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Apply the actual perfect Cartan identity to the second image

PDF p. 11: Lemma5.3, second Chern additivity

Statement

Assume the Axiom of Choice and fix the site and universe bounds of the supplied perfect K-theory, motivic filtration and Chern-operation constructions. Let $k$ be algebraically closed and let $X$ be a normal quasi-compact quasi-separated (qcqs) $k$ -scheme. The actual second motivic Chern operation restricts to a group homomorphism $c_{2} : H = F_{m o t}^{2} K_{0} (X) ⟶ H_{m o t}^{4, 2} (X, Z) .$ In particular $c_{2} (0) = 0$ and $c_{2} (- x) = - c_{2} (x)$ for $x \in H$ . Whenever $[V] - [W] \in H$ for vector bundles $V, W$ on $X$ , one has $c_{2} ([V] - [W]) = c_{2} (V) - c_{2} (W)$ , with their original bundle Chern coefficients.

Here $K_{0} (X)$ is the zeroth homotopy group of nonconnective K-theory of perfect complexes, and $H$ is the image of the second filtration stage in that group. A perfect complex is locally quasi-isomorphic to a bounded complex of finite locally free sheaves; a vector bundle is a finite locally free sheaf. Normality means that every local ring is an integrally closed domain. Quasi-compact means that every open cover has a finite subcover; quasi-separated means that intersections of affine opens are quasi-compact. No finite-type, noetherian, dimension or global bundle-representation hypothesis is required.

Facts & Assumptions

[A1]

The Axiom of Choice, as formulated in The Axiom of Choice, and the fixed site and universe bounds are inherited by the supplied perfect-category realizations, motivic constructions and local integral-chain comparisons. They are used when invoking those suppliers in steps 1.1–3.1; choosing a lift of one image element is not itself an arbitrary-index choice argument.

[F1]

Actual perfect Cartan through degree three on all qcqs field-schemes supplies coefficientwise Cartan for the actual perfect operations on every qcqs field-scheme through degree three, with zero normalization and formal inversion. Write $C_{1}$ for its first operation until it is matched with the perfect first-layer class; its second operation is the specified $c_{2}$ .

[F2]

Rank and determinant on the second piece supplies rank zero and $c_{1}^{p e r f} (x) = 0$ for every $x \in H$ in precisely this normal qcqs scope. Motivic filtration and image groups defines $H$ as an additive image subgroup.

[F3]

The globally normalized first-layer Picard map is determinant identifies the actual first-layer map on the rank-zero K-space $S$ with the perfect determinant/Picard map, including its global descent data. First Chern class factors through determinant identifies its oriented class with $c_{1}^{p e r f}$ and matches the original bundle first classes. External formal universal Chern maps and supported construction constructs the actual $C_{i}$ from those original stable universal bundle classes: their restriction to the stable free-class map is the original classifying map, including the determinant-torsor normalization in weight one. Thus on actual bundles their coefficients are the original ones.

[F4]

Local free chains detect natural perfect operations on every qcqs base supplies, on the small Zariski site of every qcqs field-scheme, detection of derived sheaf-map classes into the actual Eilenberg targets by precomposition with the rank-zero free-class map $j_{0} : B G L \to S$ . This is a local integral-chain comparison, with no dimension or global bundle-generation hypothesis.

Given: $k, X, H$ , Choice and the fixed site and universe bounds as stated.

Goal: Establish additivity for arbitrary second-image perfect classes and retain the conditional bundle-difference formula.

Proof

1.1A1F1F3F4

Restrict $C_{1}$ to $S$ and compare it with the actual first-layer determinant/Picard map, with target identified with the weight-one Eilenberg target by its specified positive orientation. After precomposition with $j_{0}$ , both maps are the oriented determinant-torsor map of the stable universal bundle by [F3], including its automorphism and gluing maps. The derived-map detection in [F4] identifies these maps on $S$ . Evaluation on $X$ therefore identifies $C_{1} (x)$ with $c_{1}^{p e r f} (x)$ for every rank-zero perfect class $x$ ; this is an equality of actual map classes, rather than a deduction from local Picard-group agreement.

2.1A1F2step 1.1

Every $x \in H$ has rank zero and $c_{1}^{p e r f} (x) = 0$ by [F2]. Step 1.1 consequently gives $C_{1} (x) = 0$ for every such $x$ . This uses the full-qcqs supplier and no global bundle representative.

3.1A1F1F2step 2.1

For $x, y \in H$ , [F1] gives $c_{2} (x + y) = c_{2} (x) + C_{1} (x) C_{1} (y) + c_{2} (y)$ . The cross term vanishes by step 2.1. Pointedness gives $c_{2} (0) = 0$ ; since $H$ is a subgroup, applying additivity to $x + (- x)$ yields $c_{2} (- x) = - c_{2} (x)$ . Thus the restriction is a group homomorphism.

4.1A1F1F2F3step 3.1∎

Suppose $[V] - [W] \in H$ . Additivity of $c_{1}^{p e r f}$ and its bundle agreement in [F3], together with [F2], give $c_{1} (V) = c_{1} (W)$ . On bundle classes the coefficients in [F1] are the original ones by [F3]. Degree-two inversion gives $c_{2} (- [W]) = c_{1} (W)^{2} - c_{2} (W)$ , and Cartan therefore gives $c_{2} ([V] - [W]) = c_{2} (V) - c_{1} (V) c_{1} (W) + c_{1} (W)^{2} - c_{2} (W)$ . The equal first classes cancel the quadratic terms, proving the stated formula. Only this conditional assertion uses the supplied bundle-difference representation.

This restores the normal qcqs scope of the supplied article's p. 3 conventions and p. 11 Lemma 5.3. The actual full-scope Cartan, first-layer and local-chain suppliers are used with their stated hypotheses; their deep constructions are not reconstructed or independently judged here. The earlier total-Cartan supplier's noetherian finite-dimensional consequence is not extended by assertion. The original article's blanket bundle-representation premise remains a source proof defect, distinct from this perfect-class proof and its conditional bundle formula.

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