-- Mathlib/RingTheory/Spectrum/Prime/FreeLocus.lean
lemma freeLocus_localization (S : Submonoid R) :
    freeLocus (Localization S) (LocalizedModule S M) =
      comap (algebraMap R _) ⁻¹' freeLocus R M

Proof state:
case h
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
⊢ p ∈ freeLocus (Localization S) (LocalizedModule S M) ↔ (comap (algebraMap R (Localization S))) p ∈ freeLocus R M

Tactic:
let p' := p.asIdeal.comap (algebraMap R _)

Result:
case h
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
⊢ p ∈ freeLocus (Localization S) (LocalizedModule S M) ↔ (comap (algebraMap R (Localization S))) p ∈ freeLocus R M

Proof state:
case h'.mk.intro.intro
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule p'.primeCompl M
this✝⁶ : Algebra (Localization S) Rₚ :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization S) Rₚ S p'.primeCompl hp'
this✝⁵ : IsScalarTower R (Localization S) Rₚ
this✝⁴ : IsLocalization.AtPrime Rₚ p.asIdeal
this✝³ : Module (Localization S) Mₚ := compHom Mₚ (algebraMap (Localization S) Rₚ)
this✝² : IsScalarTower R (Localization S) Mₚ
this✝¹ : IsScalarTower (Localization S) Rₚ Mₚ
l : LocalizedModule S M →ₗ[Localization S] LocalizedModule p'.primeCompl M :=
  LinearMap.extendScalarsOfIsLocalization S (Localization S)
    (IsLocalizedModule.liftOfLE S p'.primeCompl hp' (LocalizedModule.mkLinearMap S M)
      (LocalizedModule.mkLinearMap p'.primeCompl M))
this✝ : IsLocalizedModule p'.primeCompl (↑R l)
this : IsLocalizedModule (Algebra.algebraMapSubmonoid (Localization S) p'.primeCompl) l
x : R
s : ↥S
hx : IsLocalization.mk' (Localization S) x s ∈ p.asIdeal.primeCompl
⊢ ∃ m, m * ↑⟨IsLocalization.mk' (Localization S) x s, hx⟩ ∈ Algebra.algebraMapSubmonoid (Localization S) p'.primeCompl

Tactic:
refine ⟨algebraMap _ _ s.1, x, fun H ↦ hx ?_, by simp⟩

Result:
case h'.mk.intro.intro
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule p'.primeCompl M
this✝⁶ : Algebra (Localization S) Rₚ :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization S) Rₚ S p'.primeCompl hp'
this✝⁵ : IsScalarTower R (Localization S) Rₚ
this✝⁴ : IsLocalization.AtPrime Rₚ p.asIdeal
this✝³ : Module (Localization S) Mₚ := compHom Mₚ (algebraMap (Localization S) Rₚ)
this✝² : IsScalarTower R (Localization S) Mₚ
this✝¹ : IsScalarTower (Localization S) Rₚ Mₚ
l : LocalizedModule S M →ₗ[Localization S] LocalizedModule p'.primeCompl M :=
  LinearMap.extendScalarsOfIsLocalization S (Localization S)
    (IsLocalizedModule.liftOfLE S p'.primeCompl hp' (LocalizedModule.mkLinearMap S M)
      (LocalizedModule.mkLinearMap p'.primeCompl M))
this✝ : IsLocalizedModule p'.primeCompl (↑R l)
this : IsLocalizedModule (Algebra.algebraMapSubmonoid (Localization S) p'.primeCompl) l
x : R
s : ↥S
hx : IsLocalization.mk' (Localization S) x s ∈ p.asIdeal.primeCompl
H : x ∈ ↑p'
⊢ IsLocalization.mk' (Localization S) x s ∈ ↑p.asIdeal

Proof state:
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
⊢ freeLocus (Localization S) (LocalizedModule S M) = ⇑(comap (algebraMap R (Localization S))) ⁻¹' freeLocus R M

Tactic:
ext p

Result:
case h
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
⊢ p ∈ freeLocus (Localization S) (LocalizedModule S M) ↔ p ∈ ⇑(comap (algebraMap R (Localization S))) ⁻¹' freeLocus R M

Proof state:
case h'.mk.intro.intro
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule p'.primeCompl M
this✝⁶ : Algebra (Localization S) Rₚ :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization S) Rₚ S p'.primeCompl hp'
this✝⁵ : IsScalarTower R (Localization S) Rₚ
this✝⁴ : IsLocalization.AtPrime Rₚ p.asIdeal
this✝³ : Module (Localization S) Mₚ := compHom Mₚ (algebraMap (Localization S) Rₚ)
this✝² : IsScalarTower R (Localization S) Mₚ
this✝¹ : IsScalarTower (Localization S) Rₚ Mₚ
l : LocalizedModule S M →ₗ[Localization S] LocalizedModule p'.primeCompl M :=
  LinearMap.extendScalarsOfIsLocalization S (Localization S)
    (IsLocalizedModule.liftOfLE S p'.primeCompl hp' (LocalizedModule.mkLinearMap S M)
      (LocalizedModule.mkLinearMap p'.primeCompl M))
this✝ : IsLocalizedModule p'.primeCompl (↑R l)
this : IsLocalizedModule (Algebra.algebraMapSubmonoid (Localization S) p'.primeCompl) l
x : R
s : ↥S
hx : IsLocalization.mk' (Localization S) x s ∈ p.asIdeal.primeCompl
H : x ∈ ↑p'
⊢ IsLocalization.mk' (Localization S) x s ∈ ↑p.asIdeal

Tactic:
rw [IsLocalization.mk'_eq_mul_mk'_one]

Result:
case h'.mk.intro.intro
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule p'.primeCompl M
this✝⁶ : Algebra (Localization S) Rₚ :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization S) Rₚ S p'.primeCompl hp'
this✝⁵ : IsScalarTower R (Localization S) Rₚ
this✝⁴ : IsLocalization.AtPrime Rₚ p.asIdeal
this✝³ : Module (Localization S) Mₚ := compHom Mₚ (algebraMap (Localization S) Rₚ)
this✝² : IsScalarTower R (Localization S) Mₚ
this✝¹ : IsScalarTower (Localization S) Rₚ Mₚ
l : LocalizedModule S M →ₗ[Localization S] LocalizedModule p'.primeCompl M :=
  LinearMap.extendScalarsOfIsLocalization S (Localization S)
    (IsLocalizedModule.liftOfLE S p'.primeCompl hp' (LocalizedModule.mkLinearMap S M)
      (LocalizedModule.mkLinearMap p'.primeCompl M))
this✝ : IsLocalizedModule p'.primeCompl (↑R l)
this : IsLocalizedModule (Algebra.algebraMapSubmonoid (Localization S) p'.primeCompl) l
x : R
s : ↥S
hx : IsLocalization.mk' (Localization S) x s ∈ p.asIdeal.primeCompl
H : x ∈ ↑p'
⊢ (algebraMap R (Localization S)) x * IsLocalization.mk' (Localization S) 1 s ∈ ↑p.asIdeal