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6aac220 | 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 | import OAI.GroupTheory.PolycyclicRecognition.Algebra.BooleanPartitionMaps
import Mathlib.Data.Set.Countable
namespace OAI
/-! # Countable Boolean step rings are projective over the integers
Enumerating the Boolean coordinates gives a sequence of finite partitions.
Their modules are free, the refinement maps split, and their colimit is the
step ring itself. No freeness theorem about all bounded functions is needed.
-/
noncomputable section
namespace PolycyclicRecognition
open CategoryTheory CategoryTheory.Limits
variable {X I : Type} (a : I → X → Bool) (b : ℕ → I)
noncomputable def booleanPrefix (n : ℕ) : Finset I := by
classical
exact (Finset.range n).image b
theorem booleanPrefix_mono {i j : ℕ} (h : i ≤ j) : booleanPrefix b i ⊆ booleanPrefix b j := by
classical
exact Finset.image_subset_image (Finset.range_mono h)
theorem exists_booleanPrefix_contains (hb : Function.Surjective b) (s : Finset I) :
∃ n, s ⊆ booleanPrefix b n := by
classical
induction s using Finset.induction_on with
| empty => exact ⟨0, Finset.empty_subset _⟩
| @insert i s _ ih =>
obtain ⟨n, hn⟩ := ih
obtain ⟨m, hm⟩ := hb i
refine ⟨max n (m + 1), Finset.insert_subset_iff.mpr ⟨?_, ?_⟩⟩
· apply booleanPrefix_mono b (Nat.le_max_right n (m + 1))
exact Finset.mem_image.mpr ⟨m, Finset.mem_range.mpr (Nat.lt_succ_self m), hm⟩
· exact hn.trans (booleanPrefix_mono b (Nat.le_max_left n (m + 1)))
def booleanPartitionDiagram : SequentialRepresentationDiagram PUnit.{1} where
obj n := booleanPartitionStage a (booleanPrefix b n)
map _ _ h := booleanPartitionMap a (booleanPrefix_mono b h)
map_id n := by
apply Rep.hom_ext
apply Representation.IntertwiningMap.ext
apply LinearMap.ext
intro v
funext c
rfl
map_comp i j k hij hjk := by
apply Rep.hom_ext
apply Representation.IntertwiningMap.ext
apply LinearMap.ext
intro v
funext c
rfl
def booleanStepColimit (hb : Function.Surjective b) :
(booleanPartitionDiagram a b).ColimitCocone (booleanStepRepresentation a) where
ι n := booleanPartitionEvaluation a (booleanPrefix b n)
comm _ _ h := booleanPartitionMap_evaluation a (booleanPrefix_mono b h)
surj f := by
obtain ⟨s, hs⟩ := f.property
obtain ⟨n, hn⟩ := exists_booleanPrefix_contains b hb s
obtain ⟨v, hv⟩ := dependsOnBoolean_factorization (hs.mono hn)
refine ⟨n, v, ?_⟩
apply Subtype.ext
exact hv.symm
zero_exact i v hv := by
refine ⟨i, le_rfl, ?_⟩
have h0 : v = 0 := booleanPartitionEvaluation_injective a (booleanPrefix b i)
(hv.trans (map_zero _).symm)
rw [h0, map_zero]
theorem booleanStepRepresentation_projective (hb : Function.Surjective b) :
Projective (booleanStepRepresentation a) := by
let D := booleanPartitionDiagram a b
let r : ∀ i, D.obj (i + 1) ⟶ D.obj i := fun i =>
booleanPartitionRetraction a (booleanPrefix_mono b (Nat.le_succ i))
have hr : ∀ i, D.map i (i + 1) (Nat.le_succ i) ≫ r i = 𝟙 _ :=
fun i => booleanPartitionMap_retraction a (booleanPrefix_mono b (Nat.le_succ i))
let : ∀ i, Projective (D.obj i) := fun i => booleanPartitionStage_projective a _
exact D.split_colimit_projective r hr (booleanStepColimit a b hb)
/-- The small step-function algebra required in the countable group application. -/
theorem booleanStepRing_integer_projective [Countable I] [Nonempty I] :
Module.Projective ℤ (booleanStepRing a) := by
obtain ⟨b, hb⟩ := exists_surjective_nat I
let := booleanStepRepresentation_projective a b hb
exact integer_projective_of_projective_rep (booleanStepRepresentation a)
end PolycyclicRecognition
end
end OAI
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