File size: 2,966 Bytes
216fb31 | 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 | import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import OAI.Algebra.Finitistic.Encoding.QuadraticEncoding
import OAI.Algebra.Finitistic.Rectification.RectificationAcyclic
namespace OAI
namespace LittleFinitistic.RelationSequence
open CategoryTheory CategoryTheory.Limits
open QuadraticEncoding
variable {k : Type*} [Field k] {d t : ℕ} (P : Relations k d t)
/-- The actual length-two relation subspace, all paths, and their quotient. -/
noncomputable def sequence : ShortComplex (ModuleCat k) where
X₁ := ModuleCat.of k (relationSpace P)
X₂ := ModuleCat.of k (Paths k d)
X₃ := ModuleCat.of k (Paths k d ⧸ relationSpace P)
f := ModuleCat.ofHom
{ toFun := fun x => x.val
map_add' := by intros; rfl
map_smul' := by intros; rfl }
g := ModuleCat.ofHom
{ toFun := (relationSpace P).mkQ
map_add' := (relationSpace P).mkQ.map_add
map_smul' := (relationSpace P).mkQ.map_smul }
zero := by ext x; exact (Submodule.Quotient.mk_eq_zero _).2 x.property
/-- No additional relation appears: the path quotient sequence is genuinely exact. -/
theorem sequence_shortExact : (sequence P).ShortExact := by
apply ModuleCat.shortComplex_shortExact
· intro x
change (relationSpace P).mkQ x = 0 ↔ ∃ y : relationSpace P, (y : Paths k d) = x
constructor
· intro hx
exact ⟨⟨x,(Submodule.Quotient.mk_eq_zero _).1 hx⟩,rfl⟩
· rintro ⟨y,hy⟩
exact (Submodule.Quotient.mk_eq_zero _).2 (hy ▸ y.property)
· exact (relationSpace P).injective_subtype
· exact (relationSpace P).mkQ_surjective
/-- Over the base field this exact sequence has an actual splitting. -/
noncomputable def sequenceSplitting : (sequence P).Splitting :=
(sequence_shortExact P).splittingOfProjective
/-- The finite relation basis used by the manuscript exists for this exact subspace. -/
noncomputable def relationBasis : Module.Basis (Fin (Module.finrank k (relationSpace P))) k
(relationSpace P) := by
letI : Module.Free k (relationSpace P) := Module.Free.of_divisionRing k (relationSpace P)
exact Module.finBasis k (relationSpace P)
/-- Tensoring the split path sequence with a complex, or applying any additive
realization functor, gives a short exact sequence of actual complexes. -/
theorem realized_sequence_shortExact {C : Type*} [Category C] [Abelian C]
(F : ModuleCat k ⥤ CochainComplex C ℤ) [F.Additive] :
((sequence P).map F).ShortExact :=
((sequenceSplitting P).map F).shortExact
/-- The (0,2) associated graded total is acyclic after such a realization. -/
theorem realized_sequence_total_acyclic {C : Type*} [Category C] [Abelian C]
(F : ModuleCat k ⥤ CochainComplex C ℤ) [F.Additive] :
(RectificationTotal.total ((sequence P).map F).g ((sequence P).map F).f
(Homotopy.ofEq ((sequence P).map F).zero)).Acyclic :=
RectificationAcyclic.short_exact_total_acyclic _ (realized_sequence_shortExact P F)
end LittleFinitistic.RelationSequence
end OAI
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