normalizedTrace_comp_algHom — Mathlib

English

If f: E →_F K is an algebra homomorphism, then the normalized trace commutes with f: Tr_{F}^{K} ∘ f = Tr_{F}^{E}.

Русский

Если f: E →_F K — алгебраическая линейная гомоморфия, то нормализованный след коммутирует с f: Tr_{F}^{K} ∘ f = Tr_{F}^{E}.

LaTeX

$$$\\forall f:\\; E \\to_{F} K :\\; \\\\operatorname{normalizedTrace}_{F}^{K} \circ f = \\\\operatorname{normalizedTrace}_{F}^{E}$$$

Lean4

/-- The normalized trace commutes with (injective) maps. -/
@[simp]
theorem normalizedTrace_comp_algHom {E : Type*} [Field E] [Algebra F E] [Algebra.IsIntegral F E] (f : E →ₐ[F] K) :
    normalizedTrace F K ∘ₗ f = normalizedTrace F E :=
  LinearMap.ext <| normalizedTrace_map F K f

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