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Positive generic cycles determine the curve-supported homomorphism

PDF p. 16: Supported leading weight two

Statement

Assume the Axiom of Choice and the fixed compatible site and universe bounds of the supplied geometric and supported comparisons. In the preceding setting, let $h : G_{0} (C) \to Z [C]$ be a homomorphism. If $h$ kills every closed-point residue-field class and $h ([O_{C}]) = [C]$ , then $h ([F]) = \dim_{k (η)} F_{η} [C]$ for every coherent $F$ and every virtual class by linearity. Equivalently, to prove that a supported degree-zero map into $H_{0} z_{C}^{2} (U, ∙)$ is the positive generic-length map, it suffices to identify its generic restriction with restriction to $K_{0} (k (η)) = Z$ by dimension. A sign is fixed only once the residue-field or structure-sheaf generator has been sent to $+ 1$ ; an unspecified isomorphism of groups does not supply that condition.

Facts & Assumptions

[A1]

The assumed The Axiom of Choice discharges the coherent generic-rank supplier [F1] in step 1.1. Together with the fixed site and universe bounds it discharges the geometric and actual supported-coordinate premises [F2]–[F3] in step 2.1. Once these supplied identifications are fixed, the homomorphism argument is finite algebra.

Given: The Axiom of Choice, the stated site and universe bounds, the group and generator identifications above, and the two stated conditions on $h$ .

Goal: Isolate the exact generator/sign test needed from the external map comparison.

[F1]

Generic rank $r : G_{0} (C) \to Z$ sends $[O_{C}]$ to $1$ and has kernel generated by the closed-point residue-field classes. Generic rank and point-supported classes on a curve

[F2]

The supported cycle target is $Z [C]$ , detected isomorphically by restriction to any generic neighbourhood. Bottom supported Bloch groups on a codimension two curve

[F3]

The current supported Euler coordinate is the generic rank under devissage, with value $+ 1$ on $i_{*} O_{C} [0]$ . The actual supported Euler class has the generic length coordinate

Proof

1.1A1F1givenalgebra

For any $x \in G_{0} (C)$ , $x - r (x) [O_{C}]$ belongs to $\ker r$ . By [F1] it is a finite integral sum of closed-point classes. The first hypothesis on $h$ kills that difference, while its second hypothesis sends $r (x) [O_{C}]$ to $r (x) [C]$ . This gives the displayed formula and uniqueness.

2.1A1F1F2F3step 1.1∎

By [F2], equality of maps to the supported cycle group is tested on the coefficient of its single positive generic cycle. If that coefficient map is the residue-field dimension map, it kills closed-point classes and sends the structure sheaf to $+ 1$ , so step 1.1 applies. Transporting back by [F3] gives exactly the alternating generic-length formula on perfect-complex classes. The argument requires that specific dimension normalization and cannot extract it merely from an unnamed group isomorphism.

Depends on

Used by · 1

Smooth rank, first Chern, determinant and supported positive leading cycle

Dependency tree · 3 levels

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