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import OAI.Algebra.FrobeniusObstruction.GraphPotential
import OAI.Algebra.FrobeniusObstruction.GlobalForms
namespace OAI
noncomputable section
open scoped BigOperators
noncomputable section
open scoped BigOperators TensorProduct
namespace BoundaryOnly.FormalObstruction.Frobenius
open MixedForms
variable {ι k : Type*} [Fintype ι] [DecidableEq ι] [CommRing k]
variable (ell : ℕ) [CharP k ell]
def normalCoordinateList
(e : Ring (ι := ι) (k := k) ell ≃ₐ[k] Ring (ι := ι) (k := k) ell)
(s : Finset ι) : List (Ring (ι := ι) (k := k) ell) :=
s.toList.map (fun i => e (coordinate ell i))
omit [Fintype ι] [DecidableEq ι] [CharP k ell] in
theorem normalCoordinateList_span
(e : Ring (ι := ι) (k := k) ell ≃ₐ[k] Ring (ι := ι) (k := k) ell)
(s : Finset ι) : Ideal.span {u | u ∈ normalCoordinateList ell e s} =
chartNormalIdeal ell e s := by
congr 1
ext u
simp only [normalCoordinateList, List.mem_map, Finset.mem_toList,
Set.mem_ofPred_eq, Set.mem_image, Finset.mem_coe]
omit [Fintype ι] [DecidableEq ι] [CharP k ell] in
theorem normalCoordinateList_pow (hell : 0 < ell)
(e : Ring (ι := ι) (k := k) ell ≃ₐ[k] Ring (ι := ι) (k := k) ell)
(s : Finset ι) (u : Ring (ι := ι) (k := k) ell)
(hu : u ∈ normalCoordinateList ell e s) : u ^ (ell-1+1) = 0 := by
obtain ⟨i, _, rfl⟩ := List.mem_map.mp hu
rw [Nat.sub_add_cancel hell, ← map_pow, coordinate_pow, map_zero]
def chartNormal
(e : Ring (ι := ι) (k := k) ell ≃ₐ[k] Ring (ι := ι) (k := k) ell)
(s : Finset ι) : Forms (k := k) (A := Ring (ι := ι) (k := k) ell) (ι := ι) :=
normal (partialDeriv ell) (ell-1) (normalCoordinateList ell e s)
theorem chartNormal_d
(e : Ring (ι := ι) (k := k) ell ≃ₐ[k] Ring (ι := ι) (k := k) ell)
(s : Finset ι) : d (partialDeriv ell) (chartNormal ell e s) = 0 :=
d_normal _ (partial_commute ell) _ _
theorem chartNormal_annihilate (hell : 0 < ell)
(e : Ring (ι := ι) (k := k) ell ≃ₐ[k] Ring (ι := ι) (k := k) ell)
(s : Finset ι) (g : Ring (ι := ι) (k := k) ell)
(hg : g ∈ chartNormalIdeal ell e s) : coeff g * chartNormal ell e s = 0 :=
coeff_mul_normal_ideal _ _ _ (normalCoordinateList_pow ell hell e s) g
(by rwa [normalCoordinateList_span])
theorem chartNormal_delta (hell : 0 < ell)
(e : Ring (ι := ι) (k := k) ell ≃ₐ[k] Ring (ι := ι) (k := k) ell)
(s : Finset ι) (F : Ring (ι := ι) (k := k) ell)
(hF : F ∈ chartNormalIdeal ell e s) :
delta (partialDeriv ell) F (chartNormal ell e s) = 0 :=
delta_normal _ _ _ (normalCoordinateList_pow ell hell e s) F
(by rwa [normalCoordinateList_span])
theorem chartNormal_primitive (hell : 0 < ell)
(e : Ring (ι := ι) (k := k) ell ≃ₐ[k] Ring (ι := ι) (k := k) ell)
(s : Finset ι) (F g : Ring (ι := ι) (k := k) ell)
(h : ι → Ring (ι := ι) (k := k) ell)
(hF : F ∈ chartNormalIdeal ell e s)
(hg : g - ∑ i, h i * partialDeriv ell i F ∈ chartNormalIdeal ell e s) :
coeff g * chartNormal ell e s = delta (partialDeriv ell) F
(∑ i, coeff (h i) * contract i (chartNormal ell e s)) :=
normal_primitive _ _ _ (normalCoordinateList_pow ell hell e s) F g h
(by rwa [normalCoordinateList_span]) (by rwa [normalCoordinateList_span])
end BoundaryOnly.FormalObstruction.Frobenius
namespace BoundaryOnly.FormalObstruction
open Frobenius MixedForms
variable {k : Type*} [Field k] {d : ℕ} {n : Fin d → ℕ}
variable (ell : ℕ) (htwo : 1 < ell) [CharP k ell] [Fact ell.Prime]
abbrev ParameterForms := Forms (k := k)
(A := ParameterScalar (n := n) (k := k) ell) (ι := InternalVar n)
def parameterPartial (i : InternalVar n) :
Derivation k (ParameterScalar (n := n) (k := k) ell)
(ParameterScalar (n := n) (k := k) ell) :=
(partialDeriv (k := ThreeParameters.Ring k) ell i).restrictScalars k
omit [Fact ell.Prime] in
@[simp] theorem parameterPartial_apply (i : InternalVar n)
(x : ParameterScalar (n := n) (k := k) ell) :
parameterPartial ell i x = partialDeriv ell i x := rfl
omit [Fact ell.Prime] in
theorem parameterPartial_commute (i j : InternalVar n)
(x : ParameterScalar (n := n) (k := k) ell) :
parameterPartial ell i (parameterPartial ell j x) =
parameterPartial ell j (parameterPartial ell i x) := partial_commute ell i j x
def movingNormal (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) :
ParameterForms (n := n) (k := k) ell :=
normal (parameterPartial ell) (ell-1)
(normalCoordinateList ell (movingChartEquiv ell htwo D a ha).symm normalIndices)
theorem movingNormal_d (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) :
MixedForms.d (parameterPartial ell) (movingNormal ell htwo D a ha) = 0 :=
d_normal _ (parameterPartial_commute ell) _ _
theorem movingNormal_delta (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) :
delta (parameterPartial ell) (movingPotential ell D a ha)
(movingNormal ell htwo D a ha) = 0 :=
delta_normal _ _ _ (normalCoordinateList_pow ell (Nat.Prime.pos Fact.out) _ _)
_ (by rw [normalCoordinateList_span]; exact movingPotential_mem_normalIdeal ell htwo D a ha)
theorem movingNormal_annihilate (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0)
(g : ParameterScalar (n := n) (k := k) ell)
(hg : g ∈ chartNormalIdeal ell (movingChartEquiv ell htwo D a ha).symm normalIndices) :
coeff g * movingNormal ell htwo D a ha = 0 :=
coeff_mul_normal_ideal _ _ _ (normalCoordinateList_pow ell (Nat.Prime.pos Fact.out) _ _)
g (by rwa [normalCoordinateList_span])
theorem movingNormal_primitive (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0)
(g : MvPowerSeries (GraphVar n) k) (hg : g ∈ gradientIdeal n D.P D.Y) :
∃ eta : ParameterForms (n := n) (k := k) ell,
delta (parameterPartial ell) (movingPotential ell D a ha) eta =
coeff ((movingChartEquiv ell htwo D a ha).symm
(movingGraphSlot ell a ha false (Ideal.Quotient.mk _ g))) *
movingNormal ell htwo D a ha := by
obtain ⟨h, hh⟩ := graphIdeal_ambient_discrepancy ell htwo D a ha g hg
refine ⟨∑ i, coeff (h i) * contract i (movingNormal ell htwo D a ha), ?_⟩
apply Eq.symm
apply normal_primitive _ _ _ (normalCoordinateList_pow ell (Nat.Prime.pos Fact.out) _ _) _ _ h
· rw [normalCoordinateList_span]; exact movingPotential_mem_normalIdeal ell htwo D a ha
· rw [normalCoordinateList_span]; exact hh
theorem movingNormal_cubic_primitive (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) (i j l : Fin d) :
∃ eta : ParameterForms (n := n) (k := k) ell,
delta (parameterPartial ell) (movingPotential ell D a ha) eta =
coeff (@algebraMap (ThreeParameters.Ring k) (ParameterScalar (n := n) (k := k) ell)
inferInstance inferInstance inferInstance (a (i,0) * a (j,0) * a (l,0))) *
movingNormal ell htwo D a ha := by
obtain ⟨eta, he⟩ := movingNormal_primitive ell htwo D a ha
(slope n i * slope n j * slope n l) (D.positive_cubic i j l)
have hs : (movingChartEquiv ell htwo D a ha).symm
(movingGraphSlot ell a ha false
(Ideal.Quotient.mk _ (slope n i * slope n j * slope n l))) =
@algebraMap (ThreeParameters.Ring k) (ParameterScalar (n := n) (k := k) ell)
inferInstance inferInstance inferInstance (a (i,0) * a (j,0) * a (l,0)) := by
simp only [map_mul, slope]
change (movingChartEquiv ell htwo D a ha).symm
(movingGraphSlot ell a ha false (coordinate ell (.inl (i,0)))) *
(movingChartEquiv ell htwo D a ha).symm
(movingGraphSlot ell a ha false (coordinate ell (.inl (j,0)))) *
(movingChartEquiv ell htwo D a ha).symm
(movingGraphSlot ell a ha false (coordinate ell (.inl (l,0)))) = _
simp only [movingGraphSlot_coordinate, movingGraphCoordinates, AlgEquiv.commutes]
rw [hs] at he
exact ⟨eta, he⟩
end BoundaryOnly.FormalObstruction
namespace BoundaryOnly.FormalObstruction.Frobenius
variable {ι k l : Type*} [Fintype ι] [DecidableEq ι] [CommRing k] [CommRing l]
variable (ell : ℕ) [CharP k ell] [CharP l ell]
omit [Fintype ι] [DecidableEq ι] [CharP k ell] [CharP l ell] in
theorem series_pderiv_map (f : k →+* l) (i : ι) (x : Series (ι := ι) (k := k)) :
MvPowerSeries.map f (MvPowerSeries.pderiv i x) =
MvPowerSeries.pderiv i (MvPowerSeries.map f x) := by
ext e
simp only [MvPowerSeries.coeff_map, MvPowerSeries.coeff_pderiv, map_mul, map_add, map_natCast, map_one]
omit [DecidableEq ι] in
theorem mapCoefficients_partial (f : k →+* l) (i : ι)
(x : Ring (ι := ι) (k := k) ell) :
mapCoefficients ell f (partialDeriv ell i x) =
partialDeriv ell i (mapCoefficients ell f x) := by
obtain ⟨p,rfl⟩ := Ideal.Quotient.mk_surjective x
simp only [partial_mk, mapCoefficients_mk, series_pderiv_map]
variable [Algebra k l]
omit [DecidableEq ι] in
theorem baseChange_partial (i : ι)
(x : Ring (ι := ι) (k := k) ell) :
baseChange (l := l) ell (partialDeriv ell i x) =
partialDeriv ell i (baseChange (l := l) ell x) :=
mapCoefficients_partial ell (algebraMap k l) i x
end BoundaryOnly.FormalObstruction.Frobenius
namespace BoundaryOnly.FormalObstruction.MixedForms
open scoped TensorProduct
variable {k A B ι : Type*} [CommRing k] [CommRing A] [CommRing B]
[Algebra k A] [Algebra k B] [Fintype ι] [DecidableEq ι]
def coefficientMap (f : A →ₐ[k] B) :
Forms (k := k) (A := A) (ι := ι) →ₐ[k] Forms (k := k) (A := B) (ι := ι) :=
Algebra.TensorProduct.map f (AlgHom.id k _)
omit [Fintype ι] [DecidableEq ι] in
@[simp] theorem coefficientMap_tmul (f : A →ₐ[k] B)
(a : A) (e : Ext (k := k) (ι := ι)) :
coefficientMap f (a ⊗ₜ[k] e) = f a ⊗ₜ[k] e := rfl
omit [Fintype ι] [DecidableEq ι] in
@[simp] theorem coefficientMap_coeff (f : A →ₐ[k] B) (a : A) :
coefficientMap (ι := ι) f (coeff a) = coeff (f a) := rfl
theorem coefficientMap_d (f : A →ₐ[k] B)
(pA : ι → Derivation k A A) (pB : ι → Derivation k B B)
(hp : ∀ i a, f (pA i a) = pB i (f a))
(x : Forms (k := k) (A := A) (ι := ι)) :
coefficientMap f (d pA x) = d pB (coefficientMap f x) := by
induction x using TensorProduct.inductionOn with
| add x y hx hy => simp only [map_add, hx, hy]
| tmul a e => simp only [d_tmul, coefficientMap_tmul, map_sum, hp]
theorem coefficientMap_gradient (f : A →ₐ[k] B)
(pA : ι → Derivation k A A) (pB : ι → Derivation k B B)
(hp : ∀ i a, f (pA i a) = pB i (f a)) (F : A) :
coefficientMap f (gradient pA F) = gradient pB (f F) := by
exact coefficientMap_d f pA pB hp (coeff F)
theorem coefficientMap_delta (f : A →ₐ[k] B)
(pA : ι → Derivation k A A) (pB : ι → Derivation k B B)
(hp : ∀ i a, f (pA i a) = pB i (f a)) (F : A)
(x : Forms (k := k) (A := A) (ι := ι)) :
coefficientMap f (delta pA F x) = delta pB (f F) (coefficientMap f x) := by
simp only [delta_apply, map_mul, coefficientMap_gradient f pA pB hp]
theorem coefficientMap_cartier (f : A →ₐ[k] B)
(pA : ι → Derivation k A A) (pB : ι → Derivation k B B)
(hp : ∀ i a, f (pA i a) = pB i (f a)) (e : ℕ) (u : A) :
coefficientMap f (cartier pA e u) = cartier pB e (f u) := by
simp only [cartier, map_mul, coefficientMap_coeff, map_pow,
coefficientMap_gradient f pA pB hp]
theorem coefficientMap_normal (f : A →ₐ[k] B)
(pA : ι → Derivation k A A) (pB : ι → Derivation k B B)
(hp : ∀ i a, f (pA i a) = pB i (f a)) (e : ℕ) (us : List A) :
coefficientMap f (normal pA e us) = normal pB e (us.map f) := by
induction us with
| nil => simp only [normal, List.map_nil, List.prod_nil, map_one]
| cons u us ih =>
simpa only [normal, List.map_cons, List.prod_cons, map_mul,
coefficientMap_cartier f pA pB hp] using congrArg
(fun x => cartier pB e (f u) * x) ih
end BoundaryOnly.FormalObstruction.MixedForms
namespace BoundaryOnly.FormalObstruction
open Frobenius MixedForms
variable {k : Type*} [Field k] {d : ℕ} {n : Fin d → ℕ}
variable (ell : ℕ) (htwo : 1 < ell) [CharP k ell] [Fact ell.Prime]
theorem movingChart_inverse_specialFiber (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0)
(x : ParameterScalar (n := n) (k := k) ell) :
reduction ell ((movingChartEquiv ell htwo D a ha).symm x) =
(specialChartEquiv ell htwo D).symm (reduction ell x) := by
apply (specialChartEquiv ell htwo D).injective
rw [AlgEquiv.apply_symm_apply]
change specialChart ell D (reduction ell ((movingChartEquiv ell htwo D a ha).symm x)) = _
rw [← movingChart_specialFiber ell D a ha]
change reduction ell ((movingChartEquiv ell htwo D a ha)
((movingChartEquiv ell htwo D a ha).symm x)) = _
rw [AlgEquiv.apply_symm_apply]
def specialNormal (D : FormalData (k := k) n) :
Forms (k := k) (A := Frobenius.Ring (ι := InternalVar n) (k := k) ell)
(ι := InternalVar n) :=
chartNormal ell (specialChartEquiv ell htwo D).symm normalIndices
theorem reduce_normalList (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) :
(normalCoordinateList ell (movingChartEquiv ell htwo D a ha).symm normalIndices).map
(reduction ell) = normalCoordinateList ell (specialChartEquiv ell htwo D).symm normalIndices := by
simp only [normalCoordinateList, List.map_map]
apply List.map_congr_left
intro i hi
simp only [Function.comp_apply, movingChart_inverse_specialFiber, reduction_coordinate]
theorem reduce_movingNormal (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) :
coefficientMap (reduction ell) (movingNormal ell htwo D a ha) =
specialNormal ell htwo D := by
have hp (i : InternalVar n) (x : ParameterScalar (n := n) (k := k) ell) :
reduction ell (parameterPartial ell i x) = partialDeriv ell i (reduction ell x) :=
mapCoefficients_partial ell (ThreeParameters.augmentation k) i x
have hn := coefficientMap_normal (reduction (ι := InternalVar n) (k := k) ell)
(parameterPartial ell) (partialDeriv ell) hp (ell-1)
(normalCoordinateList ell (movingChartEquiv ell htwo D a ha).symm normalIndices)
rw [reduce_normalList ell htwo D a ha] at hn
exact hn
end BoundaryOnly.FormalObstruction
namespace BoundaryOnly.FormalObstruction
open Frobenius MixedForms
variable {k : Type*} [Field k] {d : ℕ} {n : Fin d → ℕ}
variable (ell : ℕ) (htwo : 1 < ell) [CharP k ell] [Fact ell.Prime]
def specialAmbient : Frobenius.Ring (ι := AmbientVar n) (k := k) ell →ₐ[k]
Frobenius.Ring (ι := InternalVar n) (k := k) ell :=
eval ell (Sum.elim (fun _ => 0) (coordinate ell)) (by
intro v
cases v with
| inl _ => exact zero_pow (Nat.Prime.ne_zero Fact.out)
| inr c => exact coordinate_pow ell c)
omit [CharP k ell] in
@[simp] theorem specialAmbient_slope (i : SlopeVar d) :
specialAmbient (n := n) (k := k) ell (coordinate ell (.inl i)) = 0 :=
eval_coordinate _ _ _ _
omit [CharP k ell] in
@[simp] theorem specialAmbient_internal (i : InternalVar n) :
specialAmbient (n := n) (k := k) ell (coordinate ell (.inr i)) = coordinate ell i :=
eval_coordinate _ _ _ _
theorem reduce_ambientSlot (a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0)
(x : Frobenius.Ring (ι := AmbientVar n) (k := k) ell) :
reduction ell (ambientSlot ell a ha x) = specialAmbient ell x := by
have h : (reduction ell).comp (ambientSlot ell a ha) =
specialAmbient (n := n) (k := k) ell := by
apply algHom_ext
intro v
simp only [AlgHom.comp_apply, ambientSlot_coordinate]
cases v with
| inl i => simp only [ambientCoordinates, reduction_base, ha, map_zero, specialAmbient_slope]
| inr c => simp only [ambientCoordinates, reduction_coordinate, specialAmbient_internal]
exact AlgHom.congr_fun h x
def specialPotential (D : FormalData (k := k) n) :
Frobenius.Ring (ι := InternalVar n) (k := k) ell :=
specialAmbient ell (Ideal.Quotient.mk _ (potential n D.P))
theorem reduce_movingPotential (D : FormalData (k := k) n)
(a : SlopeVar d → ThreeParameters.Ring k)
(ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) :
reduction ell (movingPotential ell D a ha) = specialPotential ell D :=
reduce_ambientSlot ell a ha _
theorem specialNormal_d (D : FormalData (k := k) n) :
MixedForms.d (partialDeriv ell) (specialNormal ell htwo D) = 0 :=
chartNormal_d ell _ _
omit [Fact ell.Prime] in
theorem parameter_reduction_partial (i : InternalVar n)
(x : ParameterScalar (n := n) (k := k) ell) :
reduction ell (parameterPartial ell i x) = partialDeriv ell i (reduction ell x) :=
mapCoefficients_partial ell (ThreeParameters.augmentation k) i x
theorem specialNormal_delta (D : FormalData (k := k) n) :
delta (partialDeriv ell) (specialPotential ell D) (specialNormal ell htwo D) = 0 := by
let a : SlopeVar d → ThreeParameters.Ring k := fun _ => 0
have ha : ∀ i, ThreeParameters.augmentation k (a i) = 0 := fun _ => map_zero _
have h := congrArg (coefficientMap (reduction ell)) (movingNormal_delta ell htwo D a ha)
rw [coefficientMap_delta _ _ _ (parameter_reduction_partial ell),
reduce_movingPotential, reduce_movingNormal, map_zero] at h
exact h
theorem specialNormal_primitive (D : FormalData (k := k) n)
(g : MvPowerSeries (GraphVar n) k) (hg : g ∈ gradientIdeal n D.P D.Y) :
∃ eta : Forms (k := k) (A := Frobenius.Ring (ι := InternalVar n) (k := k) ell)
(ι := InternalVar n),
delta (partialDeriv ell) (specialPotential ell D) eta =
coeff ((specialChartEquiv ell htwo D).symm
(graphSlot ell false (Ideal.Quotient.mk _ g))) * specialNormal ell htwo D := by
let a : SlopeVar d → ThreeParameters.Ring k := fun _ => 0
have ha : ∀ i, ThreeParameters.augmentation k (a i) = 0 := fun _ => map_zero _
obtain ⟨eta, he⟩ := movingNormal_primitive ell htwo D a ha g hg
refine ⟨coefficientMap (reduction ell) eta, ?_⟩
have h := congrArg (coefficientMap (reduction ell)) he
rw [coefficientMap_delta _ _ _ (parameter_reduction_partial ell),
reduce_movingPotential, map_mul, coefficientMap_coeff,
movingChart_inverse_specialFiber, reduce_movingNormal] at h
have hr := AlgHom.congr_fun (reduce_movingGraphSlot ell a ha false)
(Ideal.Quotient.mk _ g)
change reduction ell (movingGraphSlot ell a ha false (Ideal.Quotient.mk _ g)) = _ at hr
rw [hr] at h
exact h
end BoundaryOnly.FormalObstruction
end
end
end OAI
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