| import OAI.RingTheory.DirectFiniteness.FieldExtensions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped Classical BigOperators |
| open Module |
| namespace KaplanskyCounterexample.FiniteGeometry |
| variable {Q A B R E : Type} [Field Q] [AddCommGroup A] [Module Q A] |
| [AddCommGroup B] [Module Q B] [AddCommGroup E] [Module Q E] |
|
|
| namespace DirectionFrame |
|
|
| def transport (F : DirectionFrame Q A R E) (f : A ≃ₗ[Q] B) : DirectionFrame Q B R E where |
| w r := f.toLinearMap.comp (F.w r) |
| e j := f.toLinearMap.comp (F.e j) |
| winj r := f.injective.comp (F.winj r) |
| einj j := f.injective.comp (F.einj j) |
| direct := by |
| intro k hk j hj x hx hz |
| have hm (i : Fin k) : f.symm (x i) ∈ Sum.elim (fun r => (F.w r).range) (fun j => (F.e j).range) (j i) := by |
| have hxi := hx i |
| cases he : j i with |
| | inl r => |
| simp only [he,Sum.elim_inl] at hxi ⊢ |
| obtain ⟨a,ha⟩ := hxi |
| exact ⟨a,by rw [← ha]; exact (f.symm_apply_apply _).symm⟩ |
| | inr t => |
| simp only [he,Sum.elim_inr] at hxi ⊢ |
| obtain ⟨a,ha⟩ := hxi |
| exact ⟨a,by rw [← ha]; exact (f.symm_apply_apply _).symm⟩ |
| have hsum : ∑ i, f.symm (x i)=0 := by rw [← map_sum,hz,map_zero] |
| have hh := F.direct k hk j hj (fun i => f.symm (x i)) hm hsum |
| intro i |
| exact f.symm.injective ((hh i).trans f.symm.map_zero.symm) |
|
|
| end DirectionFrame |
|
|
| lemma exists_coordinate_frame (Q : Type) [Field Q] [Fintype Q] [CharP Q 2] |
| (R : Type) [Fintype R] |
| (hsize : Fintype.card R + Fintype.card R * Fintype.card Q ≤ (Fintype.card Q)^120) : |
| Nonempty (DirectionFrame Q (Fin 1200 → Q) R (Fin 119 → Q)) := by |
| obtain ⟨F,hF,hA,hfin,hD⟩ := exists_finite_extension Q 120 (by decide) |
| let _ := hF |
| let _ := hA |
| let _ := hfin |
| have hcard : Fintype.card F=(Fintype.card Q)^120 := by |
| rw [Module.card_eq_pow_finrank (K := Q) (V := F),hD] |
| obtain ⟨frame⟩ := exists_directionFrame (Q := Q) (F := F) (R := R) 120 (by decide) hD (by rwa [hcard]) |
| let f : (Fin 10 → F) ≃ₗ[Q] (Fin 1200 → Q) := LinearEquiv.ofFinrankEq _ _ (by |
| rw [Module.finrank_pi_fintype,Module.finrank_pi] |
| simp [hD]) |
| exact ⟨frame.transport f⟩ |
|
|
| end KaplanskyCounterexample.FiniteGeometry |
| end |
|
|
| end OAI |
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|
