Theorems · Definition · commutative algebra
Ideal.ResidueField.map
{R : Type u_1} →
{S : Type u_2} →
[inst : CommRing R] →
[inst_1 : CommRing S] →
(I : Ideal R) →
[inst_2 : I.IsPrime] →
(J : Ideal S) →
[inst_3 : J.IsPrime] → (f : R →+* S) → I = Ideal.comap f J → I.ResidueField →+* J.ResidueFieldIf I = f⁻¹(J), then there is a canonical embedding κ(I) ↪ κ(J).
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- RingHomstatement and proof · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.primeComplstatement · cited by 462
- Ideal.comapstatement and proof · cited by 443
- Localization.AtPrimestatement · cited by 299
- Ideal.ResidueFieldstatement · cited by 119
- Localization.localRingHomproof · cited by 54
- IsLocalRing.ResidueField.mapproof · cited by 16
Cited by9
Results whose statement or proof uses this declaration.
- Ideal.ResidueField.mapₐproof · cited by 21
- Ideal.ResidueField.map_algebraMapstatement and proof · cited by 2
- RingHom.SurjectiveOnStalks.residueFieldMap_bijectivestatement and proof · cited by 2
- Ideal.comap_fiberIsoOfBijectiveResidueField_symmproof · cited by 2
- Localization.exists_finite_awayMapₐ_of_surjective_awayMapₐproof · cited by 1
- Ideal.Fiber.lift_residueField_surjectiveproof · cited by 1
- Ideal.exists_not_mem_forall_mem_of_ne_of_liesOverproof · cited by 1
- Ideal.ResidueField.map.congr_simpstatement and proof · cited by 0
- Ideal.ResidueField.mapₐ_applystatement · cited by 0