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.

The actual exterior-dual class in affine perfect K zero

Statement

Assume AC. On a noetherian affine scheme $X = Spec A$ and for a rank- $r$ bundle $E$ , define $ε (E) = \sum_{i = 0}^{r} (- 1)^{i} [Λ^{i} E^{\lor}] \in K_{0} (X)$ , where $K_{0} (X)$ is the article's perfect spectral K-zero group and brackets are degree-zero bundle object classes. For $P = Γ (X, E)$ the ordinary projective-module expression $C_{r}^{K} (P) = \sum_{i} (- 1)^{i} [Λ^{i} P^{*}]$ , $P^{*} = {Hom}_{A} (P, A)$ , maps to exactly $ε (E)$ under the affine comparison. Moreover $λ_{- 1} (E) = (- 1)^{r} [\det E] ε (E)$ , so these two exterior conventions vanish simultaneously. For a regular section of $E$ , its Koszul complex uses $E^{\lor}$ and has ambient class $ε (E)$ .

Facts & Assumptions

Given: The hypotheses and data in the statement, with AC and the article's fixed universes wherever perfect K-theory is used.

Goal: Establish the asserted conclusion.

[F1]

The affine comparison identifies the module/bundle Grothendieck classes with perfect spectral object classes, and the affine equivalence preserves duals and tensor products. Isomorphism classes of vector bundles, Affine bundles and finite projective modules

[F2]

Exterior powers and determinants are functorial, finite locally free and have the increasing-wedge bases. Exterior powers and determinant of an algebraic bundle

[F3]

The exterior-duality identity is $λ_{- 1} (E) = (- 1)^{r} [\det E] λ_{- 1} (E^{\lor})$ with the determinant factor a unit. Dual conventions for the exterior Euler class

[F4]

A regular section of $E$ is a functional on $E^{\lor}$ ; its Koszul class has the dual exterior sum. Euler class of a regular bundle functional

Proof

1.1F1F2construct

The finite sum defining $ε (E)$ exists and is invariant under bundle isomorphisms by [F1] and [F2]. Sheafification identifies $P^{*}$ with $E^{\lor}$ by [F1]. The natural map from the sheaf associated to $Λ^{i} P^{*}$ to $Λ^{i} E^{\lor}$ is an isomorphism: on a principal cover trivializing $P^{*}$ both sides have the same increasing-wedge bases and the map carries each basis to itself. The affine comparison thus carries every summand of $C_{r}^{K} (P)$ to the displayed summand of $ε (E)$ .

2.1F1F3step 1.1

Tensor compatibility makes the affine comparison respect multiplication on these bundle classes; the product is tensor product on both presentations. Therefore [F3] carries its determinant-unit identity into spectral $K_{0} (X)$ and proves simultaneous vanishing.

3.1F4step 1.1∎

Apply the regular-section clause of [F4] to $E$ itself. Its complex has terms $Λ^{i} E^{\lor}$ , so its alternating ambient class is precisely $ε (E)$ . This calculation fixes the source convention but supplies no converse splitting theorem.

Depends on

Used by · 6

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.