LemmaStatement: AI-generatedProof: AI-generated

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Length orders on an integral curve

PDF p. 13: Divisor presentation for supported degree six

Statement

Assume AC. Let $T$ be an integral finite-type curve over a field, $K = k (T)$ , and $R = O_{T, x}$ for a closed point $x$ . For nonzero $a, b \in R$ set ${ord}_{x} (a / b) = ℓ_{R} (R / a R) - ℓ_{R} (R / b R)$ . These lengths are finite, the integer is independent of the fraction, and it is additive under multiplication. It vanishes for all but finitely many closed $x$ for each $f \in K^{\times}$ . Thus ${div}_{T} (f) = \sum_{x} {ord}_{x} (f) [x]$ is a homomorphism to the free zero-cycle group. A nonzero function regular near $x$ has order $ℓ_{R} (R / f R)$ ; a unit has order zero. In a discrete valuation ring (DVR), meaning a local domain whose nonzero elements are units times nonnegative powers of one uniformizer, this order is its normalized valuation. No normality is assumed.

Facts & Assumptions

Given: AC and $T, K, x, R$ as stated.

Goal: Establish the length formula and divisor homomorphism.

[F1]

Module length counts the factors of a composition series and is additive in short exact sequences of finite-length modules. Composition series and length of a module, Module length is additive in short exact sequences

[F2]

Closed-point local rings of a finite-type integral curve are one-dimensional noetherian local domains, and a proper closed subset of such a curve is finite. The dimension formula for affine domains, The Axiom of Choice

Proof

1.1F1F2algebra

For nonzero $a \in R$ , if $a$ is not a unit, the only prime containing $(a)$ is the maximal ideal $m$ . Choose generators $u_{i}$ of $m$ and exponents $e_{i} > 0$ with $u_{i}^{e_{i}} \in (a)$ . A sufficiently high power of $m$ lies in $(a)$ . The successive quotients of the resulting finite filtration of $R / a R$ are finite-dimensional over $R / m$ , and hence have finite composition series. The unit case gives the zero module.

2.1F1step 1.1algebra

For nonzero $a, c \in R$ , cancellation gives the exact sequence $0 \to R / a R \overset{c}{\to} R / a c R \to R / c R \to 0$ . Length additivity therefore gives $ℓ (R / a c R) = ℓ (R / a R) + ℓ (R / c R)$ . If $a / b = c / d$ , then $a d = b c$ , proving independence of the displayed difference. The same product formula proves additivity.

3.1F2step 2.1

On each member of a finite affine cover write $f = a / b$ with nonzero regular $a, b$ . Its order vanishes where both are units. The exceptional set is a proper closed subset of the curve, hence finite by [F2]. The finite sum is therefore defined, and step 2.1 makes it a homomorphism.

4.1F1step 2.1step 3.1∎

The regular and unit assertions follow by taking denominator one. In a DVR the filtration of $R / (π^{n})$ by powers of a uniformizer has $n$ residue-field factors; subtracting numerator and denominator exponents gives the normalized valuation.

Depends on

Used by · 4

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