Theorems · Definition · commutative algebra

IsFractionRing.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] →
                            (K : Type u_4) →
                              (L : Type u_5) →
                                [inst_9 : Field K] →
                                  [inst_10 : Field L] →
                                    [inst_11 : Algebra (A ⧸ P) K] →
                                      [inst_12 : Algebra (B ⧸ Q) L] →
                                        [inst_13 : Algebra (A ⧸ P) L] →
                                          [IsScalarTower (A ⧸ P) (B ⧸ Q) L] →
                                            [inst_15 : Algebra K L] →
                                              [IsScalarTower (A ⧸ P) K L] →
                                                [Algebra.IsInvariant A B G] →
                                                  [IsFractionRing (A ⧸ P) K] →
                                                    [IsFractionRing (B ⧸ Q) L] →
                                                      ↥(MulAction.stabilizer G Q) ⧸
                                                          Ideal.inertia (↥(MulAction.stabilizer G Q)) Q ≃*
                                                        Gal(L/K)

The isomorphism between stabilizer G Q ⧸ inertia G Q and the Galois group of the residue fields.

Defined in

Mathlib.RingTheory.Invariant.Basic

Cited by

2 results in Mathlib

Foundations

Depth 145 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraGroupFiniteMulSemiringActionSMulCommClassIdeal.IsPrimeIdeal.LiesOverFieldFieldAlgebraAlgebraAlgebraIsScalarTowerAlgebraIsScalarTowerAlgebra.IsInvariantIsFractionRingIsFractionRing