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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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.RingTheory.DirectFiniteness.PrescribedFields
namespace OAI
universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12 u_13 u_14 u_15 u_16 u_17
noncomputable section
open scoped BigOperators
open Module
namespace KaplanskyCounterexample.FiniteGeometry
def curve {F : Type u_1} [Field F] (m : ℕ) (ξ : F) : Fin m → F := fun j => ξ ^ j.val
lemma curve_relation {F : Type u_2} [Field F] {k m : ℕ}
(hk : k ≤ m) {ξ : Fin k → F} (hξ : Function.Injective ξ) (a : Fin k → F)
(h : ∑ i, a i • curve m (ξ i) = 0) : a = 0 := by
apply Matrix.eq_zero_of_forall_pow_sum_mul_pow_eq_zero hξ
intro j
have hj := congrFun h (Fin.castLE hk j)
simpa [curve, Finset.sum_apply, Pi.smul_apply] using hj
lemma curve_linearIndependent {F : Type u_3} [Field F] {k m : ℕ}
(hk : k ≤ m) {ξ : Fin k → F} (hξ : Function.Injective ξ) :
LinearIndependent F (fun i => curve m (ξ i)) := by
rw [Fintype.linearIndependent_iff]
intro a ha
exact congrFun (curve_relation hk hξ a ha)
/-- The one-dimensional extension-field line, viewed over the ground field. -/
def curveMap {Q : Type u_4} {F : Type u_5} [Field Q] [Field F] [Algebra Q F]
(m : ℕ) (ξ : F) : F →ₗ[Q] (Fin m → F) where
toFun a := a • curve m ξ
map_add' a b := add_smul _ _ _
map_smul' c a := by
ext j
simp [Pi.smul_apply, Algebra.smul_def, mul_assoc]
lemma curveMap_injective {Q : Type u_6} {F : Type u_7} [Field Q] [Field F] [Algebra Q F]
{m : ℕ} (hm : 0 < m) (ξ : F) : Function.Injective (curveMap (Q := Q) m ξ) := by
intro a b hab
have := congrFun hab ⟨0, hm⟩
simpa [curveMap, curve] using this
/-- Any at most m of the extension-field lines have direct sum over Q.
Each chosen Q-subspace can subsequently be taken inside its own line. -/
lemma curveMap_direct {Q : Type u_8} {F : Type u_9} [Field Q] [Field F] [Algebra Q F]
{k m : ℕ} (hk : k ≤ m) {ξ : Fin k → F} (hξ : Function.Injective ξ)
(x : Fin k → (Fin m → F))
(hx : ∀ i, x i ∈ (curveMap (Q := Q) m (ξ i)).range)
(hsum : ∑ i, x i = 0) : ∀ i, x i = 0 := by
choose a ha using hx
have hrel : ∑ i, a i • curve m (ξ i) = 0 := by simpa [← ha, curveMap] using hsum
have hz := curve_relation hk hξ a hrel
intro i
rw [← ha i]
simp [hz]
/-- Select any required ground-field dimension inside a curve line. -/
lemma exists_subspace_in_curve {Q : Type u_10} {F : Type u_11} [Field Q] [Field F] [Algebra Q F]
[FiniteDimensional Q F] {m : ℕ} (hm : 0 < m) (ξ : F)
(d : ℕ) (hd : d ≤ finrank Q F) :
∃ P : Submodule Q (Fin m → F), finrank Q P = d ∧
P ≤ (curveMap (Q := Q) m ξ).range := by
obtain ⟨f, hf⟩ := exists_linearIndependent_of_le_finrank (R := Q) (M := F) hd
let g := curveMap (Q := Q) m ξ
have hg : Function.Injective g := curveMap_injective hm ξ
refine ⟨Submodule.span Q (Set.range (g ∘ f)), ?_, ?_⟩
· have hgf := hf.map' g (LinearMap.ker_eq_bot.mpr hg)
simpa using finrank_span_eq_card hgf
· apply Submodule.span_le.mpr
rintro _ ⟨i, rfl⟩
exact ⟨f i, rfl⟩
/-- Direct sum for distinct selections of at most m members of a family. -/
def ShortDirect {Q : Type u_12} {A : Type u_13} {ι : Type u_14} [Field Q] [AddCommGroup A] [Module Q A]
(m : ℕ) (P : ι → Submodule Q A) : Prop :=
∀ (k : ℕ), k ≤ m → ∀ j : Fin k → ι, Function.Injective j →
∀ x : Fin k → A, (∀ i, x i ∈ P (j i)) → (∑ i, x i) = 0 → ∀ i, x i = 0
/-- The Vandermonde construction of all the required direction spaces. -/
theorem exists_shortDirect_subspaces {Q : Type u_15} {F : Type u_16} {ι : Type u_17} [Field Q] [Field F] [Algebra Q F]
[FiniteDimensional Q F] [Fintype F] [Fintype ι]
{m : ℕ} (hm : 0 < m) (d : ι → ℕ) (hd : ∀ i, d i ≤ finrank Q F)
(hcard : Fintype.card ι ≤ Fintype.card F) :
∃ P : ι → Submodule Q (Fin m → F),
(∀ i, finrank Q (P i) = d i) ∧ ShortDirect m P := by
obtain ⟨ξ⟩ := (Function.Embedding.nonempty_iff_card_le).mpr hcard
choose P hPdim hPle using fun i => exists_subspace_in_curve hm (ξ i) (d i) (hd i)
refine ⟨P, hPdim, ?_⟩
intro k hk j hj x hx hsum
exact curveMap_direct hk (ξ.injective.comp hj) x (fun i => hPle _ (hx i)) hsum
end KaplanskyCounterexample.FiniteGeometry
end
end OAI
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