Theorems · Theorem · commutative algebra
IsLocalization.algebraMap_eq_map_map_submonoid
∀ {R : Type u_1} [inst : CommSemiring R] (M : Submonoid R) (S : Type u_2) [inst_1 : CommSemiring S]
[inst_2 : Algebra R S] (Rₘ : Type u_4) (Sₘ : Type u_5) [inst_3 : CommSemiring Rₘ] [inst_4 : CommSemiring Sₘ]
[inst_5 : Algebra R Rₘ] [inst_6 : IsLocalization M Rₘ] [inst_7 : Algebra S Sₘ]
[i : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [inst_8 : Algebra Rₘ Sₘ] [inst_9 : Algebra R Sₘ]
[IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ], algebraMap Rₘ Sₘ = IsLocalization.map Sₘ (algebraMap R S) ⋯If the square below commutes, the bottom map is uniquely specified:
``
R → S
↓ ↓
Rₘ → Sₘ
``
- Defined in
- Mathlib.RingTheory.Localization.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsScalarTowerstatement and proof · cited by 3,896
- Submonoidstatement and proof · cited by 3,086
- IsLocalizationstatement and proof · cited by 636
- Submonoid.comapstatement and proof · cited by 179
- Algebra.algebraMapSubmonoidstatement and proof · cited by 137
- IsScalarTower.algebraMap_applyproof · cited by 116
- IsLocalization.mapstatement · cited by 99
Cited by3
Results whose statement or proof uses this declaration.
- FractionalIdeal.coe_extendedHom_eq_spanproof · cited by 3
- FractionalIdeal.le_one_of_extendedHom_le_oneproof · cited by 2
- IsLocalization.algebraMap_apply_eq_map_map_submonoidproof · cited by 1