Theorems · Definition · commutative algebra

Algebra.TensorProduct.quotientTensorEquiv

{R : Type u_1} →
  (S : Type u_2) →
    (T : Type u_3) →
      (A : Type u_4) →
        [inst : CommRing R] →
          [inst_1 : CommRing S] →
            [inst_2 : Algebra R S] →
              [inst_3 : CommRing T] →
                [inst_4 : Algebra R T] →
                  [inst_5 : CommRing A] →
                    [inst_6 : Algebra R A] →
                      [inst_7 : Algebra S A] →
                        [inst_8 : IsScalarTower R S A] →
                          (I : Ideal A) →
                            TensorProduct R (A ⧸ I) T ≃ₐ[S]
                              TensorProduct R A T ⧸ Ideal.map (algebraMap A (TensorProduct R A T)) I

The tensor product over R of the quotient of an S-algebra A by an ideal I with T is isomorphic (as an S-algebra) to the quotient of A ⊗[R] T by the extended ideal.

Defined in

Mathlib.RingTheory.TensorProduct.Quotient

Cited by

2 results in Mathlib

Foundations

Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraCommRingAlgebraAlgebraIsScalarTower