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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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import OAI.Algebra.Finitistic.Rectification.VertexHomology
import OAI.Algebra.Finitistic.Rectification.VertexDecomposition
namespace OAI
noncomputable section
open CategoryTheory HomologicalComplex
namespace LittleFinitistic.LeftVertexHomology
universe u
variable (k B : Type u) [Field k] [Ring B] [Algebra k B]
lemma scalar_commute (e : B) (c : k) : e * algebraMap k B c = algebraMap k B c * e :=
(Algebra.commutes c e).symm
/-- The actual left vertex functor from B-modules to k-vector spaces. -/
abbrev vertex (e : B) : ModuleCat.{u} B ⥤ ModuleCat.{u} k :=
VertexFunctor.functor (algebraMap k B) e (scalar_commute k B e)
/-- Arrow action is a natural transformation between the actual vertex functors. -/
def arrow (ei ej a : B) (ha : ej*a=a) : vertex k B ei ⟶ vertex k B ej where
app M := ModuleCat.ofHom {
toFun := fun x => ⟨a • (x.val : M), by
rw [VertexFunctor.mem_fixed]
rw [← mul_smul, ha]⟩
map_add' := fun x y => Subtype.ext (smul_add _ _ _)
map_smul' := fun c x => by
apply Subtype.ext
change a • (algebraMap k B c • (x.val : M)) =
algebraMap k B c • (a • (x.val : M))
rw [← mul_smul, ← mul_smul, scalar_commute] }
naturality M M' f := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
apply Subtype.ext
exact (f.hom.map_smul a x.val).symm
instance scalarTower (M : ModuleCat.{u} B) :
IsScalarTower k B ((ModuleCat.restrictScalars (algebraMap k B)).obj M) :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
/-- Identify the vertex functor with the actual fixed subspace of scalar restriction. -/
def spaceEquiv (e : B) (M : ModuleCat.{u} B) :
(vertex k B e).obj M ≃ₗ[k]
IdempotentTensor.leftSpace (k := k) e ((ModuleCat.restrictScalars (algebraMap k B)).obj M) where
toFun x := ⟨x.val, (VertexFunctor.mem_fixed _ _ _ M x.val).1 x.property⟩
invFun x := ⟨x.val, (VertexFunctor.mem_fixed _ _ _ M x.val).2 x.property⟩
left_inv _ := rfl
right_inv _ := rfl
map_add' _ _ := rfl
map_smul' _ _ := rfl
/-- Actual vertex homology, identified with the actual subspace of module homology. -/
def homologyEquiv (e : B) (he : e*e=e)
(X : CochainComplex (ModuleCat.{u} B) ℤ) (n : ℤ) :
(((vertex k B e).mapHomologicalComplex (.up ℤ)).obj X).homology n ≃ₗ[k]
IdempotentTensor.leftSpace (k := k) e
((ModuleCat.restrictScalars (algebraMap k B)).obj (X.homology n)) :=
(VertexHomology.iso (algebraMap k B) e he (scalar_commute k B e) X n).toLinearEquiv.trans
(spaceEquiv k B e (X.homology n))
/-- Cohomology comparison retains the original arrow action, not only dimensions. -/
theorem arrow_homology (ei ej a : B) (hi : ei*ei=ei) (hj : ej*ej=ej) (ha : ej*a=a)
(X : CochainComplex (ModuleCat.{u} B) ℤ) (n : ℤ) :
homologyMap ((NatTrans.mapHomologicalComplex (arrow k B ei ej a ha) (.up ℤ)).app X) n ≫
(VertexHomology.iso (algebraMap k B) ej hj (scalar_commute k B ej) X n).hom =
(VertexHomology.iso (algebraMap k B) ei hi (scalar_commute k B ei) X n).hom ≫
(arrow k B ei ej a ha).app (X.homology n) :=
VertexHomology.natural (algebraMap k B) ei hi (scalar_commute k B ei)
ej hj (scalar_commute k B ej) _ X n
end LittleFinitistic.LeftVertexHomology
end
end OAI
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