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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 | 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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