isDedekindDomainDvr — Mathlib

English

If A is Dedekind, then it is also DedekindDvr in the sense of localizing at primes yields DVRs.

Русский

Если A Дедекендова, то она также Дедекенд DVR в смысле локализации по пprime даёт DVR.

LaTeX

$$$\text{IsDedekindDomainDvr } A \Rightarrow \text{IsDedekindDomain } A$$$

Lean4

/-- 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. -/
instance isDedekindDomainDvr [IsDedekindDomain A] : IsDedekindDomainDvr A where
  is_dvr_at_nonzero_prime := fun _ hP _ => IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domain A hP _

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