File size: 4,686 Bytes
e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.RingTheory.DirectFiniteness.LabelSupport
namespace OAI
universe u_1 u_2 u_3 u_4
noncomputable section
open scoped Classical BigOperators
namespace KaplanskyCounterexample.FiniteGeometry
variable {Q : Type u_1} {A : Type u_2} {R : Type u_3} {E : Type u_4} [Field Q] [AddCommGroup A] [Module Q A]
[AddCommGroup E] [Module Q E]
lemma cyclic_two_apart (k : ℕ) (hk3 : 3 ≤ k) (hk5 : k ≤ 5) :
∀ i j : Fin k, i ≠ j → finRotate k i ≠ j → finRotate k j ≠ i →
finRotate k (finRotate k i) = j ∨ finRotate k (finRotate k j) = i := by
interval_cases k <;> decide
namespace DirectionFrame
variable (F : DirectionFrame Q A R E)
lemma vector_mem_U {s : PolygonStep Q R E} {j : R × Q} (hs : s.label = .inl j) :
s.vector F ∈ F.U j := by
cases s with
| inl g => cases Sum.inl.inj hs; exact ⟨(g.scalar,g.component),rfl⟩
| inr e => cases hs
lemma vector_not_mem_U {s : PolygonStep Q R E} {j : R × Q} (hs : s.label ≠ .inl j) :
s.vector F ∉ F.U j := by
intro hm
cases s with
| inl g =>
have hgj : g.label ≠ j := fun he => hs (congrArg Sum.inl he)
have hz := (Submodule.disjoint_def.mp (F.U_disjoint hgj))
((PolygonStep.vector F (.inl g))) ⟨(g.scalar,g.component),rfl⟩ hm
exact PolygonStep.vector_ne_zero F (.inl g) hz
| inr e =>
have hz := (Submodule.disjoint_def.mp (F.U_exceptional_disjoint j e.label))
((PolygonStep.vector F (.inr e))) hm ⟨e.scalar,rfl⟩
exact PolygonStep.vector_ne_zero F (.inr e) hz
lemma two_generic_bundle {k : ℕ} (s : Fin k → PolygonStep Q R E)
(hk : k ≤ 5) (hz : ∑ i, (s i).vector F = 0)
(hadj : ∀ i, (s i).label ≠ (s (finRotate k i)).label)
(hg : (genericLabels (fun i => (s i).label)).card = 2)
(r : R) (hr : ∃ i, (s i).label = .inr r) :
k = 5 ∧ ∀ j ∈ genericLabels (fun i => (s i).label), j.1 = r := by
have hc : ∀ j ∈ genericLabels (fun i => (s i).label),
(Finset.univ.filter (fun i => (s i).label = .inl j)).card = 2 := by
intro j hj
exact F.generic_occurs_twice s hk hz hadj j ((mem_genericLabels _ _).mp hj)
obtain ⟨hk5,hcr⟩ := exceptional_count_singleton_of_two_generic _ hk hc hg r hr
obtain ⟨i,i',g,g',hi,hi',hr',hr'',hne⟩ :=
F.singleton_exceptional_two_slopes s hk hz r hcr
have hne' : g.label ≠ g'.label := fun h => hne (congrArg Prod.snd h)
have hsub : {g.label,g'.label} ⊆ genericLabels (fun i => (s i).label) := by
intro j hj
simp only [Finset.mem_insert,Finset.mem_singleton] at hj
rcases hj with rfl | rfl
· exact (mem_genericLabels _ _).mpr ⟨i,by simp [hi,PolygonStep.label]⟩
· exact (mem_genericLabels _ _).mpr ⟨i',by simp [hi',PolygonStep.label]⟩
have he : genericLabels (fun i => (s i).label) = {g.label,g'.label} := by
exact (Finset.eq_of_subset_of_card_le hsub (by simp [hg,hne'])).symm
refine ⟨hk5,?_⟩
intro j hj
rw [he] at hj
simp only [Finset.mem_insert,Finset.mem_singleton] at hj
rcases hj with rfl | rfl
· exact hr'
· exact hr''
lemma distinct_generic_cosets [Fintype A] {k : ℕ} (s : Fin k → PolygonStep Q R E)
(pt : Fin k → A) (hk3 : 3 ≤ k) (hk5 : k ≤ 5)
(hpt : ∀ i, pt (finRotate k i) - pt i = (s i).vector F)
(hadj : ∀ i, (s i).label ≠ (s (finRotate k i)).label)
{i i' : Fin k} (hii : i ≠ i') {j : R × Q}
(hi : (s i).label = .inl j) (hi' : (s i').label = .inl j) :
AffineCoset.of (F.U j) (pt i) ≠ AffineCoset.of (F.U j) (pt i') := by
have hr : finRotate k i ≠ i' := fun h => hadj i (hi.trans (h ▸ hi').symm)
have hr' : finRotate k i' ≠ i := fun h => hadj i' (hi'.trans (h ▸ hi).symm)
have hh (a b : Fin k) (ha : (s a).label = .inl j)
(hab : finRotate k (finRotate k a) = b)
(hc : AffineCoset.of (F.U j) (pt a) = AffineCoset.of (F.U j) (pt b)) : False := by
have hd : pt b - pt a ∈ F.U j := (AffineCoset.of_eq_iff _ _ _).mp hc.symm
have hm : (s (finRotate k a)).vector F ∈ F.U j := by
have hsub := (F.U j).sub_mem hd (F.vector_mem_U ha)
rw [← hpt a] at hsub
rw [← hpt (finRotate k a),hab]
convert hsub using 1; abel
exact F.vector_not_mem_U (fun he => hadj a (ha.trans he.symm)) hm
intro hc
rcases cyclic_two_apart k hk3 hk5 i i' hii hr hr' with h | h
· exact hh i i' hi h hc
· exact hh i' i hi' h hc.symm
lemma closed_of_points {k : ℕ} (s : Fin k → PolygonStep Q R E) (pt : Fin k → A)
(hpt : ∀ i, pt (finRotate k i) - pt i = (s i).vector F) :
∑ i, (s i).vector F = 0 := by
simp_rw [← hpt]
rw [Finset.sum_sub_distrib,Equiv.sum_comp (finRotate k),sub_self]
end DirectionFrame
end KaplanskyCounterexample.FiniteGeometry
end
end OAI
|