Theorems · Theorem · commutative algebra

IsLocalization.tensorProduct_tensorProduct

∀ (R : Type u_7) (S : Type u_8) [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Algebra R S] {A : Type u_9}
  [inst_3 : CommSemiring A] [inst_4 : Algebra R A] (M : Submonoid A) (B : Type u_10) [inst_5 : CommSemiring B]
  [inst_6 : Algebra R B] [inst_7 : Algebra A B] [inst_8 : IsScalarTower R A B] [IsLocalization M B]
  [inst_10 : Algebra (TensorProduct R A S) (TensorProduct R B S)]
  [IsScalarTower A (TensorProduct R A S) (TensorProduct R B S)],
  (algebraMap (TensorProduct R A S) (TensorProduct R B S)).comp Algebra.TensorProduct.includeRight.toRingHom =
      Algebra.TensorProduct.includeRight.toRingHom →
    IsLocalization (Algebra.algebraMapSubmonoid (TensorProduct R A S) M) (TensorProduct R B S)

A[M⁻¹] ⊗[R] S is the localization of A ⊗[R] S at M.

Defined in

Mathlib.RingTheory.Localization.BaseChange

Cited by

3 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraCommSemiringAlgebraCommSemiringAlgebraAlgebraIsScalarTowerIsLocalizationAlgebraIsScalarTower