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Second Chern additivity on the second piece

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

Statement

Assume AC and the fixed site/universe bounds. Let $k$ be algebraically closed and $X$ a normal qcqs $k$ -scheme. For perfect-complex $K_{0}$ the actual global operation restricts to a homomorphism $c_{2} : F_{m o t}^{2} K_{0} (X) ⟶ H_{m o t}^{4, 2} (X, Z) .$ For every bundle difference $[V] - [W]$ in this piece, $c_{2} ([V] - [W]) = c_{2} (V) - c_{2} (W)$ . There is no finite-type, noetherian, finite-dimensional or global bundle-representation hypothesis. In fact the same conclusion holds on every qcqs scheme over any field with these operations and conventions.

Facts & Assumptions

[A1]

Assume the Axiom of Choice as formulated in The Axiom of Choice and the stated size bounds. Choice and the fixed size bounds are inherited from the actual first-layer comparison in [F2] at step 1.1 and the perfect Cartan and inversion operations in [F1] at steps 2.1–3.1.

Given: The stated scheme, conventions, and the second image subgroup $H$ .

Goal: Prove the claim for actual perfect classes, including those without global bundle-difference representatives.

[F1]

Actual perfect Cartan through degree three on all qcqs field-schemes supplies actual perfect Cartan through degree three on all qcqs field-schemes, with zero normalization and formal inversion.

[F2]

Motivic filtration and image groups defines the nested image subgroups. Stage projections, kernels and image groups makes the composite from the second stage to the first layer zero. The globally normalized first-layer Picard map is determinant and First Chern class factors through determinant identify this layer on the rank-zero fiber with the actual first Chern class, naturally and in full qcqs scope.

Proof

1.1A1F2given

For $x \in H$ choose a second-stage representative and map it to the first stage. Its first-layer value is zero by the consecutive fiber sequence in [F2], and its augmentation is $x$ . The actual first-layer identification in [F2] gives $c_{1} (x) = 0$ . The same holds for every element of $H$ , an additive subgroup by definition.

2.1A1F1step 1.1

For $x, y \in H$ , the degree-two identity in [F1] gives $c_{2} (x + y) = c_{2} (x) + c_{1} (x) c_{1} (y) + c_{2} (y) = c_{2} (x) + c_{2} (y)$ . Pointedness gives $c_{2} (0) = 0$ ; taking $y = - x$ gives inverse preservation. Thus this restriction is a homomorphism.

3.1A1F1F2step 1.1step 2.1∎

If $[V] - [W] \in H$ , additivity of the first class in [F2] gives $c_{1} (V) = c_{1} (W)$ . Cartan and inversion in [F1] give $c_{2} ([V] - [W]) = c_{2} (V) - c_{1} (V) c_{1} (W) + c_{1} (W)^{2} - c_{2} (W)$ , which is the stated formula. This proves the special bundle formula without assuming all perfect classes have that form.

Depends on

Used by · 3

Dependency tree · 3 levels

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Sources

Fasel and Yakerson, Vector bundles on singular surfaces and threefolds · Section 5, Lemma 5.3, page 11