File size: 3,066 Bytes
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 | import OAI.Algebra.Finitistic.Bimodule.BimoduleTensorFunctor
import OAI.Algebra.Finitistic.Localization.AdditiveSplitImage
import OAI.Algebra.Finitistic.Tensor.IdempotentTensor
namespace OAI
noncomputable section
open CategoryTheory
open scoped TensorProduct ModuleCat.Algebra
namespace LittleFinitistic.BalancedVertexImage
open ExternalTensor.Section4
universe u
variable (k B : Type u) [Field k] [Ring B] [Algebra k B]
variable (M : ModuleCat.{u} (Enveloping k B)) (e : B) (he : e*e=e)
variable (N : Type u) [AddCommGroup N] [Module k N] [Module B N]
local instance originalRightModule : Module Bᵐᵒᵖ M :=
Module.compHom M (BimoduleVertex.restriction k B)
abbrev V := (BimoduleVertex.functor k B e).obj M
abbrev W := (forgetRight k B).obj M
abbrev F := BalancedTensor.tensorFunctor k B N
def inclusion : V k B M e ⟶ W k B M := ModuleCat.ofHom
(VertexFunctor.inclusion (BimoduleVertex.restriction k B) (BimoduleVertex.vertex k B e)
(BimoduleVertex.vertex_commute k B e) M)
def projection : W k B M ⟶ V k B M e := ModuleCat.ofHom
(VertexFunctor.projection (BimoduleVertex.restriction k B) (BimoduleVertex.vertex k B e)
(BimoduleVertex.vertex_idempotent k B e he) (BimoduleVertex.vertex_commute k B e) M)
theorem split : inclusion k B M e ≫ projection k B M e he = 𝟙 _ :=
ModuleCat.hom_ext (VertexFunctor.projection_inclusion (BimoduleVertex.restriction k B)
(BimoduleVertex.vertex k B e) (BimoduleVertex.vertex_idempotent k B e he)
(BimoduleVertex.vertex_commute k B e) M)
theorem projector : ((F k B N).map (projection k B M e he ≫ inclusion k B M e)).hom =
BimoduleBalancedTensor.action k B M N e := by
apply BalancedTensor.ext
intro m n
rfl
/-- The image of the original vertex projector after tensoring is the actual left vertex. -/
def fixedEquiv :
AdditiveSplitImage.fixed (F k B N) (inclusion k B M e) (projection k B M e he) ≃ₗ[k]
IdempotentTensor.leftSpace (k := k) e (BimoduleBalancedTensor.Tensor k B M N) where
toFun x := ⟨x.val,by
change BimoduleBalancedTensor.action k B M N e x.val = x.val
rw [← projector k B M e he N]
exact (AdditiveSplitImage.mem_fixed _ _ _ x.val).1 x.property⟩
invFun x := ⟨x.val,by
exact (AdditiveSplitImage.mem_fixed (F k B N) (inclusion k B M e)
(projection k B M e he) x.val).2
((LinearMap.congr_fun (projector k B M e he N) x.val).trans x.property)⟩
left_inv _ := rfl
right_inv _ := rfl
map_add' _ _ := rfl
map_smul' _ _ := rfl
/-- Vertex extraction commutes with actual balanced tensoring, without a flatness assumption. -/
def equiv : (F k B N).obj (V k B M e) ≃ₗ[k]
IdempotentTensor.leftSpace (k := k) e (BimoduleBalancedTensor.Tensor k B M N) :=
(AdditiveSplitImage.equiv (F k B N) (inclusion k B M e) (projection k B M e he)
(split k B M e he)).trans (fixedEquiv k B M e he N)
@[simp] theorem equiv_apply_val (x : (F k B N).obj (V k B M e)) :
(equiv k B M e he N x).val = ((F k B N).map (inclusion k B M e)).hom x := rfl
end LittleFinitistic.BalancedVertexImage
end
end OAI
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