English
A full version linking eq_of_fg_of_subtype_eq' with broader linear-equivalence statements within the tensor product framework.
Русский
Полная версия, соединяющая eq_of_fg_of_subtype_eq' с общими линейно-эквивалентностями в рамке тензорного произведения.
LaTeX
$$$\\text{eq_of_fg_of_subtype_eq'} \\text{ in full generality with linear equivalences.}$$$
Lean4
theorem exists_of_fg : ∃ (A : Subalgebra R S), Subalgebra.FG A ∧ u ∈ range (rTensor N A.val.toLinearMap) :=
by
obtain ⟨P, ⟨s, hs⟩, hu⟩ := TensorProduct.exists_of_fg u
use Algebra.adjoin R s, Subalgebra.fg_adjoin_finset _
have : P ≤ (Algebra.adjoin R (s : Set S)).toSubmodule :=
by
simp only [← hs, span_le, Subalgebra.coe_toSubmodule]
exact Algebra.subset_adjoin
rw [← subtype_comp_inclusion P _ this, rTensor_comp] at hu
exact range_comp_le_range _ _ hu
