Documentation

Mathlib.RingTheory.LocalRing.ResidueField.Basic

Residue Field of local rings #

We prove basic properties of the residue field of a local ring.

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theorem IsLocalRing.residue_def {R : Type u_1} [CommRing R] [IsLocalRing R] (x : R) :
(residue R) x = (Ideal.Quotient.mk (maximalIdeal R)) x
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theorem IsLocalRing.ker_residue {R : Type u_1} [CommRing R] [IsLocalRing R] :
RingHom.ker (residue R) = maximalIdeal R
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@[simp]
theorem IsLocalRing.residue_eq_zero_iff {R : Type u_1} [CommRing R] [IsLocalRing R] (x : R) :
(residue R) x = 0 ↔ x ∈ maximalIdeal R
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theorem IsLocalRing.residue_ne_zero_iff_isUnit {R : Type u_1} [CommRing R] [IsLocalRing R] (x : R) :
(residue R) x ≠ 0 ↔ IsUnit x
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theorem IsLocalRing.residue_surjective {R : Type u_1} [CommRing R] [IsLocalRing R] :
Function.Surjective ⇑(residue R)
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@[implicit_reducible]
instance IsLocalRing.ResidueField.algebra (R : Type u_1) [CommRing R] [IsLocalRing R] {R₀ : Type u_4} [CommRing R₀] [Algebra R₀ R] :
Algebra R₀ (ResidueField R)
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instance IsLocalRing.instIsScalarTowerResidueField (R : Type u_1) [CommRing R] [IsLocalRing R] {R₁ : Type u_4} {R₂ : Type u_5} [CommRing R₁] [CommRing R₂] [Algebra R₁ R₂] [Algebra R₁ R] [Algebra R₂ R] [IsScalarTower R₁ R₂ R] :
IsScalarTower R₁ R₂ (ResidueField R)
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@[simp]
theorem IsLocalRing.ResidueField.algebraMap_eq (R : Type u_1) [CommRing R] [IsLocalRing R] :
algebraMap R (ResidueField R) = residue R
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instance IsLocalRing.instIsLocalHomResidueFieldRingHomResidue (R : Type u_1) [CommRing R] [IsLocalRing R] :
IsLocalHom (residue R)
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@[implicit_reducible]
noncomputable instance IsLocalRing.instModuleResidueFieldOfAlgebra (R : Type u_1) [CommRing R] [IsLocalRing R] {R₀ : Type u_4} [CommRing R₀] [Algebra R₀ R] :
Module R₀ (ResidueField R)
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instance IsLocalRing.instFiniteResidueField (R : Type u_1) [CommRing R] [IsLocalRing R] {R₀ : Type u_4} [CommRing R₀] [Algebra R₀ R] [Module.Finite R₀ R] :
Module.Finite R₀ (ResidueField R)
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def IsLocalRing.ResidueField.lift {R : Type u_4} {S : Type u_5} [CommRing R] [IsLocalRing R] [Field S] (f : R →+* S) [IsLocalHom f] :
ResidueField R →+* S

A local ring homomorphism into a field can be descended onto the residue field.

Instances For
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    theorem IsLocalRing.ResidueField.lift_comp_residue {R : Type u_4} {S : Type u_5} [CommRing R] [IsLocalRing R] [Field S] (f : R →+* S) [IsLocalHom f] :
    (lift f).comp (residue R) = f
    source
    @[simp]
    theorem IsLocalRing.ResidueField.lift_residue_apply {R : Type u_4} {S : Type u_5} [CommRing R] [IsLocalRing R] [Field S] (f : R →+* S) [IsLocalHom f] (x : R) :
    (lift f) ((residue R) x) = f x
    source
    noncomputable def IsLocalRing.ResidueField.map {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R →+* S) [IsLocalHom f] :
    ResidueField R →+* ResidueField S

    The map on residue fields induced by a local homomorphism between local rings

    Instances For
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      @[simp]
      theorem IsLocalRing.ResidueField.map_id {R : Type u_1} [CommRing R] [IsLocalRing R] :
      map (RingHom.id R) = RingHom.id (ResidueField R)

      Applying IsLocalRing.ResidueField.map to the identity ring homomorphism gives the identity ring homomorphism.

      source
      theorem IsLocalRing.ResidueField.map_comp {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] (f : T →+* R) (g : R →+* S) [IsLocalHom f] [IsLocalHom g] :
      map (g.comp f) = (map g).comp (map f)

      The composite of two IsLocalRing.ResidueField.maps is the IsLocalRing.ResidueField.map of the composite.

      source
      theorem IsLocalRing.ResidueField.map_comp_residue {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R →+* S) [IsLocalHom f] :
      (map f).comp (residue R) = (residue S).comp f
      source
      @[simp]
      theorem IsLocalRing.ResidueField.map_residue {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R →+* S) [IsLocalHom f] (r : R) :
      (map f) ((residue R) r) = (residue S) (f r)
      source
      theorem IsLocalRing.ResidueField.map_id_apply {R : Type u_1} [CommRing R] [IsLocalRing R] (x : ResidueField R) :
      (map (RingHom.id R)) x = x
      source
      @[simp]
      theorem IsLocalRing.ResidueField.map_map {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] (f : R →+* S) (g : S →+* T) (x : ResidueField R) [IsLocalHom f] [IsLocalHom g] :
      (map g) ((map f) x) = (map (g.comp f)) x
      source
      noncomputable def IsLocalRing.ResidueField.mapEquiv {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R ≃+* S) :
      ResidueField R ≃+* ResidueField S

      A ring isomorphism defines an isomorphism of residue fields.

      Instances For
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        @[simp]
        theorem IsLocalRing.ResidueField.mapEquiv_apply {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R ≃+* S) (a : ResidueField R) :
        (mapEquiv f) a = (map ↑f) a
        source
        @[simp]
        theorem IsLocalRing.ResidueField.mapEquiv.symm {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R ≃+* S) :
        (mapEquiv f).symm = mapEquiv f.symm
        source
        @[simp]
        theorem IsLocalRing.ResidueField.mapEquiv_trans {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] (e₁ : R ≃+* S) (e₂ : S ≃+* T) :
        mapEquiv (e₁.trans e₂) = (mapEquiv e₁).trans (mapEquiv e₂)
        source
        @[simp]
        theorem IsLocalRing.ResidueField.mapEquiv_refl {R : Type u_1} [CommRing R] [IsLocalRing R] :
        mapEquiv (RingEquiv.refl R) = RingEquiv.refl (ResidueField R)
        source
        noncomputable def IsLocalRing.ResidueField.mapAut {R : Type u_1} [CommRing R] [IsLocalRing R] :
        RingAut R →* RingAut (ResidueField R)

        The group homomorphism from RingAut R to RingAut k where k is the residue field of R.

        Instances For
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          @[simp]
          theorem IsLocalRing.ResidueField.mapAut_apply {R : Type u_1} [CommRing R] [IsLocalRing R] (f : R ≃+* R) :
          mapAut f = mapEquiv f
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          @[implicit_reducible]
          noncomputable instance IsLocalRing.ResidueField.instMulSemiringAction {R : Type u_1} [CommRing R] [IsLocalRing R] (G : Type u_4) [Group G] [MulSemiringAction G R] :
          MulSemiringAction G (ResidueField R)

          If G acts on R as a MulSemiringAction, then it also acts on IsLocalRing.ResidueField R.

          source
          @[simp]
          theorem IsLocalRing.ResidueField.residue_smul {R : Type u_1} [CommRing R] [IsLocalRing R] (G : Type u_4) [Group G] [MulSemiringAction G R] (g : G) (r : R) :
          (residue R) (g • r) = g • (residue R) r
          source
          instance IsLocalRing.ResidueField.instLiesOverMaximalIdeal {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [Algebra R S] [IsLocalHom (algebraMap R S)] :
          (maximalIdeal S).LiesOver (maximalIdeal R)
          source
          @[implicit_reducible]
          instance IsLocalRing.ResidueField.instAlgebra {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [Algebra R S] [IsLocalHom (algebraMap R S)] :
          Algebra (ResidueField R) (ResidueField S)
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          @[simp]
          theorem IsLocalRing.ResidueField.algebraMap_residue {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [Algebra R S] [IsLocalHom (algebraMap R S)] (x : R) :
          (algebraMap (ResidueField R) (ResidueField S)) ((residue R) x) = (residue S) ((algebraMap R S) x)
          source
          instance IsLocalRing.ResidueField.instIsScalarTower {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [Algebra R S] [IsLocalHom (algebraMap R S)] {R₀ : Type u_4} [CommRing R₀] [Algebra R₀ R] [Algebra R₀ S] [IsScalarTower R₀ R S] :
          IsScalarTower R₀ (ResidueField R) (ResidueField S)
          source
          instance IsLocalRing.ResidueField.instIsScalarTower_1 {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [Algebra R S] [IsLocalHom (algebraMap R S)] {R₀ : Type u_4} [CommRing R₀] [Algebra R₀ R] [Algebra R₀ S] [IsScalarTower R₀ R S] [IsLocalRing R₀] [IsLocalHom (algebraMap R₀ R)] [IsLocalHom (algebraMap R₀ S)] :
          IsScalarTower (ResidueField R₀) (ResidueField R) (ResidueField S)
          source
          @[implicit_reducible]
          noncomputable instance IsLocalRing.ResidueField.instModule {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [Algebra R S] [IsLocalHom (algebraMap R S)] :
          Module (ResidueField R) (ResidueField S)
          source
          instance IsLocalRing.ResidueField.finite_of_module_finite {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [Algebra R S] [IsLocalHom (algebraMap R S)] [Module.Finite R S] :
          Module.Finite (ResidueField R) (ResidueField S)
          source
          theorem IsLocalRing.ResidueField.finite_of_finite {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [Algebra R S] [IsLocalHom (algebraMap R S)] [Module.Finite R S] (hfin : Finite (ResidueField R)) :
          Finite (ResidueField S)
          source
          noncomputable def IsLocalRing.ResidueField.mapAlgHom {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] [Algebra R S] [Algebra R T] (e : S →ₐ[R] T) [IsLocalHom e] :
          ResidueField S →ₐ[R] ResidueField T

          A local algebra homomorphism induces an algebra homomorphism on the residue fields.

          See mapAlgHom' for a variant where the base ring R is also quotiented.

          Instances For
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            @[simp]
            theorem IsLocalRing.ResidueField.mapAlgHom_residue {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] [Algebra R S] [Algebra R T] (e : S →ₐ[R] T) [IsLocalHom e] (x : S) :
            (mapAlgHom e) ((residue S) x) = (residue T) (e x)
            source
            noncomputable def IsLocalRing.ResidueField.mapAlgEquiv {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] [Algebra R S] [Algebra R T] (e : S ≃ₐ[R] T) :
            ResidueField S ≃ₐ[R] ResidueField T

            A local algebra isomorphism induces an algebra isomorphism on the residue fields.

            See mapAlgEquiv' for a variant where the base ring R is also quotiented.

            Instances For
              source
              @[simp]
              theorem IsLocalRing.ResidueField.mapAlgEquiv_residue {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] [Algebra R S] [Algebra R T] (e : S ≃ₐ[R] T) (x : S) :
              (mapAlgEquiv e) ((residue S) x) = (residue T) (e x)
              source
              noncomputable def IsLocalRing.ResidueField.mapAlgHom' {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] [Algebra R S] [Algebra R T] [IsLocalHom (algebraMap R S)] [IsLocalHom (algebraMap R T)] (e : S →ₐ[R] T) [IsLocalHom e] :
              ResidueField S →ₐ[ResidueField R] ResidueField T

              A local algebra homomorphism induces an algebra homomorphism on the residue fields.

              See mapAlgHom for a variant where the base ring R is not quotiented.

              Instances For
                source
                @[simp]
                theorem IsLocalRing.ResidueField.mapAlgHom'_residue {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] [Algebra R S] [Algebra R T] [IsLocalHom (algebraMap R S)] [IsLocalHom (algebraMap R T)] [IsLocalRing R] (e : S →ₐ[R] T) [IsLocalHom e] (x : S) :
                (mapAlgHom' e) ((residue S) x) = (residue T) (e x)
                source
                noncomputable def IsLocalRing.ResidueField.mapAlgEquiv' {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] [Algebra R S] [Algebra R T] [IsLocalHom (algebraMap R S)] [IsLocalHom (algebraMap R T)] (e : S ≃ₐ[R] T) :
                ResidueField S ≃ₐ[ResidueField R] ResidueField T

                A local algebra isomorphism induces an algebra isomorphism on the residue fields.

                See mapAlgEquiv for a variant where the base ring R is not quotiented.

                Instances For
                  source
                  @[simp]
                  theorem IsLocalRing.ResidueField.mapAlgEquiv'_residue {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] [Algebra R S] [Algebra R T] [IsLocalHom (algebraMap R S)] [IsLocalHom (algebraMap R T)] [IsLocalRing R] (e : S ≃ₐ[R] T) (x : S) :
                  (mapAlgEquiv' e) ((residue S) x) = (residue T) (e x)