| import Mathlib |
| import OAI.Algebra.FrobeniusObstruction.Obstruction |
| import OAI.Algebra.AlgebraicObstruction.AlgebraicGerms |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BoundaryOnly.FormalObstruction.CoefficientRing |
| open MvPowerSeries FormalCorrection |
| open scoped Classical |
| variable {K : Type} [Field K] {d : ℕ} {n : Fin d → ℕ} |
|
|
| lemma potential_germ (P : (i : Fin d) → MvPowerSeries (WallVar n i) K) |
| (hP : ∀ i, Nonempty (LocalEtaleExpansion (P i))) : |
| Nonempty (LocalEtaleExpansion (potential n P)) := by |
| apply germ_sum |
| intro i _ |
| apply germ_sub |
| · apply germ_subst _ (hP i) |
| · intro j; cases j <;> exact germ_X _ |
| · intro j; cases j <;> simp [plusCoordinates] |
| · apply germ_subst _ (hP i) |
| · intro j; cases j |
| · exact germ_zero |
| · exact germ_X _ |
| · intro j; cases j <;> simp [minusCoordinates] |
|
|
| lemma graphCoordinates_germ (Y : InternalVar n → MvPowerSeries (GraphVar n) K) |
| (hY : ∀ c, Nonempty (LocalEtaleExpansion (Y c))) : |
| ∀ i, Nonempty (LocalEtaleExpansion (graphCoordinates n Y i)) := by |
| rintro (a | c) |
| · exact germ_X _ |
| · exact hY c |
|
|
| lemma graphCoordinates_constant (Q : QuadraticData (R := K) (n := n)) : |
| ∀ i, constantCoeff (graphCoordinates n Q.Y i) = 0 := by |
| rintro (a | c) |
| · exact constantCoeff_X _ |
| · exact origin_constant (Q.centered c) |
|
|
| lemma restrictedGradient_germ (Q : QuadraticData (R := K) (n := n)) |
| (hP : ∀ i, Nonempty (LocalEtaleExpansion (Q.P i))) |
| (hY : ∀ c, Nonempty (LocalEtaleExpansion (Q.Y c))) (c : InternalVar n) : |
| Nonempty (LocalEtaleExpansion (restrictedGradient n Q.P Q.Y c)) := |
| germ_subst _ (germ_pderiv (potential_germ Q.P hP) _) (graphCoordinates_germ Q.Y hY) |
| (graphCoordinates_constant Q) |
|
|
| lemma negativeSlopeValue_germ (Q : QuadraticData (R := K) (n := n)) |
| (hP : ∀ i, Nonempty (LocalEtaleExpansion (Q.P i))) |
| (hY : ∀ c, Nonempty (LocalEtaleExpansion (Q.Y c))) (i : Fin d) : |
| Nonempty (LocalEtaleExpansion (negativeSlopeValue n Q.P Q.Y i)) := by |
| apply germ_subst _ _ (fun j ↦ hY (false,⟨i,j⟩)) |
| (fun j ↦ origin_constant (Q.centered (false,⟨i,j⟩))) |
| apply germ_subst _ (germ_pderiv (hP i) _) |
| · intro j; cases j |
| · exact germ_zero |
| · exact germ_X _ |
| · intro j; cases j <;> simp [zeroSlopes] |
|
|
| theorem QuadraticData.etale_witnesses (Q : QuadraticData (R := K) (n := n)) |
| (hP : ∀ i, Nonempty (LocalEtaleExpansion (Q.P i))) |
| (hY : ∀ c, Nonempty (LocalEtaleExpansion (Q.Y c))) : |
| ∃ W : QuadraticWitnesses Q, |
| (∀ c e, Nonempty (LocalEtaleExpansion (W.b c e))) ∧ |
| (∀ t c, Nonempty (LocalEtaleExpansion (W.pos t c))) ∧ |
| (∀ t c, Nonempty (LocalEtaleExpansion (W.neg t c))) := by |
| have hgrad := restrictedGradient_germ Q hP hY |
| have hvalue := germ_subst (graphCoordinates n Q.Y) (potential_germ Q.P hP) |
| (graphCoordinates_germ Q.Y hY) (graphCoordinates_constant Q) |
| obtain ⟨b,hb,hbg⟩ := germ_square_witness _ _ hgrad hvalue Q.graph_square |
| have hp : ∀ t : Fin d × Fin d × Fin d, |
| ∃ c : InternalVar n → MvPowerSeries (GraphVar n) K, |
| slope n t.1 * slope n t.2.1 * slope n t.2.2 = |
| ∑ e, c e * restrictedGradient n Q.P Q.Y e ∧ |
| ∀ e, Nonempty (LocalEtaleExpansion (c e)) := by |
| intro t |
| apply germ_span_witness _ _ hgrad |
| · exact germ_mul (germ_mul (germ_X _) (germ_X _)) (germ_X _) |
| · exact Q.positive_cubic t.1 t.2.1 t.2.2 |
| have hn : ∀ t : Fin d × Fin d × Fin d, |
| ∃ c : InternalVar n → MvPowerSeries (GraphVar n) K, |
| negativeSlopeValue n Q.P Q.Y t.1 * negativeSlopeValue n Q.P Q.Y t.2.1 * |
| negativeSlopeValue n Q.P Q.Y t.2.2 = |
| ∑ e, c e * restrictedGradient n Q.P Q.Y e ∧ |
| ∀ e, Nonempty (LocalEtaleExpansion (c e)) := by |
| intro t |
| apply germ_span_witness _ _ hgrad |
| · exact germ_mul (germ_mul (negativeSlopeValue_germ Q hP hY _) |
| (negativeSlopeValue_germ Q hP hY _)) (negativeSlopeValue_germ Q hP hY _) |
| · exact Q.negative_cubic t.1 t.2.1 t.2.2 |
| choose p hp hpg using hp |
| choose r hr hrg using hn |
| exact ⟨⟨b,p,r,hb,hp,hr⟩,hbg,hpg,hrg⟩ |
|
|
| end BoundaryOnly.FormalObstruction.CoefficientRing |
|
|
| namespace BoundaryOnly.FormalObstruction.CoefficientRing |
| open MvPowerSeries FormalCorrection |
| open scoped Classical |
| variable {K : Type} [Field K] [CharZero K] {d : ℕ} {n : Fin d → ℕ} |
|
|
| theorem no_algebraic_quadratic_data (hd : 5 ≤ d) |
| (Q : QuadraticData (R := K) (n := n)) |
| (hP : ∀ i, Nonempty (LocalEtaleExpansion (Q.P i))) |
| (hY : ∀ c, Nonempty (LocalEtaleExpansion (Q.Y c))) : False := by |
| obtain ⟨W,hb,hpos,hneg⟩ := Q.etale_witnesses hP hY |
| apply no_etale_quadratic_data hd Q W |
| intro e |
| cases e with |
| | wall i => exact (hP i).some |
| | graph c => exact (hY c).some |
| | quadratic c e => exact (hb c e).some |
| | positive t c => exact (hpos t c).some |
| | negative t c => exact (hneg t c).some |
|
|
| end BoundaryOnly.FormalObstruction.CoefficientRing |
|
|
| end |
|
|
| end OAI |
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|
