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The actual EM first Chern class is additive and natural

PDF p. 2: Motivic projective bundle property; PDF p. 8: Orientation for the splitting principle

Statement

For every qcqs scheme $X$ over a field, define $c_{1}$ to be the degree-two map of EM Definition5.2. Then $c_{1} : Pic (X) \to H^{2, 1} (X)$ is a natural group homomorphism. In particular $c_{1} (L \otimes M) = c_{1} (L) + c_{1} (M)$ , $c_{1} (O_{X}) = 0$ , and $c_{1} (L^{\lor}) = - c_{1} (L)$ .

Facts & Assumptions

[F1]

External EM multiplicative motivic complexes and actual Chern map supplies the actual natural map $R Γ_{N i s} (X, G_{m}) [- 1] \to C_{1} (X)$ , and its source degree two identifies with $Pic (X)$ under tensor product.

[F2]

Picard group of a scheme defines this group and identifies its inverse with duality.

Proof

Given: $X, L, M$ as stated.

Goal: Justify this definition, additivity and naturality.

1.1F1F2construct

Apply $H^{2}$ to the map in [F1]. A map in the derived category induces an abelian-group homomorphism on cohomology, so its Picard map is additive for the tensor-product group law. It is well-defined on line-bundle isomorphism classes. This defines precisely the EM class, without replacing it by a K-zero line-class map.

2.1F1F2step 1.1∎

A group homomorphism sends the identity to zero and inverses to negatives. Thus the three identities hold by [F2]. The square of complexes for any pullback commutes by [F1]; its degree-two square gives $f^{*} c_{1} (L) = c_{1} (f^{*} L)$ .

Depends on

Used by · 4

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