restrictScalars — Mathlib

English

Given a tower R ⟶ S ⟶ A and a Subalgebra U ⊆ S acting on A, one can regard U as an R-subalgebra of A by restricting scalars.

Русский

Учитывая башню R ⟶ S ⟶ A и подалгебру U ⊆ S acting на A, можно рассматривать U как R-подалгебру A через ограничение скаляров.

LaTeX

$$$U^{\uparrow S} \restriction_R A$$

Lean4

/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret
`U` as an `R`-subalgebra of `A`. -/
def restrictScalars (U : Subalgebra S A) : Subalgebra R A :=
  { U with
    algebraMap_mem' := fun x ↦ by
      rw [IsScalarTower.algebraMap_apply R S A]
      exact U.algebraMap_mem _ }

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