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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 | 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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