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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 | import OAI.RingTheory.DirectFiniteness.ChosenCriterion
namespace OAI
noncomputable section
open scoped Classical
namespace KaplanskyCounterexample.Prescription
open IncidenceDirections
variable (ℓ : ℕ) [Fact ℓ.Prime]
lemma every_direction (S : Submodule (ZMod ℓ) (Fin 2 → ZMod ℓ))
(hS : Module.finrank (ZMod ℓ) S=1) :
∃ d : Option (ZMod ℓ), LinearMap.range (directionMap d)=S := by
have hne : S ≠ ⊥ := by
intro he
rw [he, finrank_bot] at hS
omega
obtain ⟨x,hx,hx0⟩ := Submodule.exists_mem_ne_zero_of_ne_bot hne
obtain ⟨d,c,hc,hcx⟩ := directionMap_covers x hx0
refine ⟨d,Submodule.eq_of_le_of_finrank_eq ?_ ?_⟩
· have h1 : directionMap d 1 ∈ S := by
have hh := S.smul_mem c⁻¹ hx
rw [← hcx,← map_smul] at hh
simpa [hc] using hh
rintro y ⟨k,rfl⟩
have hh := S.smul_mem k h1
rw [← map_smul] at hh
simpa using hh
· rw [direction_range_finrank,hS]
lemma mem_directionSet (d : Option (ZMod ℓ)) (x : Fin 2 → ZMod ℓ) :
x ∈ directionSet ℓ d ↔ x ∈ LinearMap.range (directionMap d) := by
simp only [directionSet,Finset.mem_image,Finset.mem_univ,true_and,LinearMap.mem_range]
lemma every_direction_unique (S : Submodule (ZMod ℓ) (Fin 2 → ZMod ℓ))
(hS : Module.finrank (ZMod ℓ) S=1) :
∃! d : Option (ZMod ℓ), LinearMap.range (directionMap d)=S := by
obtain ⟨d,hd⟩ := every_direction ℓ S hS
refine ⟨d,hd,?_⟩
intro e he
apply directionSet_injective ℓ
ext x
rw [mem_directionSet,mem_directionSet,he,hd]
lemma orderedAssignment_strictMono {V F : Type} [Fintype F] [LinearOrder F]
(Inc : F → Finset V) (hdeg : ∀ v, Fintype.card {f : F // v ∈ Inc f}=ℓ+1) (v : V) :
letI := directionOrder ℓ
StrictMono (orderedAssignment ℓ Inc hdeg v) := by
let := directionOrder ℓ
exact (Fintype.orderIsoFinOfCardEq (Option (ZMod ℓ)) (by simp)).strictMono.comp
(Fintype.orderIsoFinOfCardEq {f : F // v ∈ Inc f} (hdeg v)).symm.strictMono
end KaplanskyCounterexample.Prescription
end
end OAI
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