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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 | import OAI.Algebra.Finitistic.Bar.Splitting
namespace OAI
open scoped TensorProduct DirectSum
open TensorProduct CategoryTheory TrivSqZeroExt
noncomputable section
namespace DerivedBar
universe u
/-- The direct sum of module chain complexes, with the componentwise actual differential. -/
def directSumComplex {R : Type u} [Ring R] (C : ℕ → ChainComplex (ModuleCat R) ℕ) :
ChainComplex (ModuleCat R) ℕ :=
ChainComplex.of (fun n => ModuleCat.of R (⨁ r, (C r).X n))
(fun n => ModuleCat.ofHom (DirectSum.lmap (fun r => ((C r).d (n+1) n).hom)))
(fun n => by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro z
apply DFinsupp.ext
intro r
exact LinearMap.congr_fun
(ModuleCat.hom_ext_iff.mp ((C r).d_comp_d (n+1+1) (n+1) n)) (z r))
namespace SquareZero
variable (k D X N : Type u) [Field k] [Ring D] [Algebra k D]
[AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X]
[SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X]
[AddCommGroup N] [Module k N]
[Module (TrivSqZeroExt D X) N] [IsScalarTower k (TrivSqZeroExt D X) N]
variable (hN : ∀ (x : X) (v : N), (inr x : TrivSqZeroExt D X) • v = 0)
/-- Chain-level count splitting of the actual two-sided absolute bar complex.
This is the first chain isomorphism in the proof of manuscript prop:bar-splitting.
Identification of these summands with derived tensor iterates is a separate dependency. -/
def countSplitting :
directSumComplex (twoSidedWeightComplex k D X N hN) ≅
twoSidedComplex k (TrivSqZeroExt D X) D N (fstHom k D X) :=
HomologicalComplex.Hom.isoOfComponents
(fun n => (twoSidedWeightEquiv k D X N n).toModuleIso)
(by
intro i j hij
change j+1 = i at hij
subst i
apply ModuleCat.hom_ext
apply DirectSum.linearMap_ext
intro r
apply LinearMap.ext
intro v
simp only [directSumComplex, twoSidedComplex, ChainComplex.of_d,
ModuleCat.hom_comp, ModuleCat.hom_ofHom, twoSidedWeightEquiv]
change twoSidedDifferential k (TrivSqZeroExt D X) D N (fstHom k D X) j
(DirectSum.coeLinearMap (twoSidedWeight k D X N (j+1))
(DirectSum.lof D ℕ _ r v)) =
DirectSum.coeLinearMap (twoSidedWeight k D X N j)
(DirectSum.lmap (fun r => ((twoSidedWeightComplex k D X N hN r).d (j+1) j).hom)
(DirectSum.lof D ℕ _ r v))
rw [DirectSum.lmap_lof]
erw [DirectSum.coeLinearMap_lof, DirectSum.coeLinearMap_lof]
simp only [twoSidedWeightComplex, ChainComplex.of_d, ModuleCat.hom_ofHom]
rfl)
end SquareZero
end DerivedBar
end
end OAI
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