Theorems · Theorem · ring theory

Subalgebra.centralizer_coe_image_includeLeft_eq_center_tensorProduct

∀ (R : Type u_1) [inst : CommSemiring R] (A : Type u_2) [inst_1 : Semiring A] [inst_2 : Algebra R A] (B : Type u_3)
  [inst_3 : Semiring B] [inst_4 : Algebra R B] (S : Set A) [Module.Free R B],
  Subalgebra.centralizer R (⇑Algebra.TensorProduct.includeLeft '' S) =
    (Algebra.TensorProduct.map (Subalgebra.centralizer R S).val (AlgHom.id R B)).range

Let R be a commutative ring and A, B be R-algebras where B is free as R-module. For any subset S ⊆ A, the centralizer of S ⊗ 1 ⊆ A ⊗ B is C_A(S) ⊗ B where C_A(S) is the centralizer of S in A.

Defined in

Mathlib.Algebra.Algebra.Subalgebra.Centralizer

Cited by

2 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringSemiringAlgebraSemiringAlgebraModule.Free