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External norm-divisor identity for integral curves

PDF p. 13: Divisor presentation for supported degree six

Statement

Let C,T be integral finite-type curves over a field and p:C→T proper and dominant. For g∈k(C)×, with K=k(T) and L=k(C), p∗div⁡C(g)=div⁡TNm⁡L/K(g). Here the norm is the determinant of multiplication by g on the finite-dimensional K-vector space L; p∗[y]=[κ(y):κ(p(y))][p(y)]; and orders use differences of local quotient lengths. Neither curve need be normal or smooth, and L/K need not be separable.

Source contract

Given: The field, integral curves, proper dominant morphism and nonzero rational function in the Statement.

Goal: Record the established external identity with its singular-curve coefficient convention.

[F1] Stacks Project Lemma 42.18.1 proves the pushforward identity in its dimension-function setup, which includes finite-type curves over a field.

[F2] Stacks Project Lemma 10.121.8 and Definition 10.121.2 give the weighted local norm formula using quotient-length orders without normality.

External input

Import [F1], with the coefficient convention of [F2]. This is an equality of zero-cycles before quotienting by rational equivalence. The cited proof is not claimed as a local reconstruction.

Depends on

Used by · 2

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

The Stacks Project, Principal divisors and pushforward · Lemma 42.18.1; Situation 42.7.1, tag 02QL, specialized to the spectrum of a field with dimension function zero

For a proper dominant morphism of integral finite-type curves over a field, pushforward of the principal divisor of a nonzero rational function equals the principal divisor of its field norm, with residue-degree weights.

Hypotheses: Both curves are integral and finite type over a field, have dimension one, and the morphism is proper and dominant. Normality and separability are not assumed.

Reading limits: Read the full HTML statement and displayed proof, including its reduction to the finite-domain formula. The linked finite-morphism reduction and wider intersection theory were not reconstructed.

The Stacks Project, Orders of vanishing · Lemma 10.121.8 and Definition 10.121.2, tag 02MD

For a finite extension of domains over a one-dimensional noetherian local domain, the order of a norm is the sum of upstairs local orders weighted by residue-field degrees; orders are differences of quotient lengths.

Hypotheses: The base is a one-dimensional noetherian local domain, the extension is finite and is a domain, and the element is nonzero in its fraction field. Normality and separability are not assumed.

Reading limits: Read the full statement and displayed proof of Lemma 10.121.8 and the displayed order definition. The proof of its cited determinant-length lemma was not read or reconstructed.

Sources