Theorems · Definition · commutative algebra

IsIntegralClosure.lift

(R : Type u_1) →
  (A : Type u_2) →
    (B : Type u_3) →
      [inst : CommRing R] →
        [inst_1 : CommRing A] →
          [inst_2 : CommRing B] →
            [inst_3 : Algebra R B] →
              [inst_4 : Algebra A B] →
                [IsIntegralClosure A R B] →
                  {S : Type u_4} →
                    [inst_6 : CommRing S] →
                      [inst_7 : Algebra R S] →
                        [inst_8 : Algebra S B] →
                          [IsScalarTower R S B] →
                            [inst_10 : Algebra R A] →
                              [IsScalarTower R A B] → [isIntegral : Algebra.IsIntegral R S] → S →ₐ[R] A

If B / S / R is a tower of ring extensions where S is integral over R, then S maps (uniquely) into an integral closure B / A / R.

Defined in

Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic

Cited by

7 results in Mathlib

Foundations

Depth 138 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraAlgebraIsIntegralClosureCommRingAlgebraAlgebraIsScalarTowerAlgebraIsScalarTowerAlgebra.IsIntegral