| import Mathlib |
| import OAI.Algebra.FiniteTensor.PositiveObstruction |
| import OAI.Algebra.FiniteTensor.ScalarModels |
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| namespace OAI |
|
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| /-! Closed-point specialization of finite-type tensor models. -/ |
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| noncomputable section |
| open scoped BigOperators |
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| namespace PD4Tensor.Spreading |
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| |
| theorem intJacobson : IsJacobsonRing ℤ := by |
| apply isJacobsonRing_iff_prime_eq.mpr |
| intro I hI |
| by_cases h : I = ⊥ |
| · subst I |
| apply le_antisymm |
| · intro x hx |
| have hu := Ideal.mem_jacobson_bot.mp hx (3 : ℤ) |
| rcases Int.isUnit_eq_one_or hu with h | h <;> |
| simp only [Ideal.mem_bot] <;> omega |
| · exact Ideal.le_jacobson |
| · let : I.IsMaximal := hI.isMaximal h |
| exact Ideal.jacobson_eq_self_of_isMaximal |
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| theorem field_finiteType_char_ne_zero (E : Type*) [Field E] |
| [Algebra.FiniteType ℤ E] : ringChar E ≠ 0 := by |
| let : IsJacobsonRing ℤ := intJacobson |
| let : Module.Finite ℤ E := finite_of_finite_type_of_isJacobsonRing ℤ E |
| intro hz |
| have : CharP E 0 := hz ▸ (inferInstance : CharP E (ringChar E)) |
| have : CharZero E := CharP.charP_to_charZero E |
| exact Int.not_isField (isField_of_isIntegral_of_isField |
| (FaithfulSMul.algebraMap_injective ℤ E) (Field.toIsField E)) |
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| theorem exists_odd_closed_point (A : Type*) [CommRing A] [Nontrivial A] |
| [Algebra.FiniteType ℤ A] (h2 : IsUnit (2 : A)) : |
| ∃ I : Ideal A, I.IsMaximal ∧ (ringChar (A ⧸ I)).Prime ∧ ringChar (A ⧸ I) ≠ 2 := by |
| obtain ⟨I,hI⟩ := Ideal.exists_maximal A |
| let : I.IsMaximal := hI |
| let : Field (A ⧸ I) := Ideal.Quotient.field I |
| let : Algebra.FiniteType ℤ (A ⧸ I) := Algebra.FiniteType.quotient ℤ I |
| have hp : (ringChar (A ⧸ I)).Prime := |
| (CharP.char_is_prime_or_zero (A ⧸ I) (ringChar (A ⧸ I))).resolve_right |
| (@field_finiteType_char_ne_zero (A ⧸ I) (Ideal.Quotient.field I) (by |
| convert (Algebra.FiniteType.quotient ℤ I) using 1 |
| rfl)) |
| refine ⟨I,hI,hp,?_⟩ |
| intro heq |
| have hu : IsUnit (2 : A ⧸ I) := by |
| simpa only [map_ofNat] using (Ideal.Quotient.mk I).isUnit_map h2 |
| apply hu.ne_zero |
| have hz := CharP.cast_eq_zero (A ⧸ I) (ringChar (A ⧸ I)) |
| simpa only [heq,Nat.cast_ofNat] using hz |
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| end PD4Tensor.Spreading |
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|
| namespace PD4Tensor.Spreading |
| noncomputable section |
| open scoped BigOperators |
| universe u |
| variable {l : ℕ} (σ : Fin l → Type u) [∀ i,Fintype (σ i)] [∀ i,DecidableEq (σ i)] |
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| |
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| theorem no_tensor_model_field (K : Type*) [Field K] (p : ℕ) [CharP K p] [Fact p.Prime] |
| (hp : p ≠ 2) {m d c : ℕ} (e : ((i : Fin l) × σ i) ≃ Fin d ⊕ Fin c) |
| (hm : 5 ≤ m) (active : Fin m ↪ Fin l) (M : TensorModel K σ m d c e active) : False := by |
| have h2 : (2 : K) ≠ 0 := by |
| intro hz |
| have hd : p ∣ 2 := (CharP.cast_eq_zero_iff K p 2).mp hz |
| rcases (Nat.dvd_prime Nat.prime_two).mp hd with h1 | h2 |
| · exact (Fact.out : p.Prime).ne_one h1 |
| · exact hp h2 |
| let : Invertible (2 : K) := invertibleOfNonzero h2 |
| have hc := PD4Tensor.joinedMatrix_complementary e |
| (fun i j => augmentation K m (MvPowerSeries.coeff (Finsupp.single j 1) (M.H i))) |
| (fun i j => MvPowerSeries.coeff (Finsupp.single j 1) (M.G i)) M.tangent_unit.ne_zero |
| exact PD4Tensor.TruncatedForms.formal_active_obstruction_any K σ p e hm |
| M.F active M.b M.H M.G M.H_zero M.G_zero hc.1 hc.2.1 hc.2.2 |
| M.left_vanishing M.right_vanishing M.left_triple M.right_triple |
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| theorem no_tensor_model_finiteType (A : Type*) [CommRing A] [Nontrivial A] |
| [Algebra.FiniteType ℤ A] (h2 : IsUnit (2 : A)) {m d c : ℕ} |
| (e : ((i : Fin l) × σ i) ≃ Fin d ⊕ Fin c) (hm : 5 ≤ m) (active : Fin m ↪ Fin l) : |
| IsEmpty (TensorModel A σ m d c e active) := by |
| constructor |
| intro M |
| obtain ⟨I,hI,hp,hp2⟩ := exists_odd_closed_point A h2 |
| let : I.IsMaximal := hI |
| let : Field (A ⧸ I) := Ideal.Quotient.field I |
| let : Fact (ringChar (A ⧸ I)).Prime := ⟨hp⟩ |
| exact no_tensor_model_field σ (A ⧸ I) (ringChar (A ⧸ I)) hp2 e hm active |
| (M.map (Ideal.Quotient.mk I)) |
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|
| end |
| end PD4Tensor.Spreading |
| end |
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| end OAI |
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