DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Discrete valuations

Definition

A discrete valuation on a field $K$ is a valuation $v : K \to Z \cup {\infty}$ in the sense of Valuations on a field such that the restriction $v : K^{\times} \to Z$ is surjective.

The adjective "discrete" refers to the value group $Z$ with its usual order.

Depends on

Valuations on a field

Used by

DVR uniqueness need not detect nonseparatedness Counterexample
Two DVR lifts of one diagram over the doubled-origin line Counterexample
Discrete valuation rings Definition
Order codimension one rational function Definition
An open immersion has valuative uniqueness but not existence Example
Divisor of a rational function on the projective line Example
dvrs as regular local rings Example
Principal divisors on the projective line have degree zero Example
The twists on the projective line have degree n Example
Divisors on the projective line are classified by degree Lemma
Proper normal curve rational function map Lemma
The Picard group and intersection form of a product of projective lines Lemma
The Picard group of a point blowup of the projective plane Lemma
Valuation lifts detect universal closedness Lemma
Cartier divisors on a normal Noetherian scheme give Weil divisors Theorem
Local rings at closed points of smooth curves are discrete valuation rings Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Mustata, Commutative Algebra, Definition 8.7 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., (23.1) (standard reference, not scraped)