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cfa2fd2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | 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
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