of_finitePresentation — Mathlib

English

A finitely presented algebra is equivalent to a quotient of a polynomial algebra by a finitely generated ideal.

Русский

Финитно представляемое алгебраическое пространство эквивалентно фактор-пространству над полиномами по конечгенерированной идеалу.

LaTeX

$$$\\text{FinitePresentation } R A \\iff \\exists \\iota : \\text{Type},\\; \\iota\\text{ finite},\\; \\exists f : \\mathrm{MvPolynomial}\\, \\iota\\, R \\to\\!\u200b_{\! R} A,\\; \\text{Surjective } f \\wedge (\\ker f)\\!\\! . FG$$$

Lean4

/-- A finitely presented algebra is of finite type. -/
instance of_finitePresentation [FinitePresentation R A] : FiniteType R A :=
  by
  obtain ⟨n, f, hf⟩ := FinitePresentation.out (R := R) (A := A)
  apply FiniteType.iff_quotient_mvPolynomial''.2
  exact ⟨n, f, hf.1⟩

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