| import Mathlib |
| import OAI.Algebra.GroupRing.CellModule |
| import OAI.Algebra.GroupRing.FreeActionNorm |
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| namespace OAI |
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| section |
| namespace TorsionFreeZeroDivisors.CellNorm |
| open scoped Classical BigOperators |
| open CellModule |
| noncomputable section |
| variable {H G I : Type} [CommGroup H] [Group G] [Fintype H] |
| (φ : H→*G) (hφ : Function.Injective φ) |
| def norm (x : CellModule.Module G I) : CellModule.Module G I := ∑h:H,translate (φ h) x |
| lemma norm_add (x y : CellModule.Module G I) : norm φ (x+y)=norm φ x+norm φ y := by |
| simp only [norm,map_add,Finset.sum_add_distrib] |
| lemma norm_sub (x y : CellModule.Module G I) : norm φ (x-y)=norm φ x-norm φ y := by |
| simp only [norm,map_sub,Finset.sum_sub_distrib] |
| lemma equivariance (J : Type) (f : CellModule.Module G I→+CellModule.Module G J) |
| (hf : ∀g x,f (translate g x)=translate g (f x)) (x : CellModule.Module G I) : |
| f (norm φ x)=norm φ (f x) := by |
| simp only [norm,map_sum,hf] |
| @[instance_reducible] def moduleAction : MulAction H (G×I) where |
| smul h p := (φ h*p.1,p.2) |
| one_smul p := by change (φ 1*p.1,p.2)=p; simp |
| mul_smul h k p := by change (φ (h*k)*p.1,p.2)=(φ h*(φ k*p.1),p.2); rw [map_mul,mul_assoc] |
| include hφ in |
| lemma invariant_norm (g : H) (hg : ∀h,h∈Subgroup.zpowers g) (x : CellModule.Module G I) |
| (hx : translate (φ g) x=x) : ∃y,norm φ y=x := by |
| let : MulAction H (G×I) := moduleAction φ |
| let : IsCancelSMul H (G×I) := { |
| right_cancel' h k p he := hφ (mul_right_cancel (congrArg Prod.fst he)) } |
| have hact (h : H) (y : CellModule.Module G I) : FreeActionNorm.act h y=translate (φ h) y := by |
| ext p |
| change y (φ h⁻¹*p.1,p.2)=y ((φ h)⁻¹*p.1,p.2) |
| rw [map_inv] |
| obtain ⟨y,hy⟩ := FreeActionNorm.invariant_norm g hg x ((hact g x).trans hx) |
| exact ⟨y,by simpa only [FreeActionNorm.norm,hact,norm] using hy⟩ |
| include hφ in |
| lemma kernel_norm (g : H) (hg : ∀h,h∈Subgroup.zpowers g) (x : CellModule.Module G I) |
| (hx : norm φ x=0) : ∃y,translate (φ g) y-y=x := by |
| let : MulAction H (G×I) := moduleAction φ |
| let : IsCancelSMul H (G×I) := { |
| right_cancel' h k p he := hφ (mul_right_cancel (congrArg Prod.fst he)) } |
| have hact (h : H) (y : CellModule.Module G I) : FreeActionNorm.act h y=translate (φ h) y := by |
| ext p |
| change y (φ h⁻¹*p.1,p.2)=y ((φ h)⁻¹*p.1,p.2) |
| rw [map_inv] |
| have hn : FreeActionNorm.norm (H:=H) x=0 := by simpa only [FreeActionNorm.norm,hact,norm] using hx |
| obtain ⟨y,hy⟩ := FreeActionNorm.kernel_norm g hg x hn |
| exact ⟨y,by rw [hact] at hy;exact hy⟩ |
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| def augmentation : CellModule.Module G I→+ℤ := Finsupp.liftAddHom (fun _=>AddMonoidHom.id ℤ) |
| omit [Group G] in |
| @[simp] lemma augmentation_single (p : G×I) (z : ℤ) : augmentation (Finsupp.single p z)=z := by simp [augmentation] |
| lemma augmentation_translate (g : G) (x : CellModule.Module G I) : augmentation (translate g x)=augmentation x := by |
| induction x using Finsupp.induction_linear with |
| | zero=>simp |
| | add a b ha hb=>simp only [map_add,ha,hb] |
| | single p z=>simp only [translate_single,augmentation_single] |
| lemma augmentation_norm (x : CellModule.Module G I) : augmentation (norm φ x)=(Fintype.card H:ℤ)*augmentation x := by |
| simp only [norm,map_sum,augmentation_translate,Finset.sum_const,Finset.card_univ,nsmul_eq_mul] |
| lemma augmentation_boundary {A : Type} (q : FreeGroup A→*G) (x : CellModule.Module G A) : |
| augmentation (boundary q x)=0 := by |
| induction x using Finsupp.induction_linear with |
| | zero=>simp |
| | add a b ha hb=>simp only [map_add,ha,hb,add_zero] |
| | single p z=>simp only [boundary_single,map_sub,augmentation_single,sub_self] |
| include hφ in |
| lemma invariant_augmentation (g : H) (hg : ∀h,h∈Subgroup.zpowers g) (x : CellModule.Module G I) |
| (hx : translate (φ g) x=x) : (Fintype.card H:ℤ)∣augmentation x := by |
| obtain ⟨y,hy⟩ := invariant_norm φ hφ g hg x hx |
| exact ⟨augmentation y,by rw [←hy,augmentation_norm]⟩ |
| end |
| end TorsionFreeZeroDivisors.CellNorm |
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| end |
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| end OAI |
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