| import Mathlib | |
| import OAI.Algebra.FrobeniusObstruction.MovingCharts | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace BoundaryOnly.FormalObstruction | |
| open Frobenius | |
| variable {k : Type*} [Field k] {d : ℕ} {n : Fin d → ℕ} | |
| variable (ell : ℕ) (htwo : 1 < ell) [CharP k ell] [Fact ell.Prime] | |
| abbrev ParameterScalar := Frobenius.Ring (ι := InternalVar n) (k := ThreeParameters.Ring k) ell | |
| def normalIndices : Finset (InternalVar n) := Finset.univ.filter fun c => c.1 = true | |
| @[simp] theorem mem_normalIndices (c : InternalVar n) : c ∈ normalIndices (n := n) ↔ c.1 = true := by | |
| simp [normalIndices] | |
| def normalPlane : ParameterScalar (n := n) (k := k) ell →ₐ[ThreeParameters.Ring k] | |
| ParameterScalar (n := n) (k := k) ell := | |
| planeRestriction ell (Nat.Prime.pos Fact.out) normalIndices | |
| omit [CharP k ell] in | |
| @[simp] theorem normalPlane_coordinate (b : Bool) (t : TangentVar n) : | |
| normalPlane (k := k) ell (coordinate ell (b,t)) = | |
| if b then 0 else coordinate ell (b,t) := by | |
| simp only [normalPlane, planeRestriction_coordinate, mem_normalIndices] | |
| theorem normalPlane_movingGraphSlot (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) : | |
| ((normalPlane (n := n) ell).restrictScalars k).comp (movingGraphSlot ell a ha false) = | |
| movingGraphSlot ell a ha false := by | |
| refine algHom_ext (ι := GraphVar n) (k := k) | |
| (S := ParameterScalar (n := n) (k := k) ell) ell _ _ ?_ | |
| intro v | |
| simp only [AlgHom.comp_apply, AlgHom.restrictScalars_apply, movingGraphSlot_coordinate] | |
| cases v with | |
| | inl i => exact AlgHom.commutes (normalPlane (n := n) ell) _ | |
| | inr t => simp [movingGraphCoordinates] | |
| omit [CharP k ell] in | |
| theorem normalPlane_smul (r : k) (x : ParameterScalar (n := n) (k := k) ell) : | |
| normalPlane ell (r • x) = r • normalPlane ell x := | |
| map_smul ((normalPlane ell).restrictScalars k) r x | |
| omit [CharP k ell] in | |
| theorem normalPlane_fixedComplement (D : FormalData (k := k) n) (c : InternalVar n) : | |
| normalPlane ell (baseChange (l := ThreeParameters.Ring k) ell (fixedComplement ell D c)) = 0 := by | |
| simp only [fixedComplement, map_sum, map_smul, baseChange_coordinate, | |
| normalPlane_smul, normalPlane_coordinate, ite_true, smul_zero, Finset.sum_const_zero] | |
| def ambientCoordinates (a : SlopeVar d → ThreeParameters.Ring k) : | |
| AmbientVar n → ParameterScalar (n := n) (k := k) ell | |
| | .inl i => @algebraMap (ThreeParameters.Ring k) (ParameterScalar (n := n) (k := k) ell) | |
| inferInstance inferInstance inferInstance (a i) | |
| | .inr c => coordinate ell c | |
| theorem ambientCoordinates_pow (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) (i : AmbientVar n) : | |
| ambientCoordinates (n := n) ell a i ^ ell = 0 := by | |
| cases i with | |
| | inl i => | |
| change (@algebraMap (ThreeParameters.Ring k) (ParameterScalar (n := n) (k := k) ell) | |
| inferInstance inferInstance inferInstance (a i)) ^ ell = 0 | |
| rw [← map_pow, ThreeParameters.augmentation_zero_pow k ell (a i) (ha i), map_zero] | |
| | inr c => exact coordinate_pow ell c | |
| def ambientSlot (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) : | |
| Frobenius.Ring (ι := AmbientVar n) (k := k) ell →ₐ[k] ParameterScalar (n := n) (k := k) ell := | |
| eval ell (ambientCoordinates ell a) (ambientCoordinates_pow ell a ha) | |
| @[simp] theorem ambientSlot_coordinate (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) (v : AmbientVar n) : | |
| ambientSlot ell a ha (coordinate ell v) = ambientCoordinates (n := n) ell a v := | |
| eval_coordinate ell _ _ _ | |
| theorem graphCoordinates_constant_zero (D : FormalData (k := k) n) (v : AmbientVar n) : | |
| MvPowerSeries.constantCoeff (graphCoordinates n D.Y v) = 0 := by | |
| cases v with | |
| | inl i => simp [graphCoordinates] | |
| | inr c => exact graph_constant_zero D c | |
| def formalGraph (D : FormalData (k := k) n) : | |
| Frobenius.Ring (ι := AmbientVar n) (k := k) ell →ₐ[k] | |
| Frobenius.Ring (ι := GraphVar n) (k := k) ell := | |
| substitute ell (graphCoordinates n D.Y) (graphCoordinates_constant_zero D) | |
| @[simp] theorem formalGraph_mk (D : FormalData (k := k) n) (p : MvPowerSeries (AmbientVar n) k) : | |
| formalGraph ell D (Ideal.Quotient.mk _ p) = | |
| Ideal.Quotient.mk _ (MvPowerSeries.subst (graphCoordinates n D.Y) p) := | |
| substitute_mk ell _ _ _ | |
| theorem graph_restriction_hom (D : FormalData (k := k) n) | |
| (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) : | |
| (((normalPlane (n := n) ell).comp (movingChart ell D a ha)).restrictScalars k).comp | |
| (ambientSlot ell a ha) = (movingGraphSlot ell a ha false).comp (formalGraph ell D) := by | |
| refine algHom_ext (ι := AmbientVar n) (k := k) | |
| (S := ParameterScalar (n := n) (k := k) ell) ell _ _ ?_ | |
| intro v | |
| simp only [AlgHom.comp_apply, AlgHom.restrictScalars_apply, ambientSlot_coordinate, | |
| formalGraph, substitute_coordinate] | |
| cases v with | |
| | inl i => | |
| change normalPlane ell (movingChart ell D a ha | |
| ((algebraMap (ThreeParameters.Ring k) (ParameterScalar (n := n) (k := k) ell)) (a i))) = _ | |
| rw [AlgHom.commutes, AlgHom.commutes] | |
| exact (movingGraphSlot_coordinate ell a ha false (.inl i)).symm | |
| | inr c => | |
| change normalPlane ell (movingChart ell D a ha (coordinate ell c)) = _ | |
| rw [movingChart_coordinate, movingChartCoordinate, map_add, normalPlane_fixedComplement, add_zero] | |
| exact AlgHom.congr_fun (normalPlane_movingGraphSlot ell a ha) | |
| (Ideal.Quotient.mk _ (D.Y c)) | |
| def movingPotential (D : FormalData (k := k) n) (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) : ParameterScalar (n := n) (k := k) ell := | |
| ambientSlot ell a ha (Ideal.Quotient.mk _ (potential n D.P)) | |
| theorem movingPotential_graph_zero (D : FormalData (k := k) n) | |
| (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) : | |
| normalPlane ell (movingChart ell D a ha (movingPotential ell D a ha)) = 0 := by | |
| have h := AlgHom.congr_fun (graph_restriction_hom ell D a ha) | |
| (Ideal.Quotient.mk _ (potential n D.P)) | |
| change _ = 0 | |
| simp only [AlgHom.comp_apply, formalGraph_mk, D.graph_zero, map_zero] at h | |
| exact h | |
| end BoundaryOnly.FormalObstruction | |
| namespace BoundaryOnly.FormalObstruction | |
| open Frobenius | |
| variable {k : Type*} [Field k] {d : ℕ} {n : Fin d → ℕ} | |
| variable (ell : ℕ) (htwo : 1 < ell) [CharP k ell] [Fact ell.Prime] | |
| omit [Fact (Nat.Prime ell)] in | |
| theorem partial_ambientCoordinates (a : SlopeVar d → ThreeParameters.Ring k) | |
| (i : InternalVar n) (v : AmbientVar n) : | |
| partialDeriv ell i (ambientCoordinates ell a v) = | |
| if v = .inr i then 1 else 0 := by | |
| cases v with | |
| | inl a => simp only [ambientCoordinates, Derivation.map_algebraMap, reduceCtorEq, ite_false] | |
| | inr c => | |
| simp only [ambientCoordinates, coordinate, partial_coordinate, Sum.inr.injEq] | |
| by_cases h : i = c | |
| · simp [h] | |
| · simp [h, Ne.symm h] | |
| omit [Fact (Nat.Prime ell)] in | |
| theorem parameter_chain_rule | |
| (f : Frobenius.Ring (ι := AmbientVar n) (k := k) ell →ₐ[k] | |
| ParameterScalar (n := n) (k := k) ell) | |
| (i : InternalVar n) (x : Frobenius.Ring (ι := AmbientVar n) (k := k) ell) : | |
| partialDeriv ell i (f x) = | |
| ∑ v : AmbientVar n, f (partialDeriv ell v x) * partialDeriv ell i (f (coordinate ell v)) := by | |
| let : Module k (ParameterScalar (n := n) (k := k) ell) := | |
| @Algebra.toModule k (ParameterScalar (n := n) (k := k) ell) inferInstance inferInstance inferInstance | |
| exact derivation_chain_rule (S := ParameterScalar (n := n) (k := k) ell) | |
| (k := k) (ι := AmbientVar n) ell f | |
| ((partialDeriv (k := ThreeParameters.Ring k) ell i).restrictScalars k) x | |
| theorem sum_mul_single {J R : Type*} [Fintype J] [DecidableEq J] [Semiring R] | |
| (u : J → R) (i : J) : (∑ v, u v * (if v = i then 1 else 0)) = u i := by | |
| simp only [mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ, ite_true] | |
| theorem partial_ambientSlot (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) | |
| (i : InternalVar n) (x : Frobenius.Ring (ι := AmbientVar n) (k := k) ell) : | |
| partialDeriv ell i (ambientSlot ell a ha x) = | |
| ambientSlot ell a ha (partialDeriv ell (.inr i) x) := by | |
| have h := parameter_chain_rule ell (ambientSlot ell a ha) i x | |
| simp only [ambientSlot_coordinate, partial_ambientCoordinates] at h | |
| exact h.trans (sum_mul_single (fun v => ambientSlot ell a ha (partialDeriv ell v x)) (.inr i)) | |
| theorem movingPotential_restricted_partial (D : FormalData (k := k) n) | |
| (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) (c : InternalVar n) : | |
| normalPlane ell (movingChart ell D a ha (partialDeriv ell c (movingPotential ell D a ha))) = | |
| movingGraphSlot ell a ha false (Ideal.Quotient.mk _ (restrictedGradient n D.P D.Y c)) := by | |
| rw [movingPotential, partial_ambientSlot, partial_mk] | |
| have h := AlgHom.congr_fun (graph_restriction_hom ell D a ha) | |
| (Ideal.Quotient.mk _ (MvPowerSeries.pderiv (.inr c) (potential n D.P))) | |
| simp only [AlgHom.comp_apply, AlgHom.restrictScalars_apply, formalGraph_mk] at h | |
| exact h | |
| end BoundaryOnly.FormalObstruction | |
| namespace BoundaryOnly.FormalObstruction | |
| open Frobenius | |
| variable {k : Type*} [Field k] {d : ℕ} {n : Fin d → ℕ} | |
| variable (ell : ℕ) (htwo : 1 < ell) [CharP k ell] [Fact ell.Prime] | |
| @[simp] theorem normalPlane_slot_apply (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) | |
| (x : Frobenius.Ring (ι := GraphVar n) (k := k) ell) : | |
| normalPlane ell (movingGraphSlot ell a ha false x) = movingGraphSlot ell a ha false x := | |
| AlgHom.congr_fun (normalPlane_movingGraphSlot ell a ha) x | |
| theorem movingPotential_mem_normalIdeal (D : FormalData (k := k) n) | |
| (a : SlopeVar d → ThreeParameters.Ring k) | |
| (ha : ∀ i, ThreeParameters.augmentation k (a i) = 0) : | |
| movingPotential ell D a ha ∈ | |
| chartNormalIdeal ell (movingChartEquiv ell htwo D a ha).symm normalIndices := by | |
| apply (chart_restriction_eq_zero_iff ell (Nat.Prime.pos Fact.out) | |
| (movingChartEquiv ell htwo D a ha).symm normalIndices _).mp | |
| exact movingPotential_graph_zero ell D a ha | |
| theorem graphIdeal_ambient_discrepancy (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) : | |
| ∃ h : InternalVar n → ParameterScalar (n := n) (k := k) ell, | |
| ((movingChartEquiv ell htwo D a ha).symm | |
| (movingGraphSlot ell a ha false (Ideal.Quotient.mk _ g)) - | |
| ∑ c, h c * partialDeriv ell c (movingPotential ell D a ha)) ∈ | |
| chartNormalIdeal ell (movingChartEquiv ell htwo D a ha).symm normalIndices := by | |
| obtain ⟨h, hh⟩ := Ideal.mem_span_range_iff_exists_fun.mp hg | |
| let e := movingChartEquiv ell htwo D a ha | |
| refine ⟨fun c => e.symm (movingGraphSlot ell a ha false (Ideal.Quotient.mk _ (h c))), ?_⟩ | |
| apply (chart_restriction_eq_zero_iff ell (Nat.Prime.pos Fact.out) e.symm normalIndices _).mp | |
| change normalPlane ell (e (e.symm (movingGraphSlot ell a ha false (Ideal.Quotient.mk _ g)) - | |
| ∑ c, e.symm (movingGraphSlot ell a ha false (Ideal.Quotient.mk _ (h c))) * | |
| partialDeriv ell c (movingPotential ell D a ha))) = 0 | |
| simp only [map_sub, map_sum, map_mul, AlgEquiv.apply_symm_apply, normalPlane_slot_apply] | |
| have he (x : ParameterScalar (n := n) (k := k) ell) : e x = movingChart ell D a ha x := rfl | |
| simp only [he, movingPotential_restricted_partial] | |
| apply sub_eq_zero.mpr | |
| simp only [← map_mul, ← map_sum] | |
| apply congrArg (movingGraphSlot ell a ha false) | |
| apply congrArg (Ideal.Quotient.mk (powerIdeal (k := k) ell)) | |
| exact hh.symm | |
| end BoundaryOnly.FormalObstruction | |
| end | |
| end OAI | |