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The two supported cycle degrees on a curve

PDF p. 13: Divisor presentation for supported degree six

Statement

Assume the Axiom of Choice (AC). Let $U$ be a smooth finite-type scheme of pure dimension three over a field $k$ , and let $T \subset U$ be an integral closed curve. Put $Z_{0} (T) = ⨁_{x \in T closed} Z [x]$ . Let $B_{1} (T)$ be the free abelian group on integral one-dimensional closed subschemes $W \subset T \times A_{s}^{1}$ contained in neither endpoint $s = 0$ nor $s = 1$ . The degree-zero and degree-one terms of $z_{T}^{3} (U, ∙)$ are respectively $Z_{0} (T)$ and $B_{1} (T)$ . The boundary is $b [W] = [W_{1}] - [W_{0}]$ , where $[W_{ϵ}] = \sum_{w \in W \cap {s = ϵ}} ℓ_{O_{W, w}} (O_{W, w} / (s - ϵ)) [p (w)]$ and $p : T \times A^{1} \to T$ is projection. Consequently ${C H}_{T}^{3} (U) = Z_{0} (T) / b B_{1} (T)$ . Endpoint projection is an isomorphism, so no residue-field factor is omitted.

Facts & Assumptions

Given: AC and $k, U, T$ as stated; write $s = t_{1}$ and $t_{0} = 1 - s$ on the algebraic simplex $Δ^{1}$ .

Goal: Identify both terms and the boundary with its multiplicities.

[F1]

The supported complex is generated by admissible integral codimension-three cycles supported in $T \times Δ^{n}$ ; its differential is the alternating sum of face intersections, with local intersection lengths. Cycle complexes with fixed support

[F2]

The finite-type dimension formula computes the dimensions of integral closed subschemes in the ambient products. Its use here carries AC. The dimension formula for affine domains, The Axiom of Choice

Proof

1.1F1F2given

In degree zero the codimension-three integral subschemes are closed points; the support condition selects exactly the closed points of $T$ . Thus the term is $Z_{0} (T)$ .

1.2F1F2givenalgebra

In degree one the ambient dimension is four, so supported generators have dimension one. An integral closed subscheme set-theoretically supported in $T \times A^{1}$ factors through that reduced closed subscheme. Conversely every integral one-dimensional closed subscheme there has codimension three in the ambient product. It meets an endpoint properly exactly when it is not contained in that endpoint: a proper closed subset of a noetherian integral curve is finite and zero-dimensional. Hence the degree-one term is $B_{1} (T)$ .

2.1F1step 1.2algebra

The face $t_{0} = 0$ is $s = 1$ and $t_{1} = 0$ is $s = 0$ . The equations $1 - s$ and $s - 1$ differ by a unit, so their quotient lengths agree. The face formula gives $b [W] = [W_{1}] - [W_{0}]$ with the displayed coefficients. Projection restricts to an isomorphism $T \times {ϵ} \to T$ , preserving residue fields.

3.1F1step 1.1step 2.1∎

There are no negative-degree terms, so zeroth homology is the degree-zero term modulo the degree-one boundary image. The stated quotient follows.

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