Theorems · Theorem · commutative algebra

surjective_of_isLocalization_isMaximal

∀ {R : Type u_1} {S : Type u_2} [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Algebra R S]
  (Rₚ : (p : Ideal R) → [p.IsMaximal] → Type u_3) [inst_3 : (p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Rₚ p)]
  [inst_4 : (p : Ideal R) → [inst_4 : p.IsMaximal] → Algebra R (Rₚ p)] (Sₚ : (p : Ideal R) → [p.IsMaximal] → Type u_4)
  [inst_5 : (p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Sₚ p)]
  [inst_6 : (p : Ideal R) → [inst : p.IsMaximal] → Algebra S (Sₚ p)]
  [inst_7 : (p : Ideal R) → [inst : p.IsMaximal] → Algebra (Rₚ p) (Sₚ p)]
  [inst_8 : (p : Ideal R) → [inst_8 : p.IsMaximal] → Algebra R (Sₚ p)]
  [∀ (p : Ideal R) [inst_9 : p.IsMaximal], IsScalarTower R (Rₚ p) (Sₚ p)]
  [inst_10 : ∀ (p : Ideal R) [inst_10 : p.IsMaximal], IsScalarTower R S (Sₚ p)]
  [∀ (p : Ideal R) [inst_11 : p.IsMaximal], IsLocalization.AtPrime (Rₚ p) p]
  [∀ (p : Ideal R) [inst_12 : p.IsMaximal], IsLocalizedModule.AtPrime p ↑(IsScalarTower.toAlgHom R S (Sₚ p))],
  (∀ (p : Ideal R) [inst : p.IsMaximal], Function.Surjective ⇑(algebraMap (Rₚ p) (Sₚ p))) →
    Function.Surjective ⇑(algebraMap R S)

Defined in

Mathlib.RingTheory.LocalProperties.Exactness

Cited by

2 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraCommSemiringAlgebraCommSemiringAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsLocalization.AtPrimeIsLocalizedModule.AtPrime