Theorems · Theorem · commutative algebra

Algebra.FinitePresentation.of_surjective

∀ {R : Type w₁} {A : Type w₂} {B : Type w₃} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A]
  [inst_3 : CommRing B] [inst_4 : Algebra R B] {f : A →ₐ[R] B},
  Function.Surjective ⇑f →
    (RingHom.ker f.toRingHom).FG → ∀ [Algebra.FinitePresentation R A], Algebra.FinitePresentation R B

If f : A →ₐ[R] B is surjective with finitely generated kernel and A is finitely presented, then so is B.

Defined in

Mathlib.RingTheory.FinitePresentation

Cited by

3 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraAlgebra.FinitePresentation