Theorems · Theorem · commutative algebra

IsLocalization.liesOver_of_isPrime_of_disjoint

∀ {R : Type u_1} [inst : CommSemiring R] {S : Type u_4} [inst_1 : CommSemiring S] [inst_2 : Algebra R S] {R' : Type u_5}
  {S' : Type u_6} (M : Submonoid R) (T : Submonoid S) [inst_3 : CommSemiring R'] [inst_4 : CommSemiring S']
  [inst_5 : Algebra R R'] [inst_6 : Algebra S S'] [inst_7 : Algebra R' S'] [inst_8 : Algebra R S']
  [IsScalarTower R S S'] [IsScalarTower R R' S'] [IsLocalization M R'] [IsLocalization T S'] (p : Ideal R) {P : Ideal S}
  [P.IsPrime] [P.LiesOver p], Disjoint ↑T ↑P → (Ideal.map (algebraMap S S') P).LiesOver (Ideal.map (algebraMap R R') p)

If R' (resp. S') is the localization of R (resp. S) and P lies over p then the image of P in S' lies over the image of p in R'.

Defined in

Mathlib.RingTheory.Localization.AtPrime.Basic

Cited by

1 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraCommSemiringCommSemiringAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsLocalizationIsLocalizationIdeal.IsPrimeIdeal.LiesOver