exists_fgAlgCatSkeleton — Mathlib

English

Every finitely generated R-algebra is Alg-equivalent to the evaluation of some FGAlgCatSkeleton, i.e., A ≅ P.eval.obj for some skeleton P.

Русский

Каждая конечно порожденная R-алгебра эквивалентна по Alg-эквивалентности вычислению некого FGAlgCatSkeleton: A ≅ P.eval.obj.

LaTeX

$$$\exists P\; \bigl(A \simeq_R P.\text{eval.obj}\bigr)$$$

Lean4

theorem exists_fgAlgCatSkeleton (A : Type v) [CommRing A] [Algebra R A] [h : Algebra.FiniteType R A] :
    ∃ (P : FGAlgCatSkeleton R), Nonempty (A ≃ₐ[R] P.eval.obj) :=
  by
  obtain ⟨n, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.mp h
  exact ⟨⟨n, RingHom.ker f⟩, ⟨(Ideal.quotientKerAlgEquivOfSurjective hf).symm⟩⟩

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