Theorems · Definition · ring theory

BialgEquiv.ofAlgEquiv

{R : Type u} →
  {A : Type v} →
    {B : Type w} →
      [inst : CommSemiring R] →
        [inst_1 : Semiring A] →
          [inst_2 : Semiring B] →
            [inst_3 : Bialgebra R A] →
              [inst_4 : Bialgebra R B] →
                (f : A ≃ₐ[R] B) →
                  (Bialgebra.counitAlgHom R B).comp ↑f = Bialgebra.counitAlgHom R A →
                    (Algebra.TensorProduct.map ↑f ↑f).comp (Bialgebra.comulAlgHom R A) =
                        (Bialgebra.comulAlgHom R B).comp ↑f →
                      A ≃ₐc[R] B

Construct a bialgebra equiv from an algebra equiv respecting counit and comultiplication.

Defined in

Mathlib.RingTheory.Bialgebra.Equiv

Cited by

3 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringSemiringSemiringBialgebraBialgebra