English
For any m, the map esymmAlgHom (Fin m) R n is injective provided the ambient ring is a commutative ring with unity.
Русский
Для любого m отображение esymmAlgHom (Fin m) R n инъективно при época кольца с единицей.
LaTeX
$$$\operatorname{Injective} \, (\operatorname{esymmAlgHom} (\operatorname{Fin} m)\, R\, n)$$$
Lean4
theorem esymmAlgHom_fin_injective (h : n ≤ m) : Function.Injective (esymmAlgHom (Fin m) R n) :=
by
rw [injective_iff_map_eq_zero]
refine fun p ↦ (fun hp ↦ ?_).mtr
rw [p.as_sum, map_sum (esymmAlgHom (Fin m) R n), ← Subalgebra.coe_eq_zero, AddSubmonoidClass.coe_finset_sum]
refine
sum_ne_zero_of_injOn_supDegree (D := toLex) (support_eq_empty.not.2 hp) (fun t ht ↦ ?_)
(fun t ht s hs he ↦ DFunLike.ext' <| accumulate_injective h ?_)
· rw [← esymmAlgHomMonomial, Ne, ← leadingCoeff_eq_zero toLex.injective, leadingCoeff_esymmAlgHomMonomial t h]
rwa [mem_support_iff] at ht
rw [mem_coe, mem_support_iff] at ht hs
dsimp only [Function.comp] at he
rwa [← esymmAlgHomMonomial, ← esymmAlgHomMonomial, ← ofLex_inj, DFunLike.ext'_iff,
supDegree_esymmAlgHomMonomial ht t h, supDegree_esymmAlgHomMonomial hs s h] at he
