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Every DVR is a PID
Statement
Every discrete valuation ring is a principal ideal domain.
Facts & Assumptions
Given: A discrete valuation ring .
Every nonzero ideal of a DVR is a power of its maximal ideal, hence principal (Ideals in a DVR are powers of the maximal ideal).
A principal ideal domain is an integral domain in which every ideal is principal (Principal ideal domain).
Proof
A discrete valuation ring is a domain. Its zero ideal is principal, and [L1] shows that every nonzero ideal is principal.
Therefore every ideal of the domain is principal, so [F1] makes a principal ideal domain.
Depends on
Used by
- The degree of a divisor descends to the Picard group of a normal proper curve Corollary
- Degree of a nonconstant morphism of curves Definition
- Divisors on a smooth proper curve Definition
- Ramification points, branch points and unramifiedness Definition
- The local ring A₀ of rational functions regular at q=0 Definition
- A point blowup drops pairwise intersection multiplicity by at least one Lemma
- Connected smooth quasiprojective model with proper special fibre is proper Lemma
- Dilatations and defect computation Lemma
- Fibres, pullbacks and degrees of divisors under a finite morphism of curves Lemma
- Strict henselization of a DVR and smooth sections Lemma
- Torsion-free coherent modules on a smooth curve are locally free Lemma
- Canonical bundle formula with the different Theorem
- Cartier and Weil divisors agree on a smooth curve Theorem
Dependency tree · two levels
6 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
- J. S. Milne, A Primer of Commutative Algebra, v4.03, Proposition 20.2 (standard reference, not scraped)
- Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., (23.1) (standard reference, not scraped)