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Normalization and the curve divisor quotient

PDF p. 13: Divisor presentation for supported degree six

Statement

Assume AC. Let $T$ be an integral affine finite-type curve over an algebraically closed field, $ν : C = \tilde{T} \to T$ its normalization, and $K = k (T) = k (C)$ . Write $Q (D) = Z_{0} (D) / {div}_{D} (k (D)^{\times})$ . Then $ν_{*} {div}_{C} (f) = {div}_{T} (f)$ for $f \in K^{\times}$ , and $ν_{*}$ induces $Q (C) ↠ Q (T)$ . Explicitly, ${ord}_{x} (f) = \sum_{y \in ν^{- 1} (x)} {ord}_{y} (f)$ , with DVR valuations upstairs and length orders downstairs. There is a canonical identification $Pic (C) = {C H}^{1} (C) = Q (C)$ sending $O_{C} (y)$ to $[y]$ ; hence normalization induces a surjection from this Picard group to $Q (T)$ . Here $Pic$ is the group of isomorphism classes of line bundles under tensor product and ${C H}^{1}$ is degree-zero Bloch homology.

Facts & Assumptions

Given: AC and the curve and normalization in the Statement.

Goal: Establish divisor compatibility, surjectivity and the source identification.

[F1]

Normalization is finite, smooth and integral, has the same function field, and has nonempty finite closed fibers with residue fields $k$ . Finite morphisms are proper under AC. Normalization of an affine integral curve, Residue fields and normalization point degrees, Finite morphisms are proper, The Axiom of Choice

[F2]

Proper dominant curve pushforward commutes with principal divisors and norms; closed points are weighted by residue degrees. External norm-divisor identity for integral curves

[F3]

A one-dimensional noetherian normal local domain is a DVR. Length orders agree with its normalized valuation. Equivalent characterizations of a DVR, Length orders on an integral curve

[F4]

On an integral scheme Cartier divisors modulo principal Cartier divisors map isomorphically to the Picard group via $D \mapsto O (D)$ . A Cartier divisor is specified by local nonzero rational equations whose ratios are regular units. On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group

[F5]

The codimension-one Bloch complex of the smooth curve has point cycles in degree zero and endpoint intersections in degree one; the endpoint boundary subgroup is the principal-divisor subgroup. Bloch cycle complexes and higher Chow groups, Principal divisors are endpoint boundaries

Proof

1.1F1F2given

The finite normalization is proper and dominant with field extension $K / K$ . Applying [F2], whose norm is the identity in this case, yields $ν_{*} {div}_{C} (f) = {div}_{T} (f)$ . Each point in a closed fiber has residue degree one by [F1]; comparing coefficients gives the displayed order formula.

2.1F1step 1.1construct

The equality in step 1.1 makes pushforward descend to the quotient groups. Every closed point of $T$ has a closed preimage by [F1], and each such point pushes forward with coefficient one. Lifting the finitely many points in any zero-cycle proves surjectivity.

3.1F1F3F4algebrastep 2.1

By [F1] and [F3], each closed-point local ring of $C$ is a DVR. Taking valuations of the local equations of a Cartier divisor defines a zero-cycle: the support is finite by the finiteness of rational-function divisors on a finite trivializing cover. Unit ratios make the local valuations agree. If all valuations vanish, every equation is a unit at each closed point of its domain and also at the generic point, so the Cartier divisor is zero.

4.1F3F4step 3.1construct

For any closed $y$ , a uniformizer is regular on a neighbourhood of $y$ . Remove its other finitely many zeros from that neighbourhood. The uniformizer there and the equation one on $C ∖ {y}$ have unit ratio on the overlap and define a Cartier divisor mapping to $[y]$ . Therefore the map of step 3.1 is an isomorphism from Cartier divisors to zero-cycles, carrying principal Cartier divisors to principal length divisors. By [F4], $Q (C) ≅ Pic (C)$ , with $[y]$ corresponding to $O_{C} (y)$ .

5.1F5step 2.1step 4.1∎

By [F5], degree-zero Bloch homology is the same quotient $Q (C)$ . Combining this identification with steps 2.1 and 4.1 proves the claimed source and surjection.

Depends on

Used by · 1

Top cycles supported on an integral curve

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