Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.liesOver_map_of_liesOver
∀ {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] (Rₚ : Type u_3) [inst_4 : CommRing Rₚ] [inst_5 : Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p]
[inst_7 : IsLocalRing Rₚ] (Sₚ : Type u_4) [inst_8 : CommRing Sₚ] [inst_9 : Algebra S Sₚ]
[IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] [inst_11 : Algebra Rₚ Sₚ] (P : Ideal S)
[hPp : P.LiesOver p] [inst_12 : Algebra R Sₚ] [IsScalarTower R S Sₚ] [IsScalarTower R Rₚ Sₚ] [P.IsPrime],
(Ideal.map (algebraMap S Sₚ) P).LiesOver (IsLocalRing.maximalIdeal Rₚ)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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
- 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
- IsScalarTowerstatement and proof · cited by 3,896
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.mapstatement and proof · cited by 692
- IsLocalizationstatement and proof · cited by 636
- Ideal.primeComplstatement and proof · cited by 462
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement and proof · cited by 297
Cited by2
Results whose statement or proof uses this declaration.
- IsLocalization.AtPrime.ramificationIdx_map_eq_ramificationIdxproof · cited by 1
- IsDedekindDomain.primesOverEquivPrimesOver_inertiagDeg_eqproof · cited by 0