| import OAI.Algebra.Finitistic.Ordinary.OrdinaryTriangularExactness |
| import Mathlib.Algebra.Homology.ShortComplex.ModuleCat |
| import Mathlib.Algebra.Homology.QuasiIso |
|
|
| 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) |
|
|
| abbrev term (n : ℤ) := ∀ i, (K i).X n |
|
|
| /-- A finite-column total differential, recorded independently of a chosen direct-sum presentation. -/ |
| structure Differential where |
| d : ∀ n : ℤ, term K n →ₗ[R] term K (n+1) |
| square : ∀ n, (d (n+1)).comp (d n) = 0 |
| triangular : ∀ (n : ℤ) (i : Fin l) (z : term K n), |
| (∀ j : Fin l, j.val < i.val → z j = 0) → |
| d n z i = ((K i).d n (n+1)).hom (z i) |
|
|
| variable (δ : Differential K) |
|
|
| def complex : CochainComplex (ModuleCat.{u} R) ℤ := |
| CochainComplex.of (fun n => ModuleCat.of R (term K n)) |
| (fun n => ModuleCat.ofHom (δ.d n)) (fun n => ModuleCat.hom_ext (δ.square n)) |
|
|
| @[simp] theorem complex_d (n : ℤ) : |
| (complex K δ).d n (n+1) = ModuleCat.ofHom (δ.d n) := by |
| exact CochainComplex.of_d (fun n => ModuleCat.of R (term K n)) |
| (fun n => ModuleCat.ofHom (δ.d n)) n |
|
|
| @[simp] theorem complex_d_hom (n : ℤ) : |
| ((complex K δ).d n (n+1)).hom = δ.d n := by |
| rw [complex_d] |
| rfl |
|
|
| theorem lower_single (n : ℤ) (i : Fin l) (x : (K i).X n) (j : Fin l) (hji : j.val < i.val) : |
| δ.d n (Pi.single i x : term K n) j = 0 := by |
| classical |
| have ht : ∀ t : Fin l, t.val < j.val → (Pi.single i x : term K n) t = 0 := by |
| intro t ht |
| simp [show t ≠ i from by intro h; subst t; omega] |
| rw [δ.triangular n j (Pi.single i x) ht] |
| rw [Pi.single_eq_of_ne (show j ≠ i from by intro h; subst j; omega),map_zero] |
|
|
| theorem diagonal_single (n : ℤ) (i : Fin l) (x : (K i).X n) : |
| δ.d n (Pi.single i x) i = ((K i).d n (n+1)).hom x := by |
| classical |
| have ht : ∀ j : Fin l, j.val < i.val → (Pi.single i x : term K n) j = 0 := by |
| intro j hj |
| simp [show j ≠ i from by intro h; subst j; omega] |
| rw [δ.triangular n i (Pi.single i x) ht,Pi.single_eq_same] |
|
|
| /-- Finite totalization preserves acyclicity when every column is acyclic. -/ |
| theorem acyclic (hK : ∀ i, (K i).Acyclic) : (complex K δ).Acyclic := by |
| intro n |
| have hn : n-1+1=n := by omega |
| suffices hh : (complex K δ).ExactAt (n-1+1) by simpa only [hn] using hh |
| apply (HomologicalComplex.exactAt_iff' _ (n-1) (n-1+1) ((n-1+1)+1) (by simp) (by simp)).2 |
| rw [ShortComplex.moduleCat_exact_iff] |
| have hex (i : Fin l) : Function.Exact |
| ((K i).d (n-1) (n-1+1)).hom ((K i).d (n-1+1) ((n-1+1)+1)).hom := by |
| have hh := (HomologicalComplex.exactAt_iff' (K i) (n-1) (n-1+1) ((n-1+1)+1) |
| (by simp) (by simp)).mp (hK i _) |
| exact (ShortComplex.ShortExact.moduleCat_exact_iff_function_exact _).mp hh |
| have ht := TriangularExactness.exact_of_diagonal |
| (fun i => (K i).X (n-1)) (fun i => (K i).X (n-1+1)) (fun i => (K i).X ((n-1+1)+1)) |
| (δ.d (n-1)) (δ.d (n-1+1)) |
| (fun i => ((K i).d (n-1) (n-1+1)).hom) |
| (fun i => ((K i).d (n-1+1) ((n-1+1)+1)).hom) |
| (δ.square (n-1)) (lower_single K δ (n-1)) (diagonal_single K δ (n-1)) |
| (δ.triangular (n-1+1)) hex |
| change ∀ (z : term K (n-1+1)), |
| ((complex K δ).d (n-1+1) ((n-1+1)+1)).hom z = 0 → |
| ∃ w : term K (n-1), ((complex K δ).d (n-1) (n-1+1)).hom w = z |
| simp only [complex_d_hom] |
| exact fun z hz => (ht z).mp hz |
| end LittleFinitistic.FiniteColumnComplex |
|
|
| end |
|
|
| end OAI |
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|
