Theorems · Theorem · commutative algebra
RingHom.toKerIsLocalization_isLocalizedModule
∀ {R : Type u_1} {S : Type u_2} {P : Type u_3} (Q : Type u_4) [inst : CommSemiring R] [inst_1 : CommSemiring S]
[inst_2 : CommSemiring P] [inst_3 : CommSemiring Q] {M : Submonoid R} {T : Submonoid P} [inst_4 : Algebra R S]
[inst_5 : Algebra P Q] [inst_6 : IsLocalization M S] [inst_7 : IsLocalization T Q] (g : R →+* P)
(hT : Submonoid.map g M = T), IsLocalizedModule M (RingHom.toKerIsLocalization S Q g ⋯)The canonical linear map from the kernel of g to the kernel of its localization
is localizing. In other words, localization commutes with taking kernels.
- Defined in
- Mathlib.RingTheory.Localization.Algebra
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Moduleproof · cited by 20,661
- RingHom.idproof · cited by 18,349
- AddCommMonoidproof · cited by 12,281
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- LinearMapproof · cited by 10,215
- RingHomstatement and proof · cited by 10,189
- Idealstatement · cited by 4,748
- Algebra.algebraMapproof · cited by 4,706
- LinearEquivproof · cited by 3,317
- Submonoidstatement and proof · cited by 3,086
- LinearMap.compproof · cited by 1,642
Cited by1
Results whose statement or proof uses this declaration.
- AlgHom.toKerIsLocalization_isLocalizedModuleproof · cited by 0