TheoremStatement: Literature-sourcedProof: 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.

Critical-rank cancellation used in the article

PDF p. 2: Sus77 reduction; PDF p. 12: Theorem A opening; PDF p. 17: Theorem B injectivity

Statement

Assume AC and the universes of the article's K-theory construction. Let $k$ be an algebraically closed field of arbitrary characteristic and $X = Spec A$ a nonempty affine finite-type $k$ -scheme of dimension $d$ . For any two vector bundles $E, F$ of the same constant rank $r \geq d$ , $E ≅ F$ if and only if $[E] = [F]$ in the article's perfect spectral $K_{0} (X)$ . In particular this applies to rank $d$ on every normal affine surface or threefold in the article. Here $[E]$ is the perfect-K-theory object class, and the affine comparison identifies it with the vector-bundle Grothendieck class; no coherent-sheaf G-zero identification is used.

Facts & Assumptions

Given: The data and hypotheses in the statement.

Goal: Establish the stated conclusion.

[F1]

The precise module cancellation input applies in every characteristic to this affine algebra and ranks at least d. Suslin cancellation over affine algebras

[F2]

The local application lemma proves the K-zero/isomorphism equivalence whenever the corresponding constant-rank projective modules are cancellative. Local application interface for a critical-rank cancellation theorem

[F3]

The affine perfect/bundle comparison and stable-equivalence supplier fix the actual class map. Isomorphism classes of vector bundles, Equality in affine K zero and stable isomorphism

Proof

1.1F1F2F3given

A finite-type algebra over a field is noetherian, so the local affine K-zero application applies to $X$ . The imported cancellation theorem makes every constant-rank- $r$ projective module cancellative since $r \geq d$ . Thus [F2] gives the desired equivalence for $E, F$ , using the class convention in [F3].

2.1F1F2step 1.1∎

A normal affine surface or threefold in the article is an affine finite-type scheme over the same algebraically closed field, and its critical rank is respectively $d = 2$ or $d = 3$ . Substitution $r = d$ yields exactly the original assertion. Isomorphic bundles always have equal object classes, so both directions have been retained.

Source notes

The original article invokes Sus77 on pp. 2, 12 and 17 (https://arxiv.org/abs/2610.22679v1). Its rank-d normal settings are preserved as special cases of the broader precise sourced theorem. The proof is a local application of an imported theorem; no proof acceptance is asserted.

Depends on

Used by · 2

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.

External dependencies

Manoj Kumar Keshari, Cancellation problem for projective modules over affine algebras · p. 1, Introduction: definition of cancellative and Theorem 1.1(i); p. 16, reference [20]

Over a finite-type affine algebra A of dimension d over an algebraically closed field, every finite projective A-module of constant rank at least d is cancellative: a common finite free stabilization can be removed.

Hypotheses: A commutative unital affine algebra over an algebraically closed field of arbitrary characteristic, finite Krull dimension d; finitely generated projective modules of constant rank r >= d. No smoothness, normality, reducedness or invertibility of d! is assumed in Theorem 1.1(i).

Reading limits: Read the p. 1 definition and Theorem 1.1(i), and p. 16 bibliography [20], in the downloaded arXiv v1 PDF. Also read indexed text of the author-hosted https://www.math.iitb.ac.in/~keshari/laurent.pdf introduction, whose Suslin reference is numbered [21]. Direct downloads of that host timed out. Suslin 1977 original proof was not read or reconstructed; the exact theorem is imported from this authoritative subsequent statement.

Sources

Fasel–Yakerson, Vector bundles on singular surfaces and threefolds · pp. 2, 12, 17; Sus77
Manoj Kumar Keshari, Cancellation problem for projective modules over affine algebras · p. 1, Introduction: definition of cancellative and Theorem 1.1(i); p. 16, reference [20]