mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
inst✝² : IsInvariant A B G
P Q : Ideal B
inst✝¹ : P.IsPrime
inst✝ : Q.IsPrime
hPQ : Ideal.under A P = Ideal.under A Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
N N' : OpenNormalSubgroup G
f : N ⟶ N'
x : { g // Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) Q = g • Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) P }
⊢ Ideal.under (↥(B' ↥↑N'.toOpenSubgroup)) Q =
    (QuotientGroup.map (↑N.toOpenSubgroup) N'.toOpenSubgroup.1 (MonoidHom.id G) ⋯) ↑x •
      Ideal.under (↥(B' ↥↑N'.toOpenSubgroup)) P

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
ι : Type u_3
f : ι → S
h✝ : Ideal.span (Set.range f) = ⊤
T : ι → Type u_4
inst✝⁵ : (i : ι) → CommSemiring (T i)
inst✝⁴ : (i : ι) → Algebra R (T i)
inst✝³ : (i : ι) → Algebra S (T i)
inst✝² : ∀ (i : ι), IsScalarTower R S (T i)
inst✝¹ : ∀ (i : ι), IsLocalization.Away (f i) (T i)
inst✝ : ∀ (i : ι), IsLocalIso R (T i)
q : Ideal S
hq : q.IsPrime
i : ι
hi : f i ∉ q
q' : PrimeSpectrum (T i)
hq' : PrimeSpectrum.comap (algebraMap S (T i)) q' = { asIdeal := q, isPrime := hq }
g' : T i
hg' : g' ∉ q'.asIdeal
h : IsStandardOpenImmersion R (Localization.Away g')
n : ℕ
g : S
hg : g' * (algebraMap S (T i)) (f i) ^ n = (algebraMap S (T i)) g
⊢ IsLocalization.Away g' (Localization.Away ((algebraMap S (T i)) g))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Injective

Goal state before:
pp: R : Type u
inst✝¹⁴ : CommRing R
M : Type v
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
S : Submonoid R
inst✝¹¹ : Small.{v, u} R
inst✝¹⁰ : IsNoetherianRing R
Rₛ : Type u'
inst✝⁹ : Small.{v', u'} Rₛ
inst✝⁸ : CommRing Rₛ
inst✝⁷ : Algebra R Rₛ
Mₛ : Type v'
inst✝⁶ : AddCommGroup Mₛ
inst✝⁵ : Module R Mₛ
inst✝⁴ : Module Rₛ Mₛ
inst✝³ : IsScalarTower R Rₛ Mₛ
f : M →ₗ[R] Mₛ
inst✝² : IsLocalization S Rₛ
inst✝¹ : IsLocalizedModule S f
inst✝ : Injective R M
⊢ Injective Rₛ Mₛ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
inst✝² : IsInvariant A B G
P Q : Ideal B
inst✝¹ : P.IsPrime
inst✝ : Q.IsPrime
hPQ : Ideal.under A P = Ideal.under A Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
N : OpenNormalSubgroup G
⊢ (↾fun x ↦ ⟨(QuotientGroup.map (↑N.toOpenSubgroup) N.toOpenSubgroup.1 (MonoidHom.id G) ⋯) ↑x, ⋯⟩) =
    𝟙 { g // Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) Q = g • Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) P }

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
ι : Type u_3
f : ι → S
h : Ideal.span (Set.range f) = ⊤
T : ι → Type u_4
inst✝⁵ : (i : ι) → CommSemiring (T i)
inst✝⁴ : (i : ι) → Algebra R (T i)
inst✝³ : (i : ι) → Algebra S (T i)
inst✝² : ∀ (i : ι), IsScalarTower R S (T i)
inst✝¹ : ∀ (i : ι), IsLocalization.Away (f i) (T i)
inst✝ : ∀ (i : ι), IsLocalIso R (T i)
⊢ IsLocalIso R S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
s : Set S
h✝ : Ideal.span s = ⊤
h : ∀ x ∈ s, IsLocalIso R (Localization.Away x)
⊢ Ideal.span (Set.range fun i ↦ ↑i) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
inst✝² : IsInvariant A B G
P Q : Ideal B
inst✝¹ : P.IsPrime
inst✝ : Q.IsPrime
hPQ : Ideal.under A P = Ideal.under A Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
X✝ Y✝ Z✝ : OpenNormalSubgroup G
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ (↾fun x ↦ ⟨(QuotientGroup.map (↑X✝.toOpenSubgroup) Z✝.toOpenSubgroup.1 (MonoidHom.id G) ⋯) ↑x, ⋯⟩) =
    (↾fun x ↦ ⟨(QuotientGroup.map (↑X✝.toOpenSubgroup) Y✝.toOpenSubgroup.1 (MonoidHom.id G) ⋯) ↑x, ⋯⟩) ≫
      ↾fun x ↦ ⟨(QuotientGroup.map (↑Y✝.toOpenSubgroup) Z✝.toOpenSubgroup.1 (MonoidHom.id G) ⋯) ↑x, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
s : Set S
h✝ : Ideal.span s = ⊤
h : ∀ x ∈ s, IsLocalIso R (Localization.Away x)
⊢ IsLocalIso R S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Injective

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
H : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)
I : Ideal R
x✝¹ : FinitePresentation R ↥I := finitePresentation_of_finite R ↥I
m : Ideal R
x✝ : m.IsMaximal
Rₘ : Type u := Localization.AtPrime m
Mₘ : Type (max u v) := LocalizedModule m.primeCompl M
f : M →ₗ[R] LocalizedModule m.primeCompl M := LocalizedModule.mkLinearMap m.primeCompl M
h : R →ₗ[R] Rₘ := Algebra.linearMap R Rₘ
Iₘ : Ideal Rₘ := Submodule.localized' Rₘ m.primeCompl h I
g : ↥I →ₗ[R] ↥Iₘ := Submodule.toLocalized' Rₘ m.primeCompl h I
gM : (↥I →ₗ[R] M) →ₗ[R] ↥Iₘ →ₗ[Rₘ] LocalizedModule m.primeCompl M :=
  IsLocalizedModule.mapExtendScalars m.primeCompl g f Rₘ
hM : (R →ₗ[R] M) →ₗ[R] Rₘ →ₗ[Rₘ] LocalizedModule m.primeCompl M :=
  IsLocalizedModule.mapExtendScalars m.primeCompl h f Rₘ
⊢ ↑R (Submodule.subtype Iₘ) ∘ₗ g = (IsLocalizedModule.map m.primeCompl g h) (Submodule.subtype I) ∘ₗ g

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: ι : Type u_3
R : Type u_4
S : ι → Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : (i : ι) → CommRing (S i)
inst✝² : (i : ι) → Algebra R (S i)
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), IsLocalIso R (S i)
this : (i : ι) → Algebra ((i : ι) → S i) (S i) := fun i ↦ (Pi.evalAlgHom R S i).toAlgebra
i : ι
⊢ IsLocalization.Away (Pi.single i 1) (S i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: ι : Type u_3
R : Type u_4
S : ι → Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : (i : ι) → CommRing (S i)
inst✝² : (i : ι) → Algebra R (S i)
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), IsLocalIso R (S i)
⊢ IsLocalIso R ((i : ι) → S i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Injective

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
H : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)
I : Ideal R
x✝¹ : FinitePresentation R ↥I := finitePresentation_of_finite R ↥I
m : Ideal R
x✝ : m.IsMaximal
Rₘ : Type u := Localization.AtPrime m
Mₘ : Type (max u v) := LocalizedModule m.primeCompl M
f : M →ₗ[R] LocalizedModule m.primeCompl M := LocalizedModule.mkLinearMap m.primeCompl M
h : R →ₗ[R] Rₘ := Algebra.linearMap R Rₘ
Iₘ : Ideal Rₘ := Submodule.localized' Rₘ m.primeCompl h I
g : ↥I →ₗ[R] ↥Iₘ := Submodule.toLocalized' Rₘ m.primeCompl h I
gM : (↥I →ₗ[R] M) →ₗ[R] ↥Iₘ →ₗ[Rₘ] LocalizedModule m.primeCompl M :=
  IsLocalizedModule.mapExtendScalars m.primeCompl g f Rₘ
hM : (R →ₗ[R] M) →ₗ[R] Rₘ →ₗ[Rₘ] LocalizedModule m.primeCompl M :=
  IsLocalizedModule.mapExtendScalars m.primeCompl h f Rₘ
eq'' : ↑R (Submodule.subtype Iₘ) = (IsLocalizedModule.map m.primeCompl g h) (Submodule.subtype I)
l : R →ₗ[R] M
⊢ ↑R ((↑R (LinearMap.lcomp Rₘ Mₘ (Submodule.subtype Iₘ)) ∘ₗ hM) l) ∘ₗ g =
    ↑R (((IsLocalizedModule.map m.primeCompl hM gM) (LinearMap.lcomp R M (Submodule.subtype I)) ∘ₗ hM) l) ∘ₗ g

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
inst✝² : IsInvariant A B G
P Q : Ideal B
inst✝¹ : P.IsPrime
inst✝ : Q.IsPrime
hPQ : Ideal.under A P = Ideal.under A Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
F : OpenNormalSubgroup G ⥤ Type u :=
  { obj := fun N ↦ { g // Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) Q = g • Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) P },
    map := fun {N N'} f ↦
      ↾fun x ↦ ⟨(QuotientGroup.map (↑N.toOpenSubgroup) N'.toOpenSubgroup.1 (MonoidHom.id G) ⋯) ↑x, ⋯⟩,
    map_id := ⋯, map_comp := ⋯ }
N : OpenNormalSubgroup G
⊢ Nonempty (F.obj N)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Injective

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
H : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)
I : Ideal R
x✝¹ : FinitePresentation R ↥I := finitePresentation_of_finite R ↥I
m : Ideal R
x✝ : m.IsMaximal
Rₘ : Type u := Localization.AtPrime m
Mₘ : Type (max u v) := LocalizedModule m.primeCompl M
f : M →ₗ[R] LocalizedModule m.primeCompl M := LocalizedModule.mkLinearMap m.primeCompl M
h : R →ₗ[R] Rₘ := Algebra.linearMap R Rₘ
Iₘ : Ideal Rₘ := Submodule.localized' Rₘ m.primeCompl h I
g : ↥I →ₗ[R] ↥Iₘ := Submodule.toLocalized' Rₘ m.primeCompl h I
gM : (↥I →ₗ[R] M) →ₗ[R] ↥Iₘ →ₗ[Rₘ] LocalizedModule m.primeCompl M :=
  IsLocalizedModule.mapExtendScalars m.primeCompl g f Rₘ
hM : (R →ₗ[R] M) →ₗ[R] Rₘ →ₗ[Rₘ] LocalizedModule m.primeCompl M :=
  IsLocalizedModule.mapExtendScalars m.primeCompl h f Rₘ
eq'' : ↑R (Submodule.subtype Iₘ) = (IsLocalizedModule.map m.primeCompl g h) (Submodule.subtype I)
eq' :
  ↑R (LinearMap.lcomp Rₘ Mₘ (Submodule.subtype Iₘ)) =
    (IsLocalizedModule.map m.primeCompl hM gM) (LinearMap.lcomp R M (Submodule.subtype I))
⊢ LinearMap.lcomp Rₘ Mₘ (Submodule.subtype Iₘ) =
    (IsLocalizedModule.mapExtendScalars m.primeCompl hM gM Rₘ) (LinearMap.lcomp R M (Submodule.subtype I))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalIso R S
inst✝ : IsLocalIso S T
s : Set S := {g | IsStandardOpenImmersion R (Localization.Away g)}
T' : S → Type u_3 := fun g ↦ Localization.Away ((algebraMap S T) g)
this✝ : (g : S) → Algebra (Localization.Away g) (T' g) := fun g ↦ localizationAlgebra (Submonoid.powers g) T
T'' : S → T → Type u_3 := fun g x ↦ Localization.Away ((algebraMap T (T' g)) x)
t : S → Set T := fun g ↦ {x | IsStandardOpenImmersion (Localization.Away g) (T'' g x)}
ι : Type (max u_3 u_2) := (i : ↑s) × ↑(t ↑i)
this : ∀ (i : ι), IsStandardOpenImmersion (Localization.Away ↑i.fst) (T'' ↑i.fst ↑i.snd)
h : Ideal.span (Set.range fun i ↦ (algebraMap S T) ↑i.fst * ↑i.snd) = ⊤
⊢ IsLocalIso R T

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalIso R S
inst✝ : IsLocalIso S T
s : Set S := {g | IsStandardOpenImmersion R (Localization.Away g)}
T' : S → Type u_3 := fun g ↦ Localization.Away ((algebraMap S T) g)
this✝ : (g : S) → Algebra (Localization.Away g) (T' g) := fun g ↦ localizationAlgebra (Submonoid.powers g) T
T'' : S → T → Type u_3 := fun g x ↦ Localization.Away ((algebraMap T (T' g)) x)
t : S → Set T := fun g ↦ {x | IsStandardOpenImmersion (Localization.Away g) (T'' g x)}
ι : Type (max u_3 u_2) := (i : ↑s) × ↑(t ↑i)
this : ∀ (i : ι), IsStandardOpenImmersion (Localization.Away ↑i.fst) (T'' ↑i.fst ↑i.snd)
h1 :
  Ideal.map (algebraMap S T) (Ideal.span {g | IsStandardOpenImmersion R (Localization.Away g)}) =
    Ideal.map (algebraMap S T) ⊤
⊢ Ideal.span (Set.range fun i ↦ (algebraMap S T) ↑i.fst * ↑i.snd) = ⊤

Tactic:
rw [Ideal.map_top, Ideal.map_span] at h1

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalIso R S
inst✝ : IsLocalIso S T
s : Set S := {g | IsStandardOpenImmersion R (Localization.Away g)}
T' : S → Type u_3 := fun g ↦ Localization.Away ((algebraMap S T) g)
this✝ : (g : S) → Algebra (Localization.Away g) (T' g) := fun g ↦ localizationAlgebra (Submonoid.powers g) T
T'' : S → T → Type u_3 := fun g x ↦ Localization.Away ((algebraMap T (T' g)) x)
t : S → Set T := fun g ↦ {x | IsStandardOpenImmersion (Localization.Away g) (T'' g x)}
ι : Type (max u_3 u_2) := (i : ↑s) × ↑(t ↑i)
this : ∀ (i : ι), IsStandardOpenImmersion (Localization.Away ↑i.fst) (T'' ↑i.fst ↑i.snd)
h1 : Ideal.span (⇑(algebraMap S T) '' {g | IsStandardOpenImmersion R (Localization.Away g)}) = ⊤
⊢ Ideal.span (Set.range fun i ↦ (algebraMap S T) ↑i.fst * ↑i.snd) = ⊤
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
inst✝² : IsInvariant A B G
P Q : Ideal B
inst✝¹ : P.IsPrime
inst✝ : Q.IsPrime
hPQ : Ideal.under A P = Ideal.under A Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
F : OpenNormalSubgroup G ⥤ Type u :=
  { obj := fun N ↦ { g // Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) Q = g • Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) P },
    map := fun {N N'} f ↦
      ↾fun x ↦ ⟨(QuotientGroup.map (↑N.toOpenSubgroup) N'.toOpenSubgroup.1 (MonoidHom.id G) ⋯) ↑x, ⋯⟩,
    map_id := ⋯, map_comp := ⋯ }
this : ∀ (N : OpenNormalSubgroup G), Nonempty (F.obj N)
s : (j : OpenNormalSubgroup G) → F.obj j
hs : s ∈ F.sections
a : ↑(ProfiniteGrp.of G).toProfinite.toTop :=
  (ProfiniteGrp.Hom.hom (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv) ⟨fun N ↦ ↑(s N), ⋯⟩
N : OpenNormalSubgroup G
⊢ ↑a = ↑(s N)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Injective

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
H : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)
⊢ Injective R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalIso R S
inst✝ : IsLocalIso S T
s : Set S := {g | IsStandardOpenImmersion R (Localization.Away g)}
T' : S → Type u_3 := fun g ↦ Localization.Away ((algebraMap S T) g)
this✝² : (g : S) → Algebra (Localization.Away g) (T' g) := fun g ↦ localizationAlgebra (Submonoid.powers g) T
T'' : S → T → Type u_3 := fun g x ↦ Localization.Away ((algebraMap T (T' g)) x)
t : S → Set T := fun g ↦ {x | IsStandardOpenImmersion (Localization.Away g) (T'' g x)}
ι : Type (max u_3 u_2) := (i : ↑s) × ↑(t ↑i)
this✝¹ : ∀ (i : ι), IsStandardOpenImmersion (Localization.Away ↑i.fst) (T'' ↑i.fst ↑i.snd)
h1 : Ideal.span (⇑(algebraMap S T) '' {g | IsStandardOpenImmersion R (Localization.Away g)}) = ⊤
g : S
hg : IsStandardOpenImmersion R (Localization.Away g)
x : T
hx : IsStandardOpenImmersion S (Localization.Away x)
this✝ : Algebra (Localization.Away x) (T'' g x) := localizationAlgebra (Submonoid.powers x) (T' g)
this : IsScalarTower S (Localization.Away x) (T'' g x)
⊢ algebraMapSubmonoid (Localization.Away x) (Submonoid.powers g) =
    algebraMapSubmonoid (Localization.Away x) (Submonoid.powers ((algebraMap S T) g))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Injective

Goal state before:
pp: R : Type u
inst✝¹² : CommRing R
M : Type v
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u'
inst✝⁹ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)
inst✝⁸ : ∀ (P : Ideal R) [inst : P.IsMaximal], Small.{v', u'} (Rₚ P)
inst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)
inst✝⁶ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P
Mₚ : (P : Ideal R) → [P.IsMaximal] → Type v'
inst✝⁵ : (P : Ideal R) → [inst : P.IsMaximal] → AddCommGroup (Mₚ P)
inst✝⁴ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)
inst✝³ : (P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)
inst✝² : ∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)
f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P
inst : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
H : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Rₚ I) (Mₚ I)
⊢ Injective R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.InjectiveDimension

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
S : Submonoid R
X : ModuleCat R
inj : Injective X
⊢ Injective ((localizedModuleFunctor S).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.InjectiveDimension

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
inst✝² : Small.{v, u} R
inst✝¹ : IsNoetherianRing R
n : ℕ
S : Submonoid R
M : ModuleCat R
inst✝ : HasInjectiveDimensionLE M n
⊢ HasInjectiveDimensionLE (M.localizedModule S) n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.InjectiveDimension

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
S : Submonoid R
M : ModuleCat R
n : ℕ
⊢ injectiveDimension M ≤ ↑n → injectiveDimension (M.localizedModule S) ≤ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.InjectiveDimension

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
S : Submonoid R
M : ModuleCat R
⊢ injectiveDimension (M.localizedModule S) ≤ injectiveDimension M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalIso R S
inst✝ : IsLocalIso S T
s : Set S := {g | IsStandardOpenImmersion R (Localization.Away g)}
T' : S → Type u_3 := fun g ↦ Localization.Away ((algebraMap S T) g)
this✝² : (g : S) → Algebra (Localization.Away g) (T' g) := fun g ↦ localizationAlgebra (Submonoid.powers g) T
T'' : S → T → Type u_3 := fun g x ↦ Localization.Away ((algebraMap T (T' g)) x)
t : S → Set T := fun g ↦ {x | IsStandardOpenImmersion (Localization.Away g) (T'' g x)}
ι : Type (max u_3 u_2) := (i : ↑s) × ↑(t ↑i)
this✝¹ : ∀ (i : ι), IsStandardOpenImmersion (Localization.Away ↑i.fst) (T'' ↑i.fst ↑i.snd)
h1 : Ideal.span (⇑(algebraMap S T) '' {g | IsStandardOpenImmersion R (Localization.Away g)}) = ⊤
g : S
hg : IsStandardOpenImmersion R (Localization.Away g)
x : T
hx : IsStandardOpenImmersion S (Localization.Away x)
this✝ : Algebra (Localization.Away x) (T'' g x) := localizationAlgebra (Submonoid.powers x) (T' g)
this : IsScalarTower S (Localization.Away x) (T'' g x)
⊢ IsLocalization (algebraMapSubmonoid (Localization.Away x) (Submonoid.powers g)) (T'' g x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.InjectiveDimension

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
M : ModuleCat R
h : ∀ (m : MaximalSpectrum R), Injective (M.localizedModule m.asIdeal.primeCompl)
p : Ideal R
hp : p.IsMaximal
this : Small.{v, u} (Localization.AtPrime p) := small_of_surjective Localization.mkHom_surjective
⊢ Module.Injective (Localization.AtPrime p) ↑(M.localizedModule p.primeCompl)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalIso R S
inst✝ : IsLocalIso S T
⊢ IsLocalIso R T

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
T : Type u_3
inst✝² : CommSemiring T
inst✝¹ : Algebra R T
e : S ≃ₐ[R] T
inst✝ : IsLocalIso R S
⊢ IsLocalIso R T

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
T : Type u_3
inst✝² : CommSemiring T
inst✝¹ : Algebra R T
inst✝ : IsLocalIso R S
⊢ IsLocalIso T (T ⊗[R] S)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.InjectiveDimension

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
n : ℕ
M : ModuleCat R
⊢ HasInjectiveDimensionLE M n ↔
    ∀ (m : MaximalSpectrum R), HasInjectiveDimensionLE (M.localizedModule m.asIdeal.primeCompl) n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.InjectiveDimension

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
M : ModuleCat R
n : ℕ
⊢ injectiveDimension M ≤ ↑n ↔ ⨆ p, injectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.InjectiveDimension

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
M : ModuleCat R
⊢ injectiveDimension M = ⨆ p, injectiveDimension (M.localizedModule p.asIdeal.primeCompl)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.InjectiveDimension

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
M : ModuleCat R
n : ℕ
⊢ injectiveDimension M ≤ ↑n ↔ ⨆ m, injectiveDimension (M.localizedModule m.asIdeal.primeCompl) ≤ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.ProjectiveDimension

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : Small.{v, u} R
S : Submonoid R
X : ModuleCat R
proj : Projective X
⊢ Projective ((localizedModuleFunctor S).obj X)

Tactic:
by

Goal state after: