English
For R-algebras A,B, the map sending an alg hom to its underlying linear map is linearly independent over B; i.e., AlgHom.toLinearMap is LI.
Русский
Для алгебраических гладких отображений соответствующая линейная карта является линейно независимой.
LaTeX
$$$\\text{LinearIndependent } B (AlgHom.toLinearMap : (A \\to_A[R] B) \\to A \\to_L[R] B)$$$
Lean4
/-- `MulSemiringAction.toAlgHom` is bijective. -/
theorem toAlgHom_bijective [Finite G] [FaithfulSMul G F] :
Function.Bijective (MulSemiringAction.toAlgHom _ _ : G → F →ₐ[subfield G F] F) :=
by
cases nonempty_fintype G
rw [Fintype.bijective_iff_injective_and_card]
constructor
· exact MulSemiringAction.toAlgHom_injective _ F
· apply le_antisymm
· exact Fintype.card_le_of_injective _ (MulSemiringAction.toAlgHom_injective _ F)
· rw [← finrank_eq_card G F]
exact LE.le.trans_eq (finrank_algHom _ F) (finrank_linearMap_self _ _ _)
