| import Mathlib |
| 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 |
|
|
