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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 | import OAI.Algebra.Finitistic.Bar.DirectSumCoproduct
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
namespace OAI
open CategoryTheory CategoryTheory.Limits HomologicalComplex
open scoped DirectSum
noncomputable section
namespace DerivedBar
universe u
variable (D : Type u) [Ring D]
(S : ℕ → ChainComplex (ModuleCat.{u} D) ℕ)
theorem directSumComplex_d (i j : ℕ) :
((directSumComplex S).d i j).hom = DirectSum.lmap (fun r => ((S r).d i j).hom) := by
by_cases h : j+1=i
· subst i
simp only [directSumComplex, ChainComplex.of_d, ModuleCat.hom_ofHom]
rfl
· have hrel : ¬ (ComplexShape.down ℕ).Rel i j := h
rw [(directSumComplex S).shape i j hrel]
have hf : (fun r => ((S r).d i j).hom) = (fun r => (0 : (S r).X i →ₗ[D] (S r).X j)) := by
funext r
rw [(S r).shape i j hrel, ModuleCat.hom_zero]
rw [hf]
apply LinearMap.ext
intro v
apply DFinsupp.ext
intro r
rfl
/-- The original componentwise direct sum is exact at a degree whenever every
summand is exact there. This includes the degree-zero boundary. -/
theorem directSumComplex_exactAt (n : ℕ) (h : ∀ r, (S r).ExactAt n) :
(directSumComplex S).ExactAt n := by
rw [HomologicalComplex.exactAt_iff, ShortComplex.moduleCat_exact_iff_range_eq_ker]
change LinearMap.range (((directSumComplex S).d ((ComplexShape.down ℕ).prev n) n).hom) =
LinearMap.ker (((directSumComplex S).d n ((ComplexShape.down ℕ).next n)).hom)
erw [directSumComplex_d, directSumComplex_d, DirectSum.range_lmap, DirectSum.ker_lmap]
congr 2
funext r
exact ShortComplex.Exact.moduleCat_range_eq_ker (h r)
theorem directSumComplex_homology_zero (n : ℕ) (h : ∀ r, IsZero ((S r).homology n)) :
IsZero ((directSumComplex S).homology n) := by
rw [← HomologicalComplex.exactAt_iff_isZero_homology]
exact directSumComplex_exactAt D S n
(fun r => (HomologicalComplex.exactAt_iff_isZero_homology _ _).mpr (h r))
end DerivedBar
end
end OAI
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