Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.equivQuotientMapOfIsMaximal_symm_apply_mk
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)
[inst_3 : p.IsPrime] (Sₚ : Type u_4) [inst_4 : CommRing Sₚ] [inst_5 : Algebra S Sₚ]
[inst_6 : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] (P : Ideal S) [hPp : P.LiesOver p]
[inst_7 : P.IsMaximal] (x : S) (s : ↥(Algebra.algebraMapSubmonoid S p.primeCompl)),
(IsLocalization.AtPrime.equivQuotientMapOfIsMaximal p Sₚ P).symm
((Ideal.Quotient.mk (Ideal.map (algebraMap S Sₚ) P)) (IsLocalization.mk' Sₚ x s)) =
(Ideal.Quotient.mk P) x * ((Ideal.Quotient.mk P) ↑s)⁻¹- Cited by
- 1 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.
Cites38
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- mul_oneproof · cited by 3,885
- Submonoidstatement · cited by 3,086
- HasQuotient.Quotientstatement and proof · cited by 2,301
- mul_assocproof · cited by 1,667
- RingEquivstatement · cited by 1,147
- map_mulproof · cited by 1,137
Cited by1
Results whose statement or proof uses this declaration.
- IsLocalization.AtPrime.algebraMap_equivQuotMaximalIdeal_symm_applyproof · cited by 1