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 source Chern classes are the supplied projective-relation classes

PDF p. 8: Actual Chern and Whitney identities; PDF p. 11: Total virtual Chern class

Statement

Assume the Axiom of Choice and the fixed site-size and universe bounds of the supplied motivic constructions. Let $k$ be a field, $S$ a qcqs $k$ -scheme, $Y$ a smooth qcqs $S$ -scheme, and $V$ a finite locally free sheaf of constant rank $r \geq 0$ on $Y$ . Use the supplied non- $A^{1}$ EM model $E_{S}$ with its actual additive orientation and natural diagonal identifications $E_{S}^{2 i, i} (Y) ≅ H_{m o t}^{2 i, i} (Y, Z)$ for $i \geq 0$ .

For $r > 0$ the coefficients of Annala–Iwasa Definition 4.4.1 agree under these identifications with the supplied projective-relation coefficients $c_{i} (V)$ . Extend the source-side polynomial by zero coefficients above rank and, for the zero bundle, by ${\tilde{c}}_{t} (0) = 1$ . With these explicit local conventions the agreement holds for every $i \geq 0$ and every constant rank, including zero. Definition 4.4.1 itself starts at positive rank. For a line the first coefficient is the original additive EM orientation and all higher coefficients are zero.

Facts & Assumptions

[A1]

Assume the Axiom of Choice as in The Axiom of Choice and the stated site-size and universe bounds. This supplies the foundational hypotheses of the model and Chern-coefficient constructions invoked in steps 1.1 and 2.1; no additional bundle or affine-torsor choices are made here.

[F1]

External non-A¹ projective-relation filtered and Whitney theorem specifies the source quotient-projective relation. An actual non-A¹ P¹-spectrum for the supplied EM complexes and Actual EM orientation and projective bundle maps on that model identify its groups, products and actual first class.

[F2]

Motivic Chern classes and Chern coefficients of the projective relation give the unique coefficients of the supplied quotient-projective relation, including rank-zero conventions.

Proof

Given: $k, S, Y, V$ , the Axiom of Choice and the stated bounds, and the explicit source-side extensions in the statement.

Goal: Identify the actual coefficients and signs.

1.1A1F1F2given

For $r > 0$ the source relation is $\sum_{j = 0}^{r} (- 1)^{j} ξ^{j} π^{*} {\tilde{c}}_{r - j} (V) = 0$ , with ${\tilde{c}}_{0} = 1$ . Multiply by $(- 1)^{r}$ and set $i = r - j$ . It becomes $ξ^{r} + \sum_{i = 1}^{r} (- 1)^{i} π^{*} {\tilde{c}}_{i} (V) ξ^{r - i} = 0$ . The orientation and quotient $O (1)$ are the same by [F1]. Hence coefficient uniqueness in [F2] gives ${\tilde{c}}_{i} (V) = c_{i} (V)$ for $1 \leq i \leq r$ ; both zeroth coefficients are the unit.

2.1A1F1F2step 1.1∎

For rank1, the quotient projective bundle is $Y$ , its tautological quotient is $V$ , and the relation is $ξ - c_{1} (V) = 0$ . It therefore identifies the first coefficient with the original orientation, which is additive under tensor product. Above positive rank the declared zero-coefficient extension agrees with the supplied convention. In rank zero the source-side series is $1$ by our explicit local extension of Definition 4.4.1, and the supplied series is $1$ by [F2]; no relation on the empty projective bundle is used. These statements identify the whole actual series, with no virtual extension used.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Annala-Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · Definition 4.4.1 and Lemma 4.4.3, printed pages 30-31; section 2.1, page 14