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ae00908 | 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 | import OAI.Algebra.Finitistic.Rectification.VertexColumnComplex
import OAI.Algebra.Finitistic.Rectification.TensorComplex
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
open scoped TensorProduct ModuleCat.Algebra
namespace LittleFinitistic.CornerColumnComplex
universe u
variable {k V W U : Type u} [Field k] [AddCommGroup V] [Module k V]
[AddCommGroup W] [Module k W] [AddCommGroup U] [Module k U]
variable (μ : W →ₗ[k] V →ₗ[k] U)
local notation "T" => ThreeVertexAlgebra μ
attribute [local instance] ExternalTensor.rightModule ExternalTensor.bimodule
/-- Embed an actual corner into the principal left ideal of its source vertex. -/
def cornerToIdeal (i j : Fin 3) : CornerSpaces.corner μ i j →ₗ[k]
ExternalTensor.leftIdeal T (ThreeVertexAlgebra.vertex μ j) where
toFun x := ⟨x.val,⟨x.val,CornerProjection.corner_right_fixed μ i j x⟩⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
/-- The corner equivalence retains the actual tensor insertion on pure tensors. -/
theorem corner_symm_tmul_val (N : ModuleCat.{u} (ThreeVertexAlgebra μ)ᵐᵒᵖ)
(i j : Fin 3) (x : CornerSpaces.corner μ i j) (n : N) :
((VertexColumnModels.cornerEquiv μ N i j).symm (x ⊗ₜ[k] n)).val =
cornerToIdeal μ i j x ⊗ₜ[k] n := by
rfl
/-- Every actual vertex column is its concrete corner-tensor complex. -/
def complexIso (i j : Fin 3)
(D : CochainComplex (ModuleCat.{u} (ThreeVertexAlgebra μ)ᵐᵒᵖ) ℤ) :
TensorComplex.complex D (ModuleCat.of k (CornerSpaces.corner μ i j)) ≅
((VertexColumnComplex.vertexColumn μ i j).mapHomologicalComplex (.up ℤ)).obj D :=
HomologicalComplex.Hom.isoOfComponents
(fun n => (VertexColumnModels.cornerEquiv μ (D.X n) i j).symm.toModuleIso) (by
intro n m hnm
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
induction x using TensorProduct.inductionOn with
| tmul c v => rfl
| add x y hx hy =>
exact (map_add _ _ _).trans
((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm))
end LittleFinitistic.CornerColumnComplex
end
end OAI
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