [F1], $E ≅ F$ .

2.1F2step 1.1∎

If $E ≅ F$ , the spectral object-class map assigns the same element to both; equivalently [F2] applies with $N = 0$ . Hence the converse holds too.

Depends on

Used by · 1

Classical smooth critical-rank classifications

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.