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Third Chern isomorphism

PDF p. 1: Abstract; PDF p. 17: Corollary 6.9

Statement

Assume the Axiom of Choice and the site and universe size bounds required by the cited motivic filtration and support constructions. Let $k$ be an algebraically closed field, of arbitrary characteristic, and let $X$ be a normal affine finite-type $k$ -scheme of Krull dimension three. Then the composite $F_{m o t}^{3} K_{0} (X) \overset{c_{3}}{\to} H_{m o t}^{6, 3} (X, Z) \overset{ϵ_{3}}{\to} F_{m o t}^{3} K_{0} (X)$ is multiplication by $2$ , and $c_{3}$ is an isomorphism of abelian groups.

Here normal means that every local ring is an integrally closed domain; affine finite type means $X = Spec A$ for a finitely generated $k$ -algebra $A$ ; and Krull dimension is the supremum of lengths of chains of distinct irreducible closed subsets. A perfect complex is a complex locally quasi-isomorphic to a bounded complex of finite locally free sheaves. Write $K (X)$ for the nonconnective $K$ -theory spectrum of perfect complexes and $K_{0} (X) = π_{0} K (X)$ . The group $F_{m o t}^{3} K_{0} (X)$ is the image of $π_{0} F_{m o t}^{3} K (X) \to K_{0} (X)$ , where $F_{m o t}^{3} K (X)$ is the third stage of the motivic filtration. The group $H_{m o t}^{6, 3} (X, Z)$ is degree-six cohomology of the weight-three motivic complex, and $c_{3}$ is the actual third Chern operation on perfect $K$ -classes. The edge map $ϵ_{3}$ is the inverse of the actual third-stage-to-graded isomorphism followed by the stage-to- $K_{0}$ map with codomain restricted to its image. The Axiom of Choice says that every family of nonempty sets has a choice function; the size bounds fix the sites and universes used by the constructions.

Statement references

Motivic filtration and image groups

Third edge homomorphism

Facts & Assumptions

Given: The field, scheme, choice and size hypotheses in the statement.

Goal: Prove the displayed composite is multiplication by $2$ and that $c_{3}$ is an isomorphism.

[F1]

Smooth points generate the third motivic piece identifies $F_{m o t}^{3} K_{0} (X)$ with the subgroup generated by classes of smooth closed points on the three-dimensional components, extended by zero. In the pure-dimensional case this is the subgroup generated by all smooth closed points. A point is smooth when $X \to Spec k$ is smooth in a neighbourhood of it; its generator is the class of its perfect residue sheaf.

[F2]

Third Chern additivity on the third piece says that the actual $c_{3}$ is a group homomorphism on the third image piece. Third edge homomorphism defines $ϵ_{3}$ as a group homomorphism.

[F3]

Apply Factor two in the supported third Chern map on the pure threefold $X_{3}$ , the union of the three-dimensional clopen components of $X$ . For every smooth closed point $x \in X_{3}$ , it identifies the positive supported residue class $u_{x}$ and positive motivic point class $δ_{x}$ and proves $c_{3, x} (u_{x}) = 2 δ_{x}$ and $ϵ_{3, x} (δ_{x}) = u_{x}$ . Both operations commute with support-to-global maps on $X_{3}$ . No point on a lower-dimensional component is used.

[F4]

Unique divisibility of the smooth-point subgroup says that the top-dimensional smooth-point subgroup is uniquely divisible: multiplication by each positive integer is an isomorphism. Third edge isomorphism says that the specified $ϵ_{3}$ is an isomorphism for the given normal affine finite-type threefold. The cited constructions use the stated choice and size conventions; The Axiom of Choice states the choice assumption.

[F5]

Normal dimension strata and the actual residue class model makes $X_{3}$ a pure normal affine finite-type threefold and identifies zero extension of its residue classes. Finite clopen restrictions split the actual motivic image filtration gives the actual clopen product decompositions of the augmented stages and their image subgroups. Its graded triangles give the corresponding product decomposition of motivic cohomology. The actual Chern operations and the edge map commute with these restrictions; the former are pointed and the latter is defined from the natural stage-to-graded and augmentation maps in Third edge homomorphism. Thus an identity for a zero-extended top-dimensional point class can be checked on $X_{3}$ and its clopen complement.

Proof

technique · Compute on smooth-point generators and use the edge isomorphism and unique divisibility

1.1F1F2given

Set $G = F_{m o t}^{3} K_{0} (X)$ and $H = H_{m o t}^{6, 3} (X, Z)$ . By [F1], $G$ is generated by the residue-sheaf classes of smooth closed points on the three-dimensional components, with the stated zero extensions. By [F2], $c_{3} : G \to H$ and $ϵ_{3} : H \to G$ are homomorphisms.

2.1F2F3F5step 1.1

Let $g_{x}$ be one of these generators, so $x \in X_{3}$ . On the pure threefold $X_{3}$ , [F3] gives $u_{x} \mapsto 2 δ_{x} \mapsto 2 u_{x}$ in the supported groups, and the support-to-global squares give $ϵ_{3} c_{3} (g_{x} ∣_{X_{3}}) = 2 g_{x} ∣_{X_{3}}$ . On the clopen complement the class is zero; pointedness of $c_{3}$ and additivity of the edge make its composite zero there. By the product decompositions and naturality in [F5], these componentwise equalities give $ϵ_{3} c_{3} (g_{x}) = 2 g_{x}$ on $X$ . This remains true if $g_{x}$ vanishes globally.

3.1F1F2step 2.1algebra

Every element of $G$ is a finite integer linear combination of these generators. Since the composite is a homomorphism, step 2.1 shows that $ϵ_{3} c_{3} = 2 {id}_{G}$ on all of $G$ .

4.1F1F4step 3.1algebra∎

By [F4] and the equality of groups in [F1], multiplication by $2$ is an automorphism of $G$ , and $ϵ_{3}$ is an isomorphism from $H$ to $G$ . Thus $c_{3} = ϵ_{3}^{- 1} \circ (2 {id}_{G})$ is a composite of isomorphisms. Together with step 3.1 this proves both assertions.

Depends on

Used by · 3

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

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 1, abstract; p. 17, Corollary 6.9 and proof

The third Chern class followed by the third edge map is multiplication by two on the third motivic image piece, and the third Chern class is an isomorphism.

Hypotheses: X is a normal affine threefold over an algebraically closed field.

Reading limits: The cited passages were read in the supplied article extraction. The proof below uses the stated completed supplier contracts; their proofs and external inputs were not independently reconstructed for this item.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 1, abstract; p. 17, Corollary 6.9 and proof