| import OAI.Algebra.AuslanderReiten.Resolutions.CoverStepCoordinates |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ArExplicit |
| open CategoryTheory |
| open CochainComplex CochainComplex.HomComplex CompleteComparison |
| universe u w |
| variable (k : Type u) [Field k] |
| variable {C : Type w} [Category.{u} C] [Preadditive C] [Linear k C] |
| variable {K : CochainComplex C ℤ} |
| local notation "HC" => HomotopyCategory C (ComplexShape.up ℤ) |
| private abbrev sourceObject (K : CochainComplex C ℤ) := |
| (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K |
| local notation "X₀" => sourceObject K |
|
|
| private theorem fullHom_coordinates_degree {a b i j : ℤ} (h : a=b) |
| {Y : HC} (x : X₀ ⟶ Y⟦i⟧) (y : Y ⟶ X₀⟦j⟧) |
| (ha : j+i=a) (hb : j+i=b) : |
| (fullHomShiftedHomotopyEquiv k K K b).symm (ShiftedHom.comp x y hb) = |
| TateKoszul.degreeEquiv (k := k) (fun d => CohomologyClass K K d) h |
| ((fullHomShiftedHomotopyEquiv k K K a).symm (ShiftedHom.comp x y ha)) := by |
| subst b |
| rfl |
|
|
| private theorem shiftedNegative_step_at (Y Z : HC) (n : ℤ) |
| (e : X₀ ≅ Y⟦n⟧) (s : Y ≅ Z⟦(1 : ℤ)⟧) (t : X₀ ≅ Z⟦n+1⟧) |
| (ht : t.hom = ShiftedHom.comp e.hom s.hom (add_comm 1 n)) |
| (a d : ℤ) (hd : d=a-n) (v : X₀ ⟶ Y⟦a⟧) : |
| ShiftedHom.comp (ShiftedHom.comp v s.hom (add_comm 1 a)) |
| (negativeShiftIso (n+1) t).hom (by omega : -(n+1)+(a+1)=d) = |
| ShiftedHom.comp v (negativeShiftIso n e).hom (by omega : -n+a=d) := by |
| subst d |
| exact shiftedNegative_step n e s t ht a v |
|
|
| theorem coverStepFullHom (Y Z : HC) (n m : ℤ) (hm : m=n+1) |
| (e : X₀ ≅ Y⟦n⟧) (s : Y ≅ Z⟦(1 : ℤ)⟧) (t : X₀ ≅ Z⟦m⟧) |
| (ht : t.hom = ShiftedHom.comp e.hom s.hom (by omega : 1+n=m)) |
| (a : ℤ) (v : X₀ ⟶ Y⟦a⟧) : |
| (fullHomShiftedHomotopyEquiv k K K (a+1-m)).symm |
| (shiftedNegativeHomEquiv m t k X₀ (a+1) |
| (ShiftedHom.comp v s.hom (add_comm 1 a))) = |
| TateKoszul.degreeEquiv (k := k) (fun d => CohomologyClass K K d) |
| (by omega : a-n=a+1-m) |
| ((fullHomShiftedHomotopyEquiv k K K (a-n)).symm |
| (shiftedNegativeHomEquiv n e k X₀ a v)) := by |
| subst m |
| simp only [shiftedNegativeHomEquiv_comp] |
| have hh := shiftedNegative_step_at Y Z n e s t ht a (a+1-(n+1)) (by omega) v |
| rw [hh] |
| exact fullHom_coordinates_degree k (by omega) v (negativeShiftIso n e).hom _ _ |
|
|
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
|
|
