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Independent determinant and second-Chern prescription

PDF p. 2: Paragraph following Theorem A

Statement

Assume the Axiom of Choice and the site-size and universe bounds required for the motivic and perfect K-theory constructions in Motivic Chern classification on normal affine surfaces. Let $k$ be an algebraically closed field and $X$ a normal affine finite-type $k$ -scheme of Krull dimension two. Then $E ⟼ (\det E, c_{2} (E))$ induces a bijection from ${V e c t}_{2} (X)$ to $Pic (X) \times H_{m o t}^{4, 2} (X, Z)$ . Thus two rank-two bundles with the same determinant and second motivic Chern class are isomorphic, and every pair of such invariants occurs.

A vector bundle is a finite locally free $O_{X}$ -module; rank two means locally isomorphic to $O_{X}^{2}$ . The set ${V e c t}_{2} (X)$ consists of their isomorphism classes. A line bundle is a rank-one vector bundle, $\det E = ⋀^{2} E$ , and $Pic (X)$ is the group of line-bundle isomorphism classes under tensor product. Normal means that every local ring is an integrally closed domain; Krull dimension is the supremum of lengths of chains of irreducible closed subsets. The displayed motivic group and $c_{2}$ are the integral weight-two motivic cohomology group and the second motivic Chern class used in the supplier. Choice means that every family of nonempty sets has a choice function; the size bounds fix the sites and ambient universes used in those constructions.

Facts & Assumptions

Given: $k$ , $X$ , Choice and the site-size and universe bounds in the statement.

Goal: Prove that determinant and the second motivic Chern class give the stated bijection.

[F1]

Under these hypotheses, the map $(c_{1}, c_{2})$ from ${V e c t}_{2} (X)$ to $H_{m o t}^{2, 1} (X, Z) \times H_{m o t}^{4, 2} (X, Z)$ is bijective. The same supplier identifies $Pic (X)$ with $H_{m o t}^{2, 1} (X, Z)$ by first Chern class and identifies $c_{1} (E)$ with the image of $\det E$ . Motivic Chern classification on normal affine surfaces

Proof

technique · Transport the first coordinate of the supplied classification along the Picard isomorphism

1.1F1given

Let $γ : Pic (X) \to H_{m o t}^{2, 1} (X, Z)$ be the isomorphism in [F1]. For every rank-two bundle $E$ , [F1] gives $γ ([\det E]) = c_{1} (E)$ . Consequently the map in the statement is the composite of the bijection $(c_{1}, c_{2})$ with $(γ^{- 1}, i d)$ . In particular it is well-defined on isomorphism classes.

2.1F1step 1.1∎

Both maps in this composite are bijections, so their composite is a bijection. Its injectivity gives uniqueness up to isomorphism, and its surjectivity gives a bundle for every prescribed pair. This proves the stated classification and independent prescription.

Depends on

Dependency tree · 3 levels

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External dependencies

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 2, paragraph following Theorem A

A rank-two bundle is uniquely determined by its determinant and second motivic Chern class, and these invariants can be prescribed independently.

Hypotheses: A normal affine surface over the standing algebraically closed field; the article conventions take affine schemes to be of finite type.

Reading limits: The statement and its immediate context in the supplied extracted article text were read. The classification theorem is used through the named completed supplier; its proof is not reconstructed here.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 2, paragraph following Theorem A