Theorems · Theorem · commutative algebra
FractionRing.isSeparable_of_isLocalization
∀ {R : Type u_1} (S : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : IsDomain R] [inst_3 : IsDomain S]
[inst_4 : Algebra R S] (Rₘ : Type u_4) (Sₘ : Type u_5) [inst_5 : CommRing Rₘ] [inst_6 : CommRing Sₘ]
[inst_7 : Algebra R Rₘ] [inst_8 : Module.IsTorsionFree R S] [Algebra.IsSeparable (FractionRing R) (FractionRing S)]
{M : Submonoid R} [IsLocalization M Rₘ] [inst_11 : Algebra Rₘ Sₘ] [inst_12 : Algebra S Sₘ] [inst_13 : Algebra R Sₘ]
[IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ]
[inst_17 : Algebra (FractionRing Rₘ) (FractionRing Sₘ)] [IsScalarTower Rₘ (FractionRing Rₘ) (FractionRing Sₘ)],
M ≤ nonZeroDivisors R → Algebra.IsSeparable (FractionRing Rₘ) (FractionRing Sₘ)- Cited by
- 0 results in Mathlib
- Foundations
- Depth 125 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites32
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- RingHom.idproof · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomproof · cited by 10,189
- Algebra.algebraMapproof · cited by 4,706
- IsScalarTowerstatement and proof · cited by 3,896
- Submonoidstatement and proof · cited by 3,086
- IsDomainstatement and proof · cited by 2,196
- nonZeroDivisorsstatement and proof · cited by 895
- IsFractionRingproof · cited by 738
- IsLocalizationstatement and proof · cited by 636
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