Theorems · Definition · commutative algebra

IsFractionRing.stabilizerHom

{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] →
              [inst_4 : MulSemiringAction G B] →
                [SMulCommClass G A B] →
                  (P : Ideal A) →
                    (Q : Ideal B) →
                      [inst_6 : Q.LiesOver P] →
                        (K : Type u_4) →
                          (L : Type u_5) →
                            [inst_7 : Field K] →
                              [inst_8 : Field L] →
                                [inst_9 : Algebra (A ⧸ P) K] →
                                  [inst_10 : Algebra (B ⧸ Q) L] →
                                    [inst_11 : Algebra (A ⧸ P) L] →
                                      [IsScalarTower (A ⧸ P) (B ⧸ Q) L] →
                                        [inst_13 : Algebra K L] →
                                          [IsScalarTower (A ⧸ P) K L] →
                                            [IsFractionRing (A ⧸ P) K] →
                                              [IsFractionRing (B ⧸ Q) L] → ↥(MulAction.stabilizer G Q) →* Gal(L/K)

If Q lies over P, then the stabilizer of Q acts on Frac(B/Q)/Frac(A/P).

Defined in

Mathlib.RingTheory.Invariant.Basic

Cited by

8 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraGroupMulSemiringActionSMulCommClassIdeal.LiesOverFieldFieldAlgebraAlgebraAlgebraIsScalarTowerAlgebraIsScalarTowerIsFractionRingIsFractionRing