Theorems · Theorem · commutative algebra

Ideal.Quotient.exists_algHom_fixedPoint_quotient_under

∀ {A : Type u_1} {B : Type u_2} {k : Type u_3} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B]
  (G : Type u_4) [Finite G] [inst_4 : Group G] [inst_5 : MulSemiringAction G B] [Algebra.IsInvariant A B G]
  (P : Ideal A) (Q : Ideal B) [inst_7 : Q.LiesOver P] [inst_8 : CommRing k] [inst_9 : Algebra (A ⧸ P) k]
  [inst_10 : Algebra (B ⧸ Q) k] [IsScalarTower (A ⧸ P) (B ⧸ Q) k] [IsDomain k] [FaithfulSMul (B ⧸ Q) k]
  (σ : k →ₐ[A ⧸ P] k), ∃ τ, ∀ (x : B ⧸ Q), (algebraMap (B ⧸ Q) k) (τ x) = σ ((algebraMap (B ⧸ Q) k) x)

For any domain k containing B ⧸ Q, any endomorphism of k can be restricted to an endomorphism of B ⧸ Q. This is basically the fact that L/K normal implies κ(Q)/κ(P) normal in the Galois setting.

Defined in

Mathlib.RingTheory.Invariant.Basic

Cited by

1 results in Mathlib

Foundations

Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraFiniteGroupMulSemiringActionAlgebra.IsInvariantIdeal.LiesOverCommRingAlgebraAlgebraIsScalarTowerIsDomainFaithfulSMul