| import OAI.Algebra.Finitistic.Ordinary.OrdinaryFiniteColumnComplex | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| namespace LittleFinitistic.FiniteColumnComplex | |
| universe u | |
| variable {R : Type u} [Ring R] {l : ℕ} | |
| variable (K : Fin l → CochainComplex (ModuleCat.{u} R) ℤ) | |
| /-- The vertical differential on the finite direct sum of the columns. -/ | |
| def diagonal (n : ℤ) : term K n →ₗ[R] term K (n+1) := | |
| LinearMap.pi (fun i => ((K i).d n (n+1)).hom.comp (LinearMap.proj i)) | |
| @[simp] theorem diagonal_apply (n : ℤ) (z : term K n) (i : Fin l) : | |
| diagonal K n z i = ((K i).d n (n+1)).hom (z i) := rfl | |
| theorem diagonal_square (n : ℤ) : (diagonal K (n+1)).comp (diagonal K n) = 0 := by | |
| apply LinearMap.ext | |
| intro z | |
| funext i | |
| exact congrArg (fun f => f.hom (z i)) ((K i).d_comp_d n (n+1) ((n+1)+1)) | |
| /-- A horizontal differential on signed columns, with its actual square and mixed relation. -/ | |
| structure Horizontal where | |
| d : ∀ n : ℤ, term K n →ₗ[R] term K (n+1) | |
| square : ∀ n, (d (n+1)).comp (d n) = 0 | |
| mixed : ∀ n, (diagonal K (n+1)).comp (d n) + (d (n+1)).comp (diagonal K n) = 0 | |
| strictUpper : ∀ (n : ℤ) (i : Fin l) (z : term K n), | |
| (∀ j : Fin l, j.val < i.val → z j = 0) → d n z i = 0 | |
| variable (H : Horizontal K) | |
| /-- The full differential is the sum of the signed vertical and horizontal differentials. -/ | |
| def totalDifferential : Differential K where | |
| d n := diagonal K n + H.d n | |
| square n := by | |
| apply LinearMap.ext | |
| intro z | |
| funext i | |
| have hv := congrArg (fun f : term K n →ₗ[R] term K ((n+1)+1) => f z i) (diagonal_square K n) | |
| have hh := congrArg (fun f : term K n →ₗ[R] term K ((n+1)+1) => f z i) (H.square n) | |
| have hm := congrArg (fun f : term K n →ₗ[R] term K ((n+1)+1) => f z i) (H.mixed n) | |
| simp only [LinearMap.comp_apply,LinearMap.add_apply,LinearMap.zero_apply,Pi.add_apply,Pi.zero_apply] at hv hh hm ⊢ | |
| simp only [map_add,Pi.add_apply] | |
| simp only [diagonal_apply] at hv hh hm ⊢ | |
| rw [hv,hh] | |
| simpa only [zero_add,add_zero] using hm | |
| triangular n i z hz := by | |
| change diagonal K n z i + H.d n z i = _ | |
| rw [H.strictUpper n i z hz,add_zero] | |
| rfl | |
| def total : CochainComplex (ModuleCat.{u} R) ℤ := complex K (totalDifferential K H) | |
| theorem total_acyclic (hK : ∀ i, (K i).Acyclic) : (total K H).Acyclic := | |
| acyclic K (totalDifferential K H) hK | |
| end LittleFinitistic.FiniteColumnComplex | |
| end | |
| end OAI | |