Theorems · Theorem · commutative algebra

IsIntegralClosure.tower_top

∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {B : Type u_6}
  {C : Type u_7} [inst_3 : CommSemiring C] [inst_4 : CommRing B] [inst_5 : Algebra R B] [inst_6 : Algebra A B]
  [inst_7 : Algebra C B] [IsScalarTower R A B] [IsIntegralClosure C R B] [Algebra.IsIntegral R A],
  IsIntegralClosure C A B

If R → A → B is an algebra tower, C is the integral closure of R in B and A is integral over R, then C is the integral closure of A in B.

Defined in

Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic

Cited by

1 results in Mathlib

Foundations

Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommSemiringCommRingAlgebraAlgebraAlgebraIsScalarTowerIsIntegralClosureAlgebra.IsIntegral