Theorems · Theorem · ring theory

Subalgebra.centralizer_coe_map_includeRight_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 : Subalgebra R B) [Module.Free R A],
  Subalgebra.centralizer R ↑(Subalgebra.map Algebra.TensorProduct.includeRight S) =
    (Algebra.TensorProduct.map (AlgHom.id R A) (Subalgebra.centralizer R ↑S).val).range

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

Defined in

Mathlib.Algebra.Algebra.Subalgebra.Centralizer

Cited by

1 results in Mathlib

Foundations

Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringSemiringAlgebraSemiringAlgebraModule.Free