comp_surjective — Mathlib

English

Equivalences of finite presentation via mvPolynomial quotient representation.

Русский

Эквивалентности конечной презентации через представление через квоти mvPolynomial.

LaTeX

$$$\\text{FinitePresentation } R A \\iff \\exists ι (\\text{Fintype } ι) (f : \\mathrm{MvPolynomial } ι R \\to_A A), \\; \\text{Surjective } f \\wedge (\\ker f).FG$$$

Lean4

theorem comp_surjective {f : A →+* B} {g : B →+* C} (hf : f.FinitePresentation) (hg : Surjective g)
    (hker : (RingHom.ker g).FG) : (g.comp f).FinitePresentation :=
  by
  algebraize [f, g.comp f]
  exact
    Algebra.FinitePresentation.of_surjective (f :=
      { g with
        toFun := g
        commutes' := fun _ => rfl })
      hg hker

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