Theorems · Theorem · commutative algebra
Module.IsTorsionFree.of_isLocalization
∀ (R : Type u_1) [inst : CommRing R] (S : Type u_2) [inst_1 : CommRing S] [inst_2 : Algebra R S] [IsDomain R]
[IsDomain S] {Rₚ : Type u_3} {Sₚ : Type u_4} [inst_5 : CommRing Rₚ] [IsDomain Rₚ] [inst_7 : CommRing Sₚ]
[inst_8 : Algebra R Rₚ] [inst_9 : Algebra R Sₚ] [inst_10 : Algebra S Sₚ] [inst_11 : Algebra Rₚ Sₚ]
[IsScalarTower R S Sₚ] [IsScalarTower R Rₚ Sₚ] {M : Submonoid R},
M ≤ nonZeroDivisors R →
∀ [IsLocalization M Rₚ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₚ] [Module.IsTorsionFree R S],
Module.IsTorsionFree Rₚ Sₚ- Defined in
- Mathlib.RingTheory.Localization.Ideal
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- SetLike.coeproof · cited by 8,199
- Algebra.algebraMapproof · cited by 4,706
- IsScalarTowerstatement and proof · cited by 3,896
- LE.le.transproof · cited by 3,151
- Submonoidstatement and proof · cited by 3,086
- IsDomainstatement and proof · cited by 2,196
- nonZeroDivisorsstatement and proof · cited by 895
- IsLocalizationstatement and proof · cited by 636
- Module.IsTorsionFreestatement and proof · cited by 600
Cited by0
Results whose statement or proof uses this declaration.
Nothing cites this yet.