| import OAI.RingTheory.DirectFiniteness.Rectangular |
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| namespace OAI |
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| universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 |
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| noncomputable section |
| open scoped Classical BigOperators |
| namespace KaplanskyCounterexample.CharacterTwist |
| open CharacterIdempotents AlgebraicPacking |
| variable {A : Type u_1} {N : Type u_2} {C : Type u_3} {V : Type u_4} {m : Type u_5} {K : Type u_6} [CommGroup A] [Group N] [CommGroup C] |
| variable [Field K] [Fintype A] [Fintype C] [Fintype V] [Fintype m] |
| variable (ι : V → A →* N) (hi : ∀ v, Function.Injective (ι v)) (v₀ : V) |
| variable (α : V → A →* C) (ξ : C →* Kˣ) |
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| lemma map_pushed {G : Type u_7} {H : Type u_8} [Group G] [Group H] |
| (f : G →* H) (g : A →* G) (χ : A →* Kˣ) : |
| MonoidAlgebra.mapDomainAlgHom K K f (pushed g χ) = pushed (f.comp g) χ := by |
| unfold pushed |
| rw [MonoidAlgebra.mapDomainAlgHom_comp] |
| rfl |
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| lemma defect_ne_zero (X : Matrix m V K) (hlt : Fintype.card m < Fintype.card V) |
| (hA : (Fintype.card A : K) ≠ 0) (hC : (Fintype.card C : K) ≠ 0) : |
| Matrix.diagonal (fun _ : V => e ι hi v₀ α ξ) * |
| (Rectangular.T X (P ι hi v₀ α) * Rectangular.U X (P ι hi v₀ α) - |
| Matrix.diagonal (P ι hi v₀ α)) ≠ 0 := by |
| let pH (v : V) : MonoidAlgebra K (N × C) := |
| pushed ((MonoidHom.inl N C).comp (ι v)) 1 |
| let eH : MonoidAlgebra K (N × C) := pushed (MonoidHom.inr N C) ξ |
| let f := MonoidAlgebra.mapDomainAlgHom K K (emb ι hi v₀ α) |
| have hf : Function.Injective f := MonoidAlgebra.mapDomain_injective (emb_injective ι hi v₀ α) |
| have hp (v : V) : f (pH v) = P ι hi v₀ α v := map_pushed _ _ _ |
| have he : f eH = e ι hi v₀ α ξ := map_pushed _ _ _ |
| let ρ : (N × C) →* K := (Units.coeHom K).comp (ξ.comp (MonoidHom.snd N C)) |
| let ε := MonoidAlgebra.lift K K (N × C) ρ |
| have hεp (v : V) : ε (pH v) = 1 := by |
| apply lift_pushed _ _ ρ _ hA |
| intro a |
| simp [ρ] |
| have hεe : ε eH = 1 := by |
| apply lift_pushed _ _ ρ _ hC |
| intro c |
| rfl |
| intro hz |
| apply transpose_mul_ne_one X hlt |
| ext v w |
| have hzvw := congrFun (congrFun hz v) w |
| rw [Matrix.diagonal_mul] at hzvw |
| change e ι hi v₀ α ξ * ((∑ i : m, (X i v • P ι hi v₀ α v) * |
| (X i w • P ι hi v₀ α w)) - (if v = w then P ι hi v₀ α v else 0)) = 0 at hzvw |
| have hpre : eH * ((∑ i : m, (X i v • pH v) * (X i w • pH w)) - |
| (if v = w then pH v else 0)) = 0 := by |
| apply hf |
| simp only [map_mul, map_sub, map_sum, map_smul, hp, he, map_zero] |
| simpa only [apply_ite, map_zero, hp] using hzvw |
| have hs := congrArg ε hpre |
| simp only [map_mul, map_sub, map_sum, map_smul, hεp, hεe, one_mul, |
| smul_eq_mul, mul_one, map_zero, apply_ite] at hs |
| apply sub_eq_zero.mp |
| simpa only [Matrix.mul_apply, Matrix.transpose_apply, Matrix.one_apply, apply_ite] using hs |
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| theorem square_witnesses {F : Type u_9} [Fintype F] |
| (X : Matrix m V K) (Inc : F → Finset V) |
| (lineSum : F → MonoidAlgebra K (L ι hi v₀ α)) |
| (hdec : ∀ v, P (K := K) ι hi v₀ α v = |
| 1 + ∑ f : {f : F // v ∈ Inc f}, (lineSum f.val - 1)) |
| (hglobal : X * X.transpose = 1) |
| (hlocal : ∀ f i j, ∑ v ∈ Inc f, X i v * X j v = 0) |
| (hlt : Fintype.card m < Fintype.card V) |
| (hA : (Fintype.card A : K) ≠ 0) (hC : (Fintype.card C : K) ≠ 0) |
| (slot : V → m) (hξ : Function.Injective ξ) |
| (hα : ∀ v w, v ≠ w → slot v = slot w → α v ≠ α w) : |
| ∃ a b : Matrix m m (MonoidAlgebra K (L ι hi v₀ α)), a * b = 1 ∧ b * a ≠ 1 := by |
| have hp (v : V) : P (K := K) ι hi v₀ α v * P ι hi v₀ α v = P ι hi v₀ α v := |
| pushed_idempotent _ _ hA |
| obtain ⟨hUP,hPT,hUT⟩ := Rectangular.identities X (P ι hi v₀ α) lineSum Inc hp hdec hglobal hlocal |
| apply square_of_projections (Rectangular.U X (P ι hi v₀ α)) |
| (Rectangular.T X (P ι hi v₀ α)) (P ι hi v₀ α) (Q ι hi v₀ α ξ) |
| (u ι hi v₀ α) (uInv ι hi v₀ α) slot (e ι hi v₀ α ξ) |
| (pushed_idempotent _ _ hC) (e_central ι hi v₀ α ξ) hp hUP hPT hUT |
| (uInv_mul_u ι hi v₀ α) (conjugate_projection ι hi v₀ α ξ) |
| _ (defect_ne_zero ι hi v₀ α ξ X hlt hA hC) |
| intro v w hvw hslot |
| apply pushed_orthogonal |
| intro heq |
| apply hα v w hvw hslot |
| apply MonoidHom.ext |
| intro a |
| apply hξ |
| have heq' := congrArg (fun χ : A →* Kˣ => χ a) heq |
| exact inv_injective heq' |
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| end KaplanskyCounterexample.CharacterTwist |
| end |
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| end OAI |
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