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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 | import OAI.RingTheory.DirectFiniteness.DirectionFrame
namespace OAI
universe u_1 u_2 u_3
noncomputable section
open scoped Classical BigOperators
open Module
namespace KaplanskyCounterexample.FiniteGeometry
theorem exists_directionFrame {Q : Type u_1} {F : Type u_2} {R : Type u_3} [Field Q] [Field F] [Algebra Q F]
[FiniteDimensional Q F] [Fintype Q] [Fintype F] [Fintype R]
(b : ℕ) (hb : 2 ≤ b) (hdim : finrank Q F = b)
(hsize : Fintype.card R + Fintype.card R * Fintype.card Q ≤ Fintype.card F) :
Nonempty (DirectionFrame Q (Fin 10 → F) R (Fin (b-1) → Q)) := by
let ι := R ⊕ (R × Q)
let dim : ι → ℕ := Sum.elim (fun _ => 2) (fun _ => b-1)
obtain ⟨P,hd,hP⟩ := exists_shortDirect_subspaces (Q := Q) (F := F)
(ι := ι) (by decide : 0 < 10) dim (by
intro i
cases i <;> simp [dim,hdim,hb]) (by simpa [ι] using hsize)
let W (r : R) : (Q × Q) ≃ₗ[Q] P (.inl r) :=
LinearEquiv.ofFinrankEq _ _ (by simpa [dim] using (hd (.inl r)).symm)
let E (j : R × Q) : (Fin (b-1) → Q) ≃ₗ[Q] P (.inr j) :=
LinearEquiv.ofFinrankEq _ _ (by simpa [dim] using (hd (.inr j)).symm)
let w (r : R) := (P (.inl r)).subtype.comp (W r).toLinearMap
let e (j : R × Q) := (P (.inr j)).subtype.comp (E j).toLinearMap
refine ⟨{w := w, e := e, winj := ?_, einj := ?_, direct := ?_}⟩
· intro r
exact Subtype.val_injective.comp (W r).injective
· intro j
exact Subtype.val_injective.comp (E j).injective
· intro k hk j hj x hx hz
apply hP k hk j hj x _ hz
intro i
have hh := hx i
cases hji : j i with
| inl r =>
simp only [hji, Sum.elim_inl] at hh
obtain ⟨a,ha⟩ := hh
rw [← ha]
exact (W r a).property
| inr t =>
simp only [hji, Sum.elim_inr] at hh
obtain ⟨a,ha⟩ := hh
rw [← ha]
exact (E t a).property
end KaplanskyCounterexample.FiniteGeometry
end
end OAI
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