Discrete valuation rings

Depends on

• Discrete valuations
• Valuation rings

Used by

• The affine line with doubled origin is not separated Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• DVR uniqueness need not detect nonseparatedness Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• Two DVR lifts of one diagram over the doubled-origin line Counterexample
• A p-modular system is a characteristic-zero fraction field over a complete discrete valuation ring with residue field of characteristic p Definition
• Good reduction of an abelian variety over a Dedekind scheme Definition
• Order codimension one rational function Definition
• Principal parts of an invertible sheaf on a curve Definition
• Ramification index of a morphism of curves Definition
• The local ring A₀ of rational functions regular at q=0 Definition
• Uniformising parameters Definition
• Uniformising parameters at smooth points of a plane curve 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 p-adic valuation ring Example
• The twists on the projective line have degree n Example
• A meromorphic unit has locally finite nonzero order support Lemma
• DVR cellular Jantzen filtration and exact Gram valuation Lemma
• Intersection with a line is the order of vanishing of the restricted equation Lemma
• Intersection with a smooth curve is a vanishing order Lemma
• Laurent binomial valuation at one over any field Lemma
• Local q=1 deformation proves integrable highest weight universality Lemma
• Prime-to-p roots lift uniquely in a complete DVR Lemma
• Proper normal curve rational function map Lemma
• Rational maps from a smooth curve to a proper scheme are morphisms Lemma
• Rational-local quantum halves and torus specialize at q=1 Lemma
• The order function of a one-dimensional Noetherian local domain Lemma
• Valuation lifts detect universal closedness Lemma
• The valuative criterion quantifies over all valuation rings Remark
• A normal curve over a perfect field is nonsingular Theorem
• Cartier divisors on a normal Noetherian scheme give Weil divisors Theorem
• Every nonzero fraction is a unit times a power of a uniformiser Theorem
• Existence of Neron models for abelian varieties over a discrete valuation ring Theorem
• Irreducible Brauer characters are independent on p-regular elements Theorem
• Local rings at closed points of smooth curves are discrete valuation rings Theorem
• The Neron-Ogg-Shafarevich criterion in residue characteristic prime to l Theorem
• Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme Theorem

Dependency tree · two levels

3 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 8.8 (standard reference, not scraped)
• Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., (23.1) (standard reference, not scraped)