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Motivic Chern classification on normal affine threefolds

PDF p. 1: Abstract; PDF p. 2: Theorem B; PDF p. 17: Proof of Theorem B

Statement

Assume the Axiom of Choice and the site and universe size bounds required by the cited motivic filtration and Chern-class 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 $(c_{1}, c_{2}, c_{3}) : {V e c t}_{3} (X) ⟶ \prod_{i = 1}^{3} H_{m o t}^{2 i, i} (X, Z)$ is a bijection.

A vector bundle is a finite locally free $O_{X}$ -module; ${V e c t}_{3} (X)$ is the set of isomorphism classes of bundles of constant rank three. Normal means that every local ring is an integrally closed domain; affine finite type means $X = Spec A$ for a finitely generated $k$ -algebra $A$ . Krull dimension is the supremum of lengths of strict chains of irreducible closed subsets. An algebraically closed field is a field in which every nonconstant one-variable polynomial has a root. A perfect complex is locally quasi-isomorphic to a bounded complex of vector bundles, and $K_{0} (X)$ is the zeroth homotopy group of its nonconnective $K$ -theory spectrum. The class of a bundle $E$ , viewed as a complex in degree zero, is denoted $[E]$ . The group $H_{m o t}^{2 i, i} (X, Z)$ is degree- $2 i$ cohomology of the weight- $i$ motivic complex, and $c_{i}$ is the specified motivic Chern operation, evaluated on bundle classes. Choice asserts that every family of nonempty sets has a choice function; the size bounds fix the sites and ambient universes used by the suppliers.

Facts & Assumptions

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

Goal: Prove that the displayed map is injective and surjective.

[F1]

The motivic image filtration $F^{j} K_{0} (X)$ is the image of the $j$ th filtration stage in $K_{0} (X)$ and is decreasing. Its actual zeroth and first projections have respective kernels $F^{1} K_{0} (X)$ and $F^{2} K_{0} (X)$ . The zeroth projection is pointwise rank, and the first projection on $F^{1}$ is the actual first Chern class. The kernel assertions use the descended projections and the stage exact sequence. Motivic filtration and image groups Stage projections, kernels and image groups Zeroth projection is rank First projection is first Chern class

[F2]

On $F^{2} K_{0} (X)$ , the actual second projection is $- c_{2}$ and has kernel $F^{3} K_{0} (X)$ . Second projection is minus second Chern class

[F3]

The bundle total Chern series is multiplicative under direct sum. On the affine scheme $X$ , its representative-independent virtual extension assigns $c_{t} ([E] - [F]) = c_{t} (E) c_{t} (F)^{- 1}$ ; it agrees with the actual Chern operations on these classes. Whitney identities and virtual Chern arithmetic Isomorphism classes of vector bundles

[F4]

The actual third Chern class is an isomorphism from $F^{3} K_{0} (X)$ to $H_{m o t}^{6, 3} (X, Z)$ . Third Chern isomorphism

[F5]

Every triple in the displayed target occurs as the first three Chern classes of a rank-three bundle. Prescribing three Chern classes successively

[F6]

For bundles of constant rank at least the dimension on a nonempty affine finite-type scheme over an algebraically closed field, equality of their perfect $K_{0}$ classes is equivalent to isomorphism. The supplier assumes Choice and its stated universes. Critical-rank cancellation used in the article The Axiom of Choice

Proof

1.1F3givenalgebra

Let $E$ and $F$ be rank-three bundles with $c_{i} (E) = c_{i} (F)$ for $i = 1, 2, 3$ , and put $α = [E] - [F] \in K_{0} (X)$ . By [F3], the coefficients through degree three of $c_{t} (α) = c_{t} (E) c_{t} (F)^{- 1}$ vanish: the two bundle series agree through degree three, so their quotient is $1$ modulo $t^{4}$ . Thus $c_{1} (α) = c_{2} (α) = c_{3} (α) = 0$ . Both bundles have rank three at every point, so $α$ has pointwise rank zero.

2.1F1step 1.1

The zeroth projection of $α$ is zero by its rank, and [F1] therefore puts $α$ in $F^{1} K_{0} (X)$ . Its first projection is $c_{1} (α) = 0$ ; the first kernel assertion in [F1] consequently puts $α$ in $F^{2} K_{0} (X)$ .

3.1F2F4step 1.1step 2.1

By [F2], the second projection of $α$ is $- c_{2} (α) = 0$ , so $α \in F^{3} K_{0} (X)$ . The third Chern class is injective on this group by [F4], and $c_{3} (α) = 0$ by step 1.1. Hence $α = 0$ in $K_{0} (X)$ .

4.1F6step 3.1

Since $X$ has dimension three it is nonempty. The bundles $E$ and $F$ have constant rank three, equal to $\dim X$ , and step 3.1 gives $[E] = [F]$ . Applying [F6] yields $E ≅ F$ . Thus the displayed map is injective on isomorphism classes.

5.1F5step 4.1∎

Given any triple in the target, [F5] provides a rank-three bundle having exactly those three classes. This proves surjectivity. Together with step 4.1, the displayed map is a bijection.

Depends on

Used by · 1

Rank-two bundles with prescribed first two classes

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. 2, Theorem B; p. 17, proof of Theorem B

The first three motivic Chern classes classify rank-three vector bundles on a normal affine threefold over an algebraically closed field.

Hypotheses: Normal affine threefold of finite type over an algebraically closed field.

Reading limits: The cited article passages were read in the supplied extracted text. The proof uses the named completed suppliers on their stated contracts; their underlying 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. 2, Theorem B; p. 17, proof of Theorem B