| 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 |
|
|
