English
If a given solution M arises from generators and relations and satisfies the core presentation condition, then M is presented by that data; equivalently, the quotient of the free generator module by the identified relations is canonically isomorphic to M.
Русский
Если данное решение M получено как порожденники и отношения и удовлетворяет ядру презентации, то M имеет презентацию этой данными; эквивалентно, фактор-модуль свободной модуля на отношениях изоморфен M.
LaTeX
$$$\\exists e:\\ (\\text{relations.Quotient}) \\cong_A M$, i.e. there is a linear isomorphism between the quotient by the relations and the module $M$.$$
Lean4
theorem isPresentation {solution : relations.Solution M} (h : IsPresentationCore.{max u v w₀} solution) :
solution.IsPresentation where
bijective :=
by
let e : relations.Quotient ≃ₗ[A] M :=
LinearEquiv.ofLinear solution.fromQuotient ((down.{v} h).desc (ofQuotient relations))
((down.{max u w₀} h).postcomp_injective (by aesop)) (by aesop)
exact e.bijective
