Theorems · Theorem · commutative algebra

Algebra.IsFiniteSplit.exists_tensorProduct_of_etale

∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Algebra.Etale R S]
  [Module.Finite R S] {n : ℕ},
  Module.rankAtStalk S = ↑n →
    ∃ T x x_1,
      ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Algebra.Etale R T),
        Algebra.IsFiniteSplit T (TensorProduct R T S)

If S is finite étale over R of (constant) rank n, there exists a finite faithfully flat, étale R-algebra T such that T ⊗[R] S is split of rank n over T. This is the commutative algebra version of [Lenstra, Galois theory for schemes, 5.10][lenstraGSchemes].

Defined in

Mathlib.RingTheory.TotallySplit

Cited by

0 results in Mathlib

Foundations

Depth 137 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraAlgebra.EtaleModule.Finite