| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators |
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| namespace BoundaryOnly.FormalObstruction |
| namespace ThreeParameters |
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| variable (k : Type*) [Field k] |
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| abbrev Poly := MvPolynomial (Fin 3) k |
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| def squareIdeal : Ideal (Poly k) := |
| Ideal.span (Set.range fun i : Fin 3 => (MvPolynomial.X i : Poly k) ^ 2) |
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| abbrev Ring := Poly k ⧸ squareIdeal k |
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| def s (i : Fin 3) : Ring k := Ideal.Quotient.mk _ (MvPolynomial.X i) |
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| def w : Ring k := s k 0 * s k 1 * s k 2 |
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| @[simp] theorem s_sq (i : Fin 3) : s k i ^ 2 = 0 := by |
| rw [s, ← map_pow] |
| apply Ideal.Quotient.eq_zero_iff_mem.mpr |
| exact Ideal.subset_span ⟨i, rfl⟩ |
|
|
| def topExponent : Fin 3 →₀ ℕ := Finsupp.equivFunOnFinite.symm (fun _ => 1) |
|
|
| @[simp] theorem topExponent_apply (i : Fin 3) : topExponent i = 1 := by |
| simp [topExponent] |
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| theorem topCoeff_ideal (p : Poly k) (hp : p ∈ squareIdeal k) : |
| p.coeff topExponent = 0 := by |
| obtain ⟨c, rfl⟩ := Ideal.mem_span_range_iff_exists_fun.mp hp |
| rw [MvPolynomial.coeff_sum] |
| apply Finset.sum_eq_zero |
| intro i _ |
| rw [MvPolynomial.X_pow_eq_monomial, MvPolynomial.coeff_mul_monomial'] |
| have hi : ¬ Finsupp.single i 2 ≤ topExponent := by |
| intro h |
| have := h i |
| simp only [Finsupp.single_eq_same, topExponent_apply] at this |
| omega |
| rw [ite_eq_right hi] |
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|
| theorem w_ne_zero : w k ≠ 0 := by |
| intro hw |
| have hmem : (MvPolynomial.X 0 * MvPolynomial.X 1 * MvPolynomial.X 2 : Poly k) |
| ∈ squareIdeal k := by |
| apply Ideal.Quotient.eq_zero_iff_mem.mp |
| simpa only [map_mul, s, w] using hw |
| have hc := topCoeff_ideal k _ hmem |
| have he : Finsupp.single (0 : Fin 3) 1 + Finsupp.single 1 1 + |
| Finsupp.single 2 1 = topExponent := by |
| ext i |
| fin_cases i <;> norm_num [Finsupp.single_apply] |
| have hprod : (MvPolynomial.X 0 * MvPolynomial.X 1 * MvPolynomial.X 2 : Poly k) = |
| MvPolynomial.monomial topExponent 1 := by |
| simp only [MvPolynomial.X, MvPolynomial.monomial_mul_monomial, one_mul, he] |
| rw [hprod, MvPolynomial.coeff_monomial, ite_eq_left rfl] at hc |
| exact one_ne_zero hc |
|
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| |
| def augmentation : Ring k →+* k := |
| Ideal.Quotient.lift _ MvPolynomial.constantCoeff (by |
| intro p hp |
| obtain ⟨c, rfl⟩ := Ideal.mem_span_range_iff_exists_fun.mp hp |
| simp) |
|
|
| @[simp] theorem augmentation_mk (p : Poly k) : |
| augmentation k (Ideal.Quotient.mk _ p) = MvPolynomial.constantCoeff p := rfl |
|
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| @[simp] theorem w_mul_s (i : Fin 3) : w k * s k i = 0 := by |
| fin_cases i |
| · calc |
| w k * s k 0 = s k 0 ^ 2 * s k 1 * s k 2 := by dsimp [w]; ring |
| _ = 0 := by simp |
| · calc |
| w k * s k 1 = s k 0 * s k 1 ^ 2 * s k 2 := by dsimp [w]; ring |
| _ = 0 := by simp |
| · calc |
| w k * s k 2 = s k 0 * s k 1 * s k 2 ^ 2 := by dsimp [w]; ring |
| _ = 0 := by simp |
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| |
| theorem w_mul (b : Ring k) : w k * b = augmentation k b • w k := by |
| obtain ⟨p, rfl⟩ := Ideal.Quotient.mk_surjective b |
| induction p using MvPolynomial.induction_on with |
| | C c => |
| simp only [augmentation_mk, MvPolynomial.constantCoeff_C] |
| change w k * algebraMap k (Ring k) c = c • w k |
| simpa only [mul_comm] using (Algebra.smul_def c (w k)).symm |
| | add p q hp hq => |
| simp only [map_add, mul_add, augmentation_mk, hp, hq, add_smul] |
| | mul_X p i hp => |
| simp only [map_mul, augmentation_mk, MvPolynomial.constantCoeff_X, mul_zero, zero_smul] |
| change w k * (Ideal.Quotient.mk _ p * s k i) = 0 |
| rw [← mul_assoc, mul_right_comm, w_mul_s, zero_mul] |
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| |
| |
| theorem residue_contradiction (b : Ring k) (hb : augmentation k b ≠ 0) |
| (hwb : w k * b = 0) : False := by |
| rw [w_mul] at hwb |
| exact w_ne_zero k ((smul_eq_zero.mp hwb).resolve_left hb) |
|
|
| end ThreeParameters |
| end BoundaryOnly.FormalObstruction |
|
|
| namespace BoundaryOnly.FormalObstruction.ThreeParameters |
| variable (k : Type*) [Field k] |
|
|
| @[simp] theorem augmentation_s (i : Fin 3) : augmentation k (s k i) = 0 := by |
| simp [s] |
|
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| @[simp] theorem augmentation_algebraMap (a : k) : |
| augmentation k (algebraMap k (Ring k) a) = a := by |
| change MvPolynomial.constantCoeff (MvPolynomial.C a) = a |
| simp |
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| |
| theorem nilpotent_sub_augmentation (b : Ring k) : |
| IsNilpotent (b - algebraMap k (Ring k) (augmentation k b)) := by |
| obtain ⟨p, rfl⟩ := Ideal.Quotient.mk_surjective b |
| induction p using MvPolynomial.induction_on with |
| | C c => |
| rw [augmentation_mk, MvPolynomial.constantCoeff_C] |
| change IsNilpotent (algebraMap k (Ring k) c - algebraMap k (Ring k) c) |
| simp |
| | add p q hp hq => |
| simp only [map_add] |
| rw [show (Ideal.Quotient.mk _ p + Ideal.Quotient.mk _ q) - |
| (algebraMap k (Ring k) (augmentation k (Ideal.Quotient.mk _ p)) + |
| algebraMap k (Ring k) (augmentation k (Ideal.Quotient.mk _ q))) = |
| (Ideal.Quotient.mk _ p - algebraMap k (Ring k) (augmentation k (Ideal.Quotient.mk _ p))) + |
| (Ideal.Quotient.mk _ q - algebraMap k (Ring k) (augmentation k (Ideal.Quotient.mk _ q))) by ring] |
| exact (Commute.all _ _).isNilpotent_add hp hq |
| | mul_X p i hp => |
| simp only [map_mul, augmentation_mk, MvPolynomial.constantCoeff_X, mul_zero, map_zero, |
| sub_zero] |
| exact (Commute.all _ _).isNilpotent_mul_left ⟨2, s_sq k i⟩ |
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| |
| theorem isUnit_of_augmentation_ne_zero (b : Ring k) (hb : augmentation k b ≠ 0) : |
| IsUnit b := by |
| have hunit : IsUnit (algebraMap k (Ring k) (augmentation k b)) := |
| (isUnit_iff_ne_zero.mpr hb).map _ |
| have hnil := nilpotent_sub_augmentation k b |
| have h := hnil.isUnit_add_left_of_commute hunit (Commute.all _ _) |
| simpa only [add_sub_cancel] using h |
|
|
| end BoundaryOnly.FormalObstruction.ThreeParameters |
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| end |
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| end OAI |
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