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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 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 98 99 100 101 102 103 104 105 106 107 108 | import OAI.Algebra.Finitistic.Bar.ModuleBlockIndex
import OAI.Algebra.Finitistic.Bar.FiniteSumSupport
namespace OAI
open CategoryTheory CategoryTheory.Limits TensorProduct
open scoped TensorProduct DirectSum
noncomputable section
namespace DerivedBar.RecursiveTotal
open DerivedBar.Multibar
universe u
local instance (r n : ℕ) : DecidableEq (Index r n) := Classical.decEq _
local instance (r n : ℕ) : DecidableEq (Multibar.Indexed.Shape r n) := Classical.decEq _
/-- The supported index change preserves the literal D/X word. -/
theorem moduleShape_toList (r n : ℕ)
(t : {t : Index (r+1) n // inputDegree (r+1) t=0}) :
Multibar.Indexed.toList (moduleShapeEquiv r n t) = leadingPattern r n t.val := by
simp [moduleShapeEquiv, moduleBlockEquiv, Multibar.Indexed.blockDegreeEquiv,
pattern_moduleBlocks]
section General
variable {D : Type u} [Ring D]
local notation "c" => ComplexShape.down ℕ
/-- A zero input term remains zero through every additive bar degree functor. -/
theorem summand_isZero
(B : ℕ → ModuleCat.{u} D ⥤ ChainComplex (ModuleCat.{u} D) ℕ)
[∀ r, (B r).Additive] (Z : ChainComplex (ModuleCat.{u} D) ℕ)
(r n : ℕ) (t : Index r n) (h : IsZero (Z.X (inputDegree r t))) :
IsZero (summand B Z r n t) := by
induction r generalizing n with
| zero => exact h
| succ r ih =>
rcases t with ⟨p,t⟩
exact Functor.map_isZero (degreeFunctor B r p.val.1) (ih p.val.2 t h)
end General
local notation "c" => ComplexShape.down ℕ
variable (k D X : Type u) [Field k] [Ring D] [Algebra k D]
[AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X]
[SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X]
local instance (N : ModuleCat.{u} D) : Module k N := BimoduleTensor.scalarModule k D N
local instance (N : ModuleCat.{u} D) : IsScalarTower k D N :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
variable (N : ModuleCat.{u} D)
local notation "Z" => (Functor.obj (ChainComplex.single₀ (ModuleCat D)) N)
/-- Positive final input degrees contribute zero, without any assumption on X. -/
theorem moduleSummand_isZero (r n : ℕ) (t : Index (r+1) n)
(h : inputDegree (r+1) t ≠ 0) :
IsZero (summand (leadingBarSequence k D X r) Z (r+1) n t) :=
summand_isZero _ Z _ _ t (HomologicalComplex.isZero_single_obj_X c 0 N _ h)
/-- Each surviving recursive-total summand is the exact previously indexed
block tensor in degree r+n; the input is the original D-module. -/
def moduleSummandIso (r n : ℕ)
(t : {t : Index (r+1) n // inputDegree (r+1) t=0}) :
summand (leadingBarSequence k D X r) Z (r+1) n t.val ≅
ModuleCat.of D (Multibar.Indexed.outerTensor k D X N (moduleShapeEquiv r n t)) := by
have h : Multibar.Indexed.tensor k D X N (moduleShapeEquiv r n t) =
patternTerm k D X N (leadingPattern r n t.val) := by
rw [Multibar.Indexed.tensor_toList, moduleShape_toList]
have e : summand (leadingBarSequence k D X r) Z (r+1) n t.val ≅
ModuleCat.of D (D ⊗[k] patternTerm k D X N (leadingPattern r n t.val)) := by
have hn : (Z).X (inputDegree (r+1) t.val) = N := by
rw [t.property]
rfl
exact leadingSummandIso k D X Z r n t.val ≪≫ eqToIso
(congrArg (fun V : ModuleCat D => ModuleCat.of D
(D ⊗[k] patternTerm k D X V (leadingPattern r n t.val))) hn)
exact e ≪≫ eqToIso (congrArg (fun V : ModuleCat k => ModuleCat.of D (D ⊗[k] V)) h.symm)
/-- The exact block tensor after reindexing supported summands by Shape. -/
def reindexedModuleSummandIso (r n : ℕ) (p : Multibar.Indexed.Shape (r+n) r) :
summand (leadingBarSequence k D X r) Z (r+1) n ((moduleShapeEquiv r n).symm p).val ≅
ModuleCat.of D (Multibar.Indexed.outerTensor k D X N p) := by
exact moduleSummandIso k D X N r n ((moduleShapeEquiv r n).symm p) ≪≫
eqToIso (congrArg (fun s : Multibar.Indexed.Shape (r+n) r =>
ModuleCat.of D (Multibar.Indexed.outerTensor k D X N s))
((moduleShapeEquiv r n).apply_symm_apply p))
/-- All recursive-total terms are identified with the manuscript's exact finite
sum of tensor blocks. This includes discarding only proved zero input-degree terms. -/
def moduleFlatCoordinates (r n : ℕ) :
(flatLeadingMultibar k D X Z r).X n ≅
ModuleCat.of D (⨁ p : Multibar.Indexed.Shape (r+n) r,
Multibar.Indexed.outerTensor k D X N p) := by
let V := summand (leadingBarSequence k D X r) Z (r+1) n
let P := fun t : Index (r+1) n => inputDegree (r+1) t=0
let e := moduleShapeEquiv r n
exact supportIso V P (moduleSummand_isZero k D X N r n) ≪≫
(DirectSum.lequivCongrLeft D (M := fun t : {t // P t} => V t.val) e).toModuleIso ≪≫
(DirectSum.congrLinearEquiv (fun p =>
(reindexedModuleSummandIso k D X N r n p).toLinearEquiv)).toModuleIso
/-- The degreewise tensor-block isomorphism for the actual recursive leading total. -/
def leadingModuleCoordinates (r n : ℕ) :
(barTotal k D (explicitMultibar k D X r Z)).X n ≅
ModuleCat.of D (⨁ p : Multibar.Indexed.Shape (r+n) r,
Multibar.Indexed.outerTensor k D X N p) :=
(HomologicalComplex.eval _ c n).mapIso (flatLeadingMultibarIso k D X Z r).symm ≪≫
moduleFlatCoordinates k D X N r n
end DerivedBar.RecursiveTotal
end
end OAI
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