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Line bundles and first motivic Chern class

PDF p. 1: Introduction, second paragraph; PDF p. 2: After Theorem A

Statement

Assume the Axiom of Choice and the site-size and universe bounds required for the weight-one comparison, Nisnevich cohomology and perfect-complex determinant below. Let $k$ be a field and $X$ a quasi-compact quasi-separated $k$ -scheme. The first motivic Chern class gives a natural group isomorphism $c_{1} : Pic (X) \tilde{\to} H_{m o t}^{2, 1} (X, Z) .$ Consequently two line bundles on $X$ are isomorphic exactly when their first motivic Chern classes agree, and every class in the displayed motivic cohomology group is the first class of a line bundle.

A line bundle is a Zariski locally free $O_{X}$ -module of rank one; $Pic (X)$ is the group of its isomorphism classes under tensor product. The group $H_{m o t}^{2, 1} (X, Z)$ is degree-two cohomology of the supplied weight-one motivic complex. The map $c_{1}$ is the supplied first motivic Chern class on an actual line bundle. Its inverse first applies the inverse of the weight-one comparison to obtain a class in $H_{N i s}^{1} (X, G_{m})$ and then takes the line bundle descended from the corresponding frame torsor. Here $G_{m}$ is the sheaf of invertible regular functions on the small Nisnevich site.

Statement references

Motivic weight one

First Chern class factors through determinant

Units and Picard groups in Nisnevich degree zero and one

Facts & Assumptions

Given: A field $k$ , a quasi-compact quasi-separated $k$ -scheme $X$ , the Axiom of Choice and the site-size and universe bounds in the cited suppliers.

Goal: Prove that the actual first motivic Chern class is the stated natural bijective group homomorphism, with the specified inverse.

[F1]

The natural weight-one comparison induces an isomorphism $H_{N i s}^{1} (X, G_{m}) \to H_{m o t}^{2, 1} (X, Z)$ under Choice and its stated site-size bounds. Motivic weight one

[F2]

Under Choice and its specified small-site and universe bounds, the frame-torsor construction identifies $Pic (X)$ with $H_{N i s}^{1} (X, G_{m})$ as a group; descent from frames gives the inverse. Units and Picard groups in Nisnevich degree zero and one

[F3]

For every qcqs scheme over a field, under the stated Choice and universe bounds, the actual first motivic Chern class on a bundle agrees with the composite of its determinant and the weight-one orientation $γ_{X} : Pic (X) \to H_{m o t}^{2, 1} (X, Z)$ . This orientation is the isomorphism obtained from the weight-one comparison and the Picard identification. First Chern class factors through determinant

[F4]

The determinant of a bundle agrees with its exterior determinant. In rank one the top exterior power is the bundle itself, canonically. Determinant descends on the actual perfect K zero

[F5]

The Axiom of Choice is assumed as required by the cited cohomological and determinant constructions. The Axiom of Choice

Proof

technique · Identify the first Chern class with a composite of the frame-torsor and weight-one isomorphisms

1.1F1F2F5

By [F2], a line bundle class maps to the class of its frame torsor, and this defines a group isomorphism from $Pic (X)$ to $H_{N i s}^{1} (X, G_{m})$ . Compose it with the isomorphism of [F1]. The resulting map $γ_{X}$ is a group isomorphism, with inverse given by the two inverse constructions in reverse order.

2.1F3F4step 1.1

Let $L$ be a line bundle. Its determinant is canonically $L$ by [F4]. Applying the actual bundle-class comparison in [F3] therefore gives $c_{1} (L) = γ_{X} ([L])$ . Thus the map of step 1.1 is precisely the stated first Chern class, including its orientation.

3.1F1F2step 1.1step 2.1∎

Since $c_{1} = γ_{X}$ , it is a natural group isomorphism. Its injectivity says that equal first classes give the same Picard class, hence isomorphic line bundles; its surjectivity gives a line bundle representing each motivic class. The inverse is the one described in the statement by steps 1.1 and 2.1. Naturality follows from the naturality of the two supplied comparison maps.

Depends on

Used by · 3

Dependency tree · 2 levels

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Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · Introduction, p. 1; equation (2.3), p. 4; Proposition 3.2 and its proof, p. 8