Theorems · Definition · commutative algebra

Ideal.Quotient.stabilizerQuotientInertiaEquiv

{A : Type u_1} →
  {B : Type u_2} →
    [inst : CommRing A] →
      [inst_1 : CommRing B] →
        [inst_2 : Algebra A B] →
          (G : Type u_3) →
            [inst_3 : Group G] →
              [Finite G] →
                [inst_5 : MulSemiringAction G B] →
                  [SMulCommClass G A B] →
                    (P : Ideal A) →
                      (Q : Ideal B) →
                        [Q.IsPrime] →
                          [inst_8 : Q.LiesOver P] →
                            [Algebra.IsInvariant A B G] →
                              ↥(MulAction.stabilizer G Q) ⧸ Ideal.inertia (↥(MulAction.stabilizer G Q)) Q ≃*
                                (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q

The isomorphism between stabilizer G Q ⧸ inertia G Q and the Galois group of the residue fields extension B ⧸ Q over A ⧸ P.

Defined in

Mathlib.RingTheory.Invariant.Basic

Cited by

1 results in Mathlib

Foundations

Depth 146 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraGroupFiniteMulSemiringActionSMulCommClassIdeal.IsPrimeIdeal.LiesOverAlgebra.IsInvariant