Documentation

Mathlib.RingTheory.DedekindDomain.Dvr

Dedekind domains #

This file defines an equivalent notion of a Dedekind domain (or Dedekind ring), namely a Noetherian integral domain where the localization at all nonzero prime ideals is a DVR (TODO: and shows that implies the main definition).

Main definitions #

Main results #

Implementation notes #

The definitions that involve a field of fractions choose a canonical field of fractions, but are independent of that choice. The ..._iff lemmas express this independence.

Often, definitions assume that Dedekind domains are not fields. We found it more practical to add a (h : ¬ IsField A) assumption whenever this is explicitly needed.

References #

Tags #

dedekind domain, dedekind ring

structure IsDedekindDomainDvr (A : Type u_2) [CommRing A] [IsDomain A] :

A Dedekind domain is an integral domain that is Noetherian, and the localization at every nonzero prime is a discrete valuation ring.

This is equivalent to IsDedekindDomain. TODO: prove the equivalence.

Instances For
    theorem Ring.DimensionLEOne.localization {R : Type u_4} (Rₘ : Type u_5) [CommRing R] [IsDomain R] [CommRing Rₘ] [Algebra R Rₘ] {M : Submonoid R} [IsLocalization M Rₘ] (hM : M nonZeroDivisors R) [h : Ring.DimensionLEOne R] :

    Localizing a domain of Krull dimension ≤ 1 gives another ring of Krull dimension ≤ 1.

    Note that the same proof can/should be generalized to preserving any Krull dimension, once we have a suitable definition.

    theorem IsLocalization.isDedekindDomain (A : Type u_2) [CommRing A] [IsDomain A] [IsDedekindDomain A] {M : Submonoid A} (hM : M nonZeroDivisors A) (Aₘ : Type u_4) [CommRing Aₘ] [IsDomain Aₘ] [Algebra A Aₘ] [IsLocalization M Aₘ] :

    The localization of a Dedekind domain is a Dedekind domain.

    The localization of a Dedekind domain at every nonzero prime ideal is a Dedekind domain.

    theorem IsLocalization.AtPrime.not_isField (A : Type u_2) [CommRing A] [IsDomain A] {P : Ideal A} (hP : P ) [pP : Ideal.IsPrime P] (Aₘ : Type u_4) [CommRing Aₘ] [Algebra A Aₘ] [IsLocalization.AtPrime Aₘ P] :

    In a Dedekind domain, the localization at every nonzero prime ideal is a DVR.

    Dedekind domains, in the sense of Noetherian integrally closed domains of Krull dimension ≤ 1, are also Dedekind domains in the sense of Noetherian domains where the localization at every nonzero prime ideal is a DVR.