| import Mathlib |
| import OAI.Algebra.FiniteTensor.DeformedGraphs |
| import OAI.Algebra.FiniteTensor.RelativeKoszul |
|
|
| namespace OAI |
|
|
| /-! The finite chart obstruction and formal evaluation of separated potentials. -/ |
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| noncomputable section |
| open scoped BigOperators |
|
|
| namespace PD4Tensor.TruncatedForms |
| noncomputable section |
| open scoped TensorProduct BigOperators |
| open Forms FrobeniusTruncation FiniteCoordinates |
| attribute [local instance] parameterCharP |
| universe u |
| variable (K : Type*) [Field K] (p : ℕ) [CharP K p] [Fact p.Prime] [Invertible (2 : K)] |
| {n : ℕ} (σ : Fin (n+1) → Type u) [∀ i,Fintype (σ i)] [∀ i,DecidableEq (σ i)] |
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| |
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| def sigmaDeformed (S b : ∀ i,A K (σ i) p) (q : Fin 3 ↪ Fin (n+1)) : |
| Ring (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p) := |
| coefficientMap K (T K 3) _ (fun _ => p) (sigmaPotential K p σ S) + |
| ∑ i,threeParameters (K:=K) q i • |
| coefficientMap K (T K 3) _ (fun _ => p) (sigmaBlock K p σ i (b i)) |
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| |
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| theorem finite_chart_obstruction {d c : ℕ} |
| (r : ((i : Fin (n+1)) × σ i) ≃ Fin (c+d)) |
| (e : ((i : Fin (n+1)) × σ i) ≃ Fin d ⊕ Fin c) |
| (q0 : Ring K ((i : Fin (n+1)) × σ i) (fun _ => p) ≃ₐ[K] |
| Ring K ((i : Fin (n+1)) × σ i) (fun _ => p)) |
| (qL : (Fin 3 ↪ Fin (n+1)) → |
| Ring (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p) ≃ₐ[T K 3] |
| Ring (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p)) |
| (S b : ∀ i,A K (σ i) p) (active : Finset (Fin (n+1))) |
| (hb : ∀ i,i∉active → b i=0) (hA : 5≤active.card) |
| (hL0 : splitRestriction (Fact.out : p.Prime).pos e q0 (sigmaPotential K p σ S)=0) |
| (hR : splitRestriction (Fact.out : p.Prime).pos (e.trans (Equiv.sumComm _ _)) q0 |
| (sigmaPotential K p σ S)=0) |
| (htripR : ∀ s : Finset (Fin (n+1)), s⊆active → s.card=3 → |
| splitRestriction (Fact.out : p.Prime).pos (e.trans (Equiv.sumComm _ _)) q0 |
| (sigmaSelected K p σ b s) ∈ Ideal.span (Set.range (fun i => |
| splitRestriction (Fact.out : p.Prime).pos (e.trans (Equiv.sumComm _ _)) q0 |
| (chartDerivative K K _ p i (sigmaPotential K p σ S))))) |
| (hspecial : ∀ q : Fin 3 ↪ Fin (n+1), (∀ j,q j∈active) → ∀ a, |
| coefficientHom K (T K 3) K _ (fun _ => p) (augmentationAlg K 3) (qL q a)= |
| q0 (coefficientHom K (T K 3) K _ (fun _ => p) (augmentationAlg K 3) a)) |
| (hL : ∀ q : Fin 3 ↪ Fin (n+1), (∀ j,q j∈active) → |
| splitRestriction (Fact.out : p.Prime).pos e (qL q) (sigmaDeformed K p σ S b q)=0) |
| (htripL : ∀ q : Fin 3 ↪ Fin (n+1), (∀ j,q j∈active) → |
| splitRestriction (Fact.out : p.Prime).pos e (qL q) |
| (algebraMap (T K 3) (Ring (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p)) |
| (t K 3 0*t K 3 1*t K 3 2)) ∈ Ideal.span (Set.range (fun i => |
| splitRestriction (Fact.out : p.Prime).pos e (qL q) |
| (chartDerivative K (T K 3) _ p i (sigmaDeformed K p σ S b q))))) : False := by |
| classical |
| obtain ⟨q,κ,hq,hκ,hpair⟩ := exists_three_graph_tensor_cycle K p σ r e q0 S b active hb hA hL0 hR htripR |
| let η := sigmaExtendedForms K (T K 3) p σ κ |
| let lam := chartNormal K (T K 3) ((i : Fin (n+1)) × σ i) p e (qL q) |
| let u : T K 3 := t K 3 0*t K 3 1*t K 3 2 |
| let J := extendFormFunctional K (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p) |
| (orientedResidue K _ p (Fact.out : p.Prime).pos r) |
| have hη : relativeKoszul K (T K 3) _ p (sigmaDeformed K p σ S b q) η=0 := by |
| change relativeKoszul K (T K 3) _ p |
| (coefficientMap K (T K 3) _ (fun _ => p) (sigmaPotential K p σ S) + |
| ∑ i,threeParameters (K:=K) q i • |
| coefficientMap K (T K 3) _ (fun _ => p) (sigmaBlock K p σ i (b i))) |
| (sigmaExtendedForms K (T K 3) p σ κ)=0 |
| rw [relativeKoszul_sigmaExtended,hκ] |
| exact (sigmaExtendedForms K (T K 3) p σ).map_zero |
| have hη' : oneForm K _ _ (fun i => |
| chartDerivative K (T K 3) _ p i (sigmaDeformed K p σ S b q))*η=0 := by |
| rw [relativeKoszul_eq] at hη |
| exact hη |
| have hB := chartNormal_boundary K (T K 3) ((i : Fin (n+1)) × σ i) p |
| (Fact.out : p.Prime).pos e (qL q) _ (hL q hq) _ (htripL q hq) |
| rw [scalar_parameter_mul] at hB |
| have hz : (u • lam)*η=0 := boundary_mul_cycle_zero _ _ _ hB hη' |
| have hzero : u * J (lam*η)=0 := by |
| change u • J (lam*η)=0 |
| rw [←map_smul,←smul_mul_assoc,hz,map_zero] |
| have hunit : IsUnit (J (lam*η)) := by |
| apply isUnit_of_augmentation_ne_zero |
| change (augmentationAlg K 3) (extendFormFunctional K (T K 3) _ (fun _ => p) |
| (orientedResidue K _ p (Fact.out : p.Prime).pos r) (lam*η))≠0 |
| rw [extendFormFunctional_specialization,map_mul] |
| have hlam := chartNormal_specialization K (T K 3) K ((i : Fin (n+1)) × σ i) p |
| (augmentationAlg K 3) e (qL q) q0 (hspecial q hq) |
| change mapCoefficients K _ _ _ _ lam=_ at hlam |
| rw [hlam] |
| change orientedResidue K _ p (Fact.out : p.Prime).pos r |
| (chartNormal K K _ p e q0 * mapCoefficients K _ _ _ _ |
| (sigmaExtendedForms K (T K 3) p σ κ))≠0 |
| rw [sigmaExtendedForms_specialization] |
| exact hpair |
| exact triple_mul_unit_ne_zero K 3 0 1 2 (by decide) (by decide) (by decide) hunit hzero |
|
|
| end |
| end PD4Tensor.TruncatedForms |
|
|
| namespace PD4Tensor.FrobeniusTruncation |
| noncomputable section |
| variable (K σ τ : Type*) [Field K] [Fintype σ] |
| [Fintype τ] (p : ℕ) |
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| |
| theorem formalEval_rename (f : σ → τ) (F : MvPowerSeries σ K) : |
| formalEval K τ p (MvPowerSeries.rename f F)=rename K σ p f (formalEval K σ p F) := by |
| unfold formalEval |
| erw [nilEval_rename] |
| have hh := nilEval_map (rename K σ p f) (coord K σ p) |
| (fun i => ⟨p,variable_pow K σ p i⟩) F |
| simpa only [rename_coord,Function.comp_def] using hh.symm |
|
|
| end |
| end PD4Tensor.FrobeniusTruncation |
|
|
| namespace PD4Tensor.FiniteCoordinates |
| noncomputable section |
| variable (K R σ : Type*) [CommRing K] [CommRing R] [Algebra K R] |
| [Fintype σ] (p : ℕ) |
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| |
| theorem nilEval_extension (F : MvPowerSeries σ K) : |
| nilEval R (coord R σ (fun _ => p)) (fun j => ⟨p,coord_pow _ _ _ j⟩) |
| (MvPowerSeries.map (algebraMap K R) F)= |
| coefficientMap K R σ (fun _ => p) |
| (nilEval K (coord K σ (fun _ => p)) (fun j => ⟨p,coord_pow _ _ _ j⟩) F) := by |
| have hh := nilEval_coefficients (algebraMap K R) |
| (coefficientMap K R σ (fun _ => p)).toRingHom |
| (fun k => by |
| change coefficientMap K R σ (fun _ => p) (algebraMap K (Ring K σ (fun _ => p)) k)=_ |
| rw [AlgHom.commutes] |
| exact IsScalarTower.algebraMap_apply K R (Ring R σ (fun _ => p)) k) |
| (coord K σ (fun _ => p)) (fun j => ⟨p,coord_pow _ _ _ j⟩) F |
| simp only [AlgHom.toRingHom_eq_coe,AlgHom.coe_toRingHom,coefficientMap_coord] at hh |
| exact hh.symm |
|
|
| end |
| end PD4Tensor.FiniteCoordinates |
|
|
| namespace PD4Tensor.TruncatedForms |
| noncomputable section |
| open scoped BigOperators |
| open FrobeniusTruncation |
| universe u |
| variable (K : Type*) [Field K] (p : ℕ) {n : ℕ} |
| (σ : Fin (n+1) → Type u) [∀ i,Fintype (σ i)] [∀ i,DecidableEq (σ i)] |
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| |
| |
| @[simp] theorem formalSigmaPotential_eval (F : ∀ i,MvPowerSeries (σ i) K) : |
| formalEval K _ p (formalSigmaPotential K σ F)= |
| sigmaPotential K p σ (fun i => formalEval K (σ i) p (F i)) := by |
| simp only [formalSigmaPotential,map_sum,formalEval_rename,sigmaPotential] |
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| omit [∀ i,DecidableEq (σ i)] in |
| @[simp] theorem formalSigmaSelected_eval (b : ∀ i,MvPowerSeries (σ i) K) |
| (s : Finset (Fin (n+1))) : |
| formalEval K _ p (formalSigmaSelected K σ b s)= |
| sigmaSelected K p σ (fun i => formalEval K (σ i) p (b i)) s := by |
| simp only [formalSigmaSelected,map_prod,formalEval_rename,sigmaSelected,sigmaBlock] |
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|
| open FiniteCoordinates in |
| @[simp] theorem formalSigmaDeformed_eval (F b : ∀ i,MvPowerSeries (σ i) K) |
| (q : Fin 3 ↪ Fin (n+1)) : |
| nilEval (T K 3) (coord (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p)) |
| (fun j => ⟨p,coord_pow _ _ _ j⟩) (formalSigmaDeformed K σ F b q)= |
| sigmaDeformed K p σ (fun i => formalEval K (σ i) p (F i)) |
| (fun i => formalEval K (σ i) p (b i)) q := by |
| simp only [formalSigmaDeformed,map_add,map_sum,map_mul,nilEval_extension] |
| unfold sigmaDeformed |
| congr 1 |
| · exact congrArg (coefficientMap K (T K 3) _ (fun _ => p)) |
| (formalSigmaPotential_eval K p σ F) |
| · apply Finset.sum_congr rfl |
| intro i hi |
| erw [show nilEval (T K 3) (coord (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p)) |
| (fun j => ⟨p,coord_pow _ _ _ j⟩) (MvPowerSeries.C (threeParameters (K:=K) q i))= |
| algebraMap (T K 3) (Ring (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p)) |
| (threeParameters (K:=K) q i) from (nilEval (T K 3) _ _).commutes _] |
| change algebraMap (T K 3) _ (threeParameters (K:=K) q i)* |
| coefficientMap K (T K 3) _ (fun _ => p) |
| (formalEval K _ p (MvPowerSeries.rename (Sigma.mk i) (b i)))=_ |
| rw [formalEval_rename] |
| exact (Algebra.smul_def _ _).symm |
|
|
| end |
| end PD4Tensor.TruncatedForms |
|
|
| namespace PD4Tensor |
| noncomputable section |
| open MvPowerSeries.WithPiTopology |
| variable {R A σ : Type*} [CommRing R] [CommRing A] [Algebra R A] [Finite σ] |
|
|
| theorem nilEval_zero (q : MvPowerSeries σ R) : |
| nilEval R (fun _ : σ => (0 : A)) (fun _ => IsNilpotent.zero) q = |
| algebraMap R A (MvPowerSeries.constantCoeff q) := by |
| let : UniformSpace R := ⊥ |
| let : UniformSpace A := ⊥ |
| let : ContinuousSMul R A := ⟨continuous_of_discreteTopology⟩ |
| have hn : MvPowerSeries.HasEval (fun _ : σ => (0 : A)) := |
| ⟨fun _ => IsNilpotent.zero.isTopologicallyNilpotent, |
| by simp only [Filter.cofinite_eq_bot]; exact Filter.tendsto_bot⟩ |
| have hc : Continuous (nilEval R (fun _ : σ => (0 : A)) |
| (fun _ => IsNilpotent.zero) : MvPowerSeries σ R → A) := MvPowerSeries.continuous_aeval hn |
| have hh := Continuous.ext_on (denseRange_toMvPowerSeries (σ:=σ) (R:=R)) hc |
| ((continuous_of_discreteTopology : Continuous (algebraMap R A)).comp |
| (continuous_constantCoeff R)) (by |
| rintro _ ⟨f,rfl⟩ |
| rw [nilEval_coe] |
| change MvPolynomial.aeval (fun _ : σ => (0 : A)) f = |
| algebraMap R A (MvPowerSeries.coeff 0 (f : MvPowerSeries σ R)) |
| rw [MvPolynomial.coeff_coe] |
| exact MvPolynomial.aeval_zero f) |
| exact congr_fun hh q |
|
|
| end |
| end PD4Tensor |
| end |
|
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| end OAI |
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|
