Theorems · Definition · commutative algebra

Algebra.FormallyUnramified.sec

(R : Type u_1) →
  (S : Type u_2) →
    [inst : CommRing R] →
      [inst_1 : CommRing S] →
        [inst_2 : Algebra R S] →
          (M : Type u_3) →
            [inst_3 : AddCommGroup M] →
              [inst_4 : Module R M] →
                [inst_5 : Module S M] →
                  [IsScalarTower R S M] →
                    [Algebra.FormallyUnramified R S] → [Algebra.EssFiniteType R S] → M →ₗ[S] TensorProduct R S M

Proposition I.2.3 of [iversen] If S is an unramified R-algebra, and M is an S-module, then the map S ⊗[R] M →ₗ[S] M taking (b, m) ↦ b • m admits an S-linear section.

Defined in

Mathlib.RingTheory.Unramified.Finite

Cited by

4 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraAddCommGroupModuleModuleIsScalarTowerAlgebra.FormallyUnramifiedAlgebra.EssFiniteType