| 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 |
|
|
