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The actual first Chern map is the Picard isomorphism on an affine curve

Statement

Assume AC. Let $C$ be a smooth integral affine curve over an algebraically closed field. The actual first Chern map $Pic (C) \to {C H}^{1} (C)$ , $[L] \mapsto c_{1} (L)$ , is an isomorphism. In particular $c_{1} (L) = 0$ if and only if $L ≅ O_{C}$ , and this is equivalent to $ε (L) = 0$ . Since $\det L = L$ , this is the determinant/line-bundle interface in dimension one.

Facts & Assumptions

Given: The hypotheses and data in the statement, with AC and the article's fixed universes wherever perfect K-theory is used.

Goal: Establish the asserted conclusion.

[F1]

The local rings of a smooth scheme over the perfect base field are regular. A one-dimensional regular local ring is a DVR, and a DVR is a PID and hence a UFD under AC. Regular equals smooth over a perfect field, one dimensional regular local rings are dvrs, Every DVR is a PID, Every principal ideal domain is a unique factorisation domain

[F2]

On a locally factorial noetherian integral scheme Cartier and Weil divisors agree, compatibly with principal divisors; Cartier divisors modulo principal ones give the Picard group. Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme, On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group

[F3]

On a curve rational equivalence of codimension-one cycles is generated by principal divisors of rational functions on the curve itself. The actual first Chern class of a line bundle is the divisor of a nonzero rational section, modulo these relations. Rational equivalence and the Chow group of cycles, Intersection with an invertible sheaf and the first Chern class

[F4]

For a line bundle, vanishing of the exterior-dual class is equivalent to triviality. Determinants detect vanishing of the rank-one exterior-dual class

Proof

1.1F1F2F3

The noetherian integral curve has a field as its generic local ring, and every other local ring is a one-dimensional regular local ring. By [F1] these rings are UFDs, so the curve is locally factorial. By [F2], Cartier divisors modulo principal divisors identify with Picard classes and with Weil divisors modulo principal divisors. By [F3] the latter quotient is precisely ${C H}^{1} (C)$ .

2.1F2F3F4step 1.1∎

This is the actual first Chern map: a rational section of $L$ defines a Cartier divisor $D$ with $O_{C} (D) ≅ L$ , and [F3] sends $L$ to the Weil divisor cycle of $D$ . Thus the composed quotient isomorphism carries $[L]$ to $c_{1} (L)$ , fixing the sign and the map, rather than identifying only groups. Its kernel is the trivial line class. Combining with [F4] gives the asserted equivalences; $Λ^{1} L = L$ gives the determinant statement.

Depends on

Used by · 1

Smooth critical-rank splitting criterion

Dependency tree · 3 levels

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