| import Mathlib |
| import OAI.Algebra.FrobeniusObstruction.Obstruction |
| import OAI.Algebra.AlgebraicObstruction.AlgebraicData |
| import OAI.Algebra.AlgebraicObstruction.RadicalPrimitives |
| import OAI.Algebra.AlgebraicObstruction.PotentialReplacement |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BoundaryOnly.FormalObstruction.FormalCorrection.AffineEtaleChart |
| open MvPowerSeries |
| variable {K α : Type} [Field K] [CharZero K] [IsAlgClosed K] [Finite α] |
|
|
| omit [CharZero K] [IsAlgClosed K] in |
| lemma completion_map_ideal [CharZero K] [IsAlgClosed K] |
| (E : AffineEtaleChart K α) (L : Ideal E.LocalRing) : |
| (L.map (algebraMap E.LocalRing |
| (AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing))).map |
| E.completionEquiv.toRingHom = L.map E.localExpansion.toRingHom := by |
| rw [Ideal.map_map] |
| congr 1 |
| apply RingHom.ext |
| intro a |
| exact E.completionEquiv_of a |
|
|
| lemma completion_mem_iff (E : AffineEtaleChart K α) (L : Ideal E.LocalRing) |
| (b : AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing) : |
| E.completionEquiv b ∈ L.map E.localExpansion.toRingHom ↔ |
| b ∈ L.map (algebraMap E.LocalRing |
| (AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing)) := by |
| rw [← E.completion_map_ideal L] |
| exact Ideal.apply_mem_of_equiv_iff |
|
|
| theorem primitive_series_approximation (E : AffineEtaleChart K α) (L : Ideal E.LocalRing) |
| (P : MvPowerSeries α K) |
| (hgrad : ∀ j : α, pderiv (R := K) j P ∈ L.map E.localExpansion.toRingHom) |
| (hjets : ∀ j : α, ∀ q : ℕ, 1 ≤ q → ∃ a : E.LocalRing, |
| pderiv (R := K) j P - E.localExpansion a ∈ (L.map E.localExpansion.toRingHom)^q) |
| (m : ℕ) (hm : 1 ≤ m) : |
| ∃ a : E.LocalRing, P - E.localExpansion a ∈ (L.map E.localExpansion.toRingHom)^m := by |
| let b := E.completionEquiv.symm P |
| have hb : E.completionEquiv b = P := E.completionEquiv.apply_symm_apply P |
| have hd (j : α) : E.completionEquiv (E.completionDeriv j b) = pderiv (R := K) j P := by |
| change E.completionEquiv (E.completionEquiv.symm (pderiv (R := K) j (E.completionEquiv b))) = _ |
| rw [E.completionEquiv.apply_symm_apply,hb] |
| obtain ⟨a,ha⟩ := E.primitive_approximation L b (fun j ↦ |
| (E.completion_mem_iff L _).mp (by rw [hd]; exact hgrad j)) (by |
| intro j q hq |
| obtain ⟨a,ha⟩ := hjets j q hq |
| refine ⟨a,?_⟩ |
| rw [← Ideal.map_pow] |
| apply (E.completion_mem_iff (L^q) _).mp |
| rw [map_sub,hd,E.completionEquiv_of,Ideal.map_pow] |
| exact ha) m hm |
| refine ⟨a,?_⟩ |
| have h := (E.completion_mem_iff (L^m) _).mpr ha |
| rwa [map_sub,hb,E.completionEquiv_of,Ideal.map_pow] at h |
|
|
| end BoundaryOnly.FormalObstruction.FormalCorrection.AffineEtaleChart |
|
|
| namespace BoundaryOnly.FormalObstruction.FormalCorrection.AffineEtaleChart |
| open MvPowerSeries |
| variable {K α : Type} [Field K] [CharZero K] [IsAlgClosed K] [Finite α] |
|
|
| omit [CharZero K] [IsAlgClosed K] in |
| lemma ideal_map_origin [CharZero K] [IsAlgClosed K] |
| (E : AffineEtaleChart K α) (L : Ideal E.LocalRing) |
| (hL : L ≤ IsLocalRing.maximalIdeal E.LocalRing) : |
| L.map E.localExpansion.toRingHom ≤ originIdeal K α := by |
| classical |
| let := Fintype.ofFinite α |
| apply Ideal.map_le_iff_le_comap.mpr |
| intro a ha |
| apply mem_origin_iff.mpr |
| by_contra hn |
| have hu : IsUnit (E.localExpansion a) := MvPowerSeries.isUnit_iff_constantCoeff.mpr |
| (isUnit_iff_ne_zero.mpr hn) |
| exact (hL ha) (isUnit_of_map_unit E.localExpansion a hu) |
|
|
| end BoundaryOnly.FormalObstruction.FormalCorrection.AffineEtaleChart |
|
|
| namespace BoundaryOnly.FormalObstruction.CoefficientRing |
| open MvPowerSeries FormalCorrection AlgebraicReplacement |
| open scoped Classical |
| variable {K : Type} [Field K] [CharZero K] [IsAlgClosed K] {d : ℕ} {n : Fin d → ℕ} |
|
|
| theorem no_algebraic_input (hd : 5 ≤ d) |
| (Q : QuadraticData (R := K) (n := n)) |
| (E : (i : Fin d) → AffineEtaleChart K (WallVar n i)) |
| (L : (i : Fin d) → Ideal (E i).LocalRing) |
| (hL : ∀ i, L i ≤ IsLocalRing.maximalIdeal (E i).LocalRing) |
| (hgradient : ∀ i w, pderiv (R := K) w (Q.P i) ∈ (L i).map (E i).localExpansion.toRingHom) |
| (hjets : ∀ i w, ∀ q : ℕ, 1 ≤ q → ∃ a : (E i).LocalRing, |
| pderiv (R := K) w (Q.P i) - (E i).localExpansion a ∈ |
| ((L i).map (E i).localExpansion.toRingHom)^q) |
| (hpulled : ∀ b i, Ideal.map (wallEval Q.Y Q.centered b i) |
| ((L i).map (E i).localExpansion.toRingHom) ≤ originIdeal K (GraphVar n)) |
| (hcube : ∀ b i, (Ideal.map (wallEval Q.Y Q.centered b i) |
| ((L i).map (E i).localExpansion.toRingHom))^3 ≤ gradientIdeal n Q.P Q.Y) |
| (hY : ∀ c, Nonempty (LocalEtaleExpansion (Q.Y c))) : False := by |
| have happ (i : Fin d) := (E i).primitive_series_approximation (L i) (Q.P i) |
| (hgradient i) (hjets i) 9 (by omega) |
| choose a ha using happ |
| let P := fun i ↦ (E i).localExpansion (a i) |
| let W := fun i ↦ (L i).map (E i).localExpansion.toRingHom |
| have hW (i : Fin d) : W i ≤ originIdeal K (WallVar n i) := |
| (E i).ideal_map_origin (L i) (hL i) |
| have hP (i : Fin d) : P i - Q.P i ∈ (W i)^9 := by |
| simpa only [neg_sub] using ((W i)^9).neg_mem (ha i) |
| let T : QuadraticData (R := K) (n := n) := Q.replace W hW hpulled hcube P hP |
| apply no_algebraic_quadratic_data hd T |
| · intro i |
| exact ⟨(E i).localGerm (a i)⟩ |
| · exact hY |
|
|
| end BoundaryOnly.FormalObstruction.CoefficientRing |
|
|
| end |
|
|
| end OAI |
|
|
