| import OAI.Algebra.AuslanderReiten.Tensor.EnvelopingBalancedExact |
| import OAI.Algebra.AuslanderReiten.Tensor.EnvelopingStableEvaluation |
| import OAI.Algebra.AuslanderReiten.Resolutions.FiniteResolutionLinear |
| import OAI.Algebra.AuslanderReiten.Complete.ProjectiveCompleteResolution |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Pretriangulated |
| open scoped TensorProduct |
|
|
| namespace ArExplicit.FiniteLeftModule |
| universe u |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] [Module.Finite k R] |
| variable (X : FiniteLeftModule k R) |
| variable (M : FiniteLeftModule k (R ⊗[k] Rᵐᵒᵖ)) |
|
|
| def evaluated : FiniteLeftModule k R where |
| carrier := ModuleCat.of R (BalancedTensor k R (EnvelopingBimodule k R M.carrier) X.carrier) |
| scalar := inferInstance |
| tower := inferInstance |
| finite := inferInstance |
|
|
| variable {M N P : FiniteLeftModule k (R ⊗[k] Rᵐᵒᵖ)} |
|
|
| def evaluationMap (f : M.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] N.carrier) : |
| (evaluated X M).carrier →ₗ[R] (evaluated X N).carrier := |
| EnvelopingBimodule.tensorMap X.carrier (ModuleCat.ofHom f) |
|
|
| omit [Module.Finite k R] in |
| @[simp] theorem evaluationMap_id : evaluationMap X (LinearMap.id : M.carrier →ₗ[_] M.carrier) = |
| LinearMap.id := EnvelopingBimodule.tensorMap_id X.carrier |
|
|
| omit [Module.Finite k R] in |
| @[simp] theorem evaluationMap_comp (f : M.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] N.carrier) |
| (g : N.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] P.carrier) : |
| evaluationMap X (g.comp f) = (evaluationMap X g).comp (evaluationMap X f) := |
| EnvelopingBimodule.tensorMap_comp X.carrier (ModuleCat.ofHom f) (ModuleCat.ofHom g) |
|
|
| def evaluationStableIso (e : StableIsomorphism (R ⊗[k] Rᵐᵒᵖ) M.carrier N.carrier) : |
| StableIsomorphism R (evaluated X M).carrier (evaluated X N).carrier where |
| hom := EnvelopingBimodule.stableEvaluation X.carrier e.hom |
| inv := EnvelopingBimodule.stableEvaluation X.carrier e.inv |
| hom_inv_id := by |
| erw [← EnvelopingBimodule.stableEvaluation_comp, e.hom_inv_id] |
| exact EnvelopingBimodule.stableEvaluation_id X.carrier |
| inv_hom_id := by |
| erw [← EnvelopingBimodule.stableEvaluation_comp, e.inv_hom_id] |
| exact EnvelopingBimodule.stableEvaluation_id X.carrier |
|
|
| omit [Module.Finite k R] in |
| theorem evaluated_projective [Module.Projective (R ⊗[k] Rᵐᵒᵖ) M.carrier] : |
| Module.Projective R (evaluated X M).carrier := by |
| let := Module.Finite.of_restrictScalars_finite k (R ⊗[k] Rᵐᵒᵖ) M.carrier |
| exact EnvelopingBimodule.tensor_projective X.carrier M.carrier |
|
|
| variable (f : M.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] N.carrier) |
| (g : N.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] P.carrier) |
| variable (hf : Function.Injective f) (hg : Function.Surjective g) (he : Function.Exact f g) |
| variable [Module.Projective Rᵐᵒᵖ (EnvelopingBimodule k R P.carrier)] |
|
|
| include hf hg he in |
| omit [Module.Finite k R] in |
|
|
| theorem evaluation_shortExact : |
| Function.Injective (evaluationMap X f) ∧ Function.Surjective (evaluationMap X g) ∧ |
| Function.Exact (evaluationMap X f) (evaluationMap X g) := by |
| have hr : LinearMap.range f = LinearMap.ker g := by |
| ext y |
| exact (he y).symm |
| obtain ⟨hi, hs, hx⟩ := EnvelopingBimodule.tensorMap_shortExact X.carrier |
| (ModuleCat.ofHom f) (ModuleCat.ofHom g) hf hg hr |
| change LinearMap.range (evaluationMap X f) = LinearMap.ker (evaluationMap X g) at hx |
| refine ⟨hi, hs, ?_⟩ |
| intro y |
| exact (show y ∈ LinearMap.ker (evaluationMap X g) ↔ |
| y ∈ LinearMap.range (evaluationMap X f) by rw [hx]) |
|
|
| variable (τ : FrobeniusTrace k R) |
|
|
| def evaluationTriangle : Triangle (HomotopyCategory (ModuleCat.{u} R) (ComplexShape.up ℤ)) := |
| resolutionTriangle τ (evaluationMap X f) (evaluationMap X g) |
| (evaluation_shortExact X f g hf hg he).1 (evaluation_shortExact X f g hf hg he).2.1 |
| (evaluation_shortExact X f g hf hg he).2.2 |
|
|
| theorem evaluationTriangle_distinguished : evaluationTriangle X f g hf hg he τ ∈ |
| distTriang (HomotopyCategory (ModuleCat.{u} R) (ComplexShape.up ℤ)) := |
| resolutionTriangle_distinguished _ _ _ _ _ _ |
|
|
| def evaluationSuccessorIso [Module.Projective (R ⊗[k] Rᵐᵒᵖ) N.carrier] : |
| (HomotopyCategory.quotient _ _).obj ((evaluated X P).completeCochain τ) ≅ |
| ((HomotopyCategory.quotient _ _).obj ((evaluated X M).completeCochain τ))⟦(1 : ℤ)⟧ := by |
| letI := evaluated_projective X (M := N) |
| exact projectiveMiddleShiftIso τ (evaluationMap X f) (evaluationMap X g) |
| (evaluation_shortExact X f g hf hg he).1 (evaluation_shortExact X f g hf hg he).2.1 |
| (evaluation_shortExact X f g hf hg he).2.2 |
|
|
| @[simp] |
| theorem evaluationSuccessorIso_hom [Module.Projective (R ⊗[k] Rᵐᵒᵖ) N.carrier] : |
| (evaluationSuccessorIso X f g hf hg he τ).hom = |
| (evaluationTriangle X f g hf hg he τ).mor₃ := by |
| let := evaluated_projective X (M := N) |
| exact projectiveMiddleShiftIso_hom _ _ _ _ _ _ |
|
|
| section Naturality |
| variable {M₁ N₁ P₁ M₂ N₂ P₂ : FiniteLeftModule k (R ⊗[k] Rᵐᵒᵖ)} |
| variable (f₁ : M₁.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] N₁.carrier) |
| (g₁ : N₁.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] P₁.carrier) |
| (f₂ : M₂.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] N₂.carrier) |
| (g₂ : N₂.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] P₂.carrier) |
| variable (hf₁ : Function.Injective f₁) (hg₁ : Function.Surjective g₁) (he₁ : Function.Exact f₁ g₁) |
| (hf₂ : Function.Injective f₂) (hg₂ : Function.Surjective g₂) (he₂ : Function.Exact f₂ g₂) |
| variable [Module.Projective Rᵐᵒᵖ (EnvelopingBimodule k R P₁.carrier)] |
| [Module.Projective Rᵐᵒᵖ (EnvelopingBimodule k R P₂.carrier)] |
| variable (a : M₁.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] M₂.carrier) |
| (b : N₁.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] N₂.carrier) |
| (c : P₁.carrier →ₗ[R ⊗[k] Rᵐᵒᵖ] P₂.carrier) |
|
|
| theorem evaluationConnecting_natural |
| (hab : b.comp f₁ = f₂.comp a) (hbc : g₂.comp b = c.comp g₁) : |
| (HomotopyCategory.quotient _ _).map (resolutionMap τ (evaluationMap X c)) ≫ |
| (evaluationTriangle X f₂ g₂ hf₂ hg₂ he₂ τ).mor₃ = |
| (evaluationTriangle X f₁ g₁ hf₁ hg₁ he₁ τ).mor₃ ≫ |
| (CategoryTheory.shiftFunctor (HomotopyCategory (ModuleCat.{u} R) (ComplexShape.up ℤ)) 1).map |
| ((HomotopyCategory.quotient _ _).map (resolutionMap τ (evaluationMap X a))) := by |
| apply resolutionConnecting_natural |
| · rw [← evaluationMap_comp, ← evaluationMap_comp, hab] |
| · rw [← evaluationMap_comp, ← evaluationMap_comp, hbc] |
| end Naturality |
|
|
| end ArExplicit.FiniteLeftModule |
|
|
| end |
|
|
| end OAI |
|
|
