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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import OAI.RingTheory.DirectFiniteness.IncidenceDirections
namespace OAI
universe u_1 u_2 u_3
noncomputable section
open scoped Classical BigOperators
namespace KaplanskyCounterexample.AssignedDirections
open IncidenceDirections
variable {V : Type u_1} {F : Type u_2} (Inc : F → Finset V) (ℓ : ℕ) [Fact ℓ.Prime]
variable (assign : ∀ v, {f : F // v ∈ Inc f} ≃ Option (ZMod ℓ))
def incidenceMap (v : V) (f : {f : F // v ∈ Inc f}) : ZMod ℓ →+ (Fin 2 → ZMod ℓ) :=
(directionMap (assign v f)).toAddMonoidHom
lemma incidenceMap_injective (v : V) (f : {f : F // v ∈ Inc f}) :
Function.Injective (incidenceMap Inc ℓ assign v f) := directionMap_injective _
lemma incidenceMap_independent (v : V) (f g : {f : F // v ∈ Inc f}) (hne : f ≠ g)
(a b : ZMod ℓ) (h : incidenceMap Inc ℓ assign v f a + incidenceMap Inc ℓ assign v g b = 0) :
a = 0 ∧ b = 0 := directionMap_independent ((assign v).injective.ne hne) a b h
lemma incidenceMap_covers (v : V) (a : Fin 2 → ZMod ℓ) (ha : a ≠ 0) :
∃ (f : F) (hf : v ∈ Inc f) (c : ZMod ℓ), c ≠ 0 ∧ incidenceMap Inc ℓ assign v ⟨f,hf⟩ c = a := by
obtain ⟨d,c,hc,hca⟩ := directionMap_covers a ha
obtain ⟨f,hf⟩ := (assign v).surjective d
exact ⟨f.val,f.property,c,hc,by simpa [incidenceMap,hf] using hca⟩
end KaplanskyCounterexample.AssignedDirections
namespace KaplanskyCounterexample.AssignedDevelopment
open GluedGroup AssignedDirections IncidenceDirections
variable {V F : Type} (Inc : F → Finset V) (ℓ : ℕ) [Fact ℓ.Prime]
variable (assign : ∀ v, {f : F // v ∈ Inc f} ≃ Option (ZMod ℓ))
variable (hInc : ∀ f, (Inc f).Nonempty)
abbrev maps := AssignedDirections.incidenceMap Inc ℓ assign
lemma covers : CoversPoints Inc ℓ (maps Inc ℓ assign) :=
AssignedDirections.incidenceMap_covers Inc ℓ assign
abbrev Base := N Inc ℓ (maps Inc ℓ assign)
def point (v : V) : PointGroup ℓ →* Base Inc ℓ assign :=
theta Inc ℓ (maps Inc ℓ assign) (covers Inc ℓ assign) v
include hInc in
lemma point_injective
(hg : 12 ≤ (Planar.incidenceGraph (fun v f => v ∈ Inc f)).egirth) (v : V) :
Function.Injective (point Inc ℓ assign v) := by
apply theta_injective _ _ _ _ hInc _ _ hg v
· intro v f hv
rw [totalMap_of_mem _ _ _ v f hv]
exact AssignedDirections.incidenceMap_injective Inc ℓ assign v ⟨f, hv⟩
· intro v f g hf hg hne a b hab
rw [totalMap_of_mem _ _ _ v f hf, totalMap_of_mem _ _ _ v g hg] at hab
exact (AssignedDirections.incidenceMap_independent Inc ℓ assign v ⟨f, hf⟩ ⟨g, hg⟩
(fun h => hne (congrArg Subtype.val h)) a b hab).1
def line (f : F) : Multiplicative (ZMod ℓ) →* Base Inc ℓ assign :=
(point Inc ℓ assign ((hInc f).choose)).comp
(AddMonoidHom.toMultiplicative (maps Inc ℓ assign (hInc f).choose
⟨f, (hInc f).choose_spec⟩))
lemma point_direction (v : V) (f : F) (hf : v ∈ Inc f) (c : ZMod ℓ) :
point Inc ℓ assign v (Multiplicative.ofAdd (maps Inc ℓ assign v ⟨f, hf⟩ c)) =
line Inc ℓ assign hInc f (Multiplicative.ofAdd c) := by
by_cases hc : c = 0
· subst c
simp
· exact (theta_generator Inc ℓ (maps Inc ℓ assign) (covers Inc ℓ assign) v f hf c hc).trans
(theta_generator Inc ℓ (maps Inc ℓ assign) (covers Inc ℓ assign) (hInc f).choose f (hInc f).choose_spec c hc).symm
lemma line_injective
(hg : 12 ≤ (Planar.incidenceGraph (fun v f => v ∈ Inc f)).egirth) (f : F) :
Function.Injective (line Inc ℓ assign hInc f) := by
apply (point_injective Inc ℓ assign hInc hg (hInc f).choose).comp
exact AssignedDirections.incidenceMap_injective Inc ℓ assign _ _
variable [Fintype F]
lemma subgroup_decomposition {K : Type u_3} [CommRing K] (v : V) :
∑ a : PointGroup ℓ, MonoidAlgebra.single (point Inc ℓ assign v a) (1 : K) =
1 + ∑ f : {f : F // v ∈ Inc f},
((∑ c : Multiplicative (ZMod ℓ), MonoidAlgebra.single (line Inc ℓ assign hInc f.val c) (1 : K)) - 1) := by
have h := sum_directions (K := ZMod ℓ) (fun a =>
MonoidAlgebra.single (point Inc ℓ assign v (Multiplicative.ofAdd a)) (1 : K))
rw [← Equiv.sum_comp Multiplicative.ofAdd
(fun a : PointGroup ℓ => MonoidAlgebra.single (point Inc ℓ assign v a) (1 : K))]
rw [h]
simp only [ofAdd_zero, map_one, ← MonoidAlgebra.one_def]
congr 1
rw [← Equiv.sum_comp (assign v)
(fun d : Option (ZMod ℓ) =>
(∑ c : ZMod ℓ, MonoidAlgebra.single
(point Inc ℓ assign v (Multiplicative.ofAdd (directionMap d c))) (1 : K)) - 1)]
apply Finset.sum_congr rfl
intro f _
congr 1
rw [← Equiv.sum_comp Multiplicative.ofAdd
(fun c : Multiplicative (ZMod ℓ) => MonoidAlgebra.single (line Inc ℓ assign hInc f.val c) (1 : K))]
apply Finset.sum_congr rfl
intro c _
exact congrArg (fun g => MonoidAlgebra.single g (1 : K))
(point_direction Inc ℓ assign hInc v f.val f.property c)
end KaplanskyCounterexample.AssignedDevelopment
end
end OAI
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