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Proper norm and graph formulas for principal divisors

PDF p. 13: Proof of Proposition 6.1: ordinary CH3 identification and degree-seven vanishing

Statement

Let $k$ be a field. For integral finite-type $k$ -schemes $P, Q$ , a proper dominant generically finite morphism $p : P \to Q$ and $h \in k (P)^{\times}$ satisfy $p_{*} {div}_{P} (h) = {div}_{Q} {Nm}_{k (P) / k (Q)} (h) .$ Divisor orders use lengths in the one-dimensional local rings, and pushforward includes residue-field degrees.

For an integral finite-type $k$ -scheme $Q$ and $f \in k (Q)^{\times}$ , let $G \subset Q \times P^{1}$ be the reduced closure of the graph on any nonempty open where $f$ is invertible. Its projection $p : G \to Q$ is proper birational. The fibers $G_{0}, G_{\infty}$ are effective Cartier divisors, allowing the empty divisor, and $p_{*} ([G_{0}] - [G_{\infty}]) = {div}_{Q} (f) .$ Neither scheme is required to be separated over $k$ .

External input

Import exactly the norm and graph formulas of Stacks Lemmas 42.18.1 and 42.18.2. The field base with dimension function zero satisfies Situation 42.7.1. The formulas concern the actual length-weighted cycles; they do not assume a higher-Chow comparison or a localization theorem. These substantial divisor inputs are cited, not locally reconstructed.

Depends on

Used by · 1

Classical Chow relations and Bloch boundaries

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

Stacks Project, Principal divisors and proper pushforward · Situation 42.7.1 (02QL); Lemma 42.18.1 (02RT) or Lemma 42.18.2 (02RQ), respectively; live online tag text retrieved and inspected 2026-11-09; no pinned commit asserted

A dominant proper morphism between integral schemes of equal dimension over a locally noetherian universally catenary base with a dimension function pushes a principal divisor to the divisor of its function-field norm.

Hypotheses: Integral schemes locally of finite type over the base in Situation 42.7.1; the norm formula requires a dominant proper map of equal dimension. Here the base is Spec k with dimension function zero. No separatedness over k, normality or separability is required.

Reading limits: Read Situation 42.7.1 and both complete lemma statements and displayed proofs in the primary Stacks pages. The underlying commutative-algebra norm identity and cited foundational lemmas were not reconstructed.

Stacks Project, Graph formula for a principal divisor · Situation 42.7.1 (02QL); Lemma 42.18.1 (02RT) or Lemma 42.18.2 (02RQ), respectively; live online tag text retrieved and inspected 2026-11-09; no pinned commit asserted

The closure of the graph of an invertible rational function in the relative projective line is proper birational; its zero-minus-pole Cartier-divisor cycle pushes forward to the principal divisor.

Sources

Stacks Project, Principal divisors and proper pushforward · Situation 42.7.1 (02QL); Lemma 42.18.1 (02RT) or Lemma 42.18.2 (02RQ), respectively; live online tag text retrieved and inspected 2026-11-09; no pinned commit asserted
Stacks Project, Graph formula for a principal divisor · Situation 42.7.1 (02QL); Lemma 42.18.1 (02RT) or Lemma 42.18.2 (02RQ), respectively; live online tag text retrieved and inspected 2026-11-09; no pinned commit asserted