Theorems · Theorem · ring theory
IsAzumaya.mulLeftRight_comp_congr
∀ (R : Type u_1) (A : Type u_2) (B : Type u_3) [inst : CommSemiring R] [inst_1 : Ring A] [inst_2 : Ring B]
[inst_3 : Algebra R A] [inst_4 : Algebra R B] (e : A ≃ₐ[R] B),
(AlgHom.mulLeftRight R B).comp ↑(Algebra.TensorProduct.congr e (AlgEquiv.op e)) =
(↑(LinearEquiv.conjAlgEquiv R ↑e)).comp (AlgHom.mulLeftRight R A)The following diagram commutes:
``
e ⊗ eᵒᵖ
A ⊗ Aᵐᵒᵖ > B ⊗ Bᵐᵒᵖ
| |
| |
| mulLeftRight R A | mulLeftRight R B
| |
V V
End R A > End R B
e.conj
``
- Defined in
- Mathlib.Algebra.Azumaya.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites33
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Equivstatement · cited by 8,337
- Ringstatement and proof · cited by 7,463
- mul_oneproof · cited by 3,885
- AlgHomstatement · cited by 3,236
- one_mulproof · cited by 2,841
- TensorProductstatement · cited by 2,545
- AlgEquivstatement and proof · cited by 1,681
- TensorProduct.tmulproof · cited by 1,182
- map_mulproof · cited by 1,137
Cited by1
Results whose statement or proof uses this declaration.
- IsAzumaya.of_AlgEquivproof · cited by 0