| import OAI.Algebra.AuslanderReiten.Complete.SelfInjectiveComplex |
| import OAI.Algebra.AuslanderReiten.Modules.CoinducedEmbedding |
| import Mathlib.Algebra.Category.ModuleCat.Basic |
| import Mathlib.RingTheory.Finiteness.Cardinality |
|
|
| namespace OAI |
|
|
| open CategoryTheory |
|
|
| namespace ArExplicit |
|
|
| universe u |
|
|
| variable (k R : Type u) [Field k] [Ring R] [Algebra k R] |
|
|
| structure FiniteLeftModule where |
| carrier : ModuleCat.{u} R |
| [scalar : Module k carrier] |
| [tower : IsScalarTower k R carrier] |
| [finite : Module.Finite k carrier] |
|
|
| attribute [instance] FiniteLeftModule.scalar FiniteLeftModule.tower FiniteLeftModule.finite |
|
|
| namespace FiniteLeftModule |
|
|
| variable {k R} [Module.Finite k R] |
| |
| structure FreeCover (M : FiniteLeftModule k R) where |
| rank : ℕ |
| map : (Fin rank → R) →ₗ[R] M.carrier |
| surjective : Function.Surjective map |
|
|
| noncomputable def cover (M : FiniteLeftModule k R) : FreeCover M := by |
| let := Module.Finite.of_restrictScalars_finite k R M.carrier |
| let h := Module.Finite.exists_fin' R M.carrier |
| exact ⟨h.choose, h.choose_spec.choose, h.choose_spec.choose_spec⟩ |
|
|
| noncomputable def syzygy (M : FiniteLeftModule k R) : FiniteLeftModule k R where |
| carrier := ModuleCat.of R (LinearMap.ker M.cover.map) |
| scalar := inferInstance |
| tower := inferInstance |
| finite := Module.Finite.of_injective |
| ((LinearMap.ker M.cover.map).subtype.restrictScalars k) Subtype.val_injective |
|
|
| noncomputable def syzygies (M : FiniteLeftModule k R) : ℕ → FiniteLeftModule k R |
| | 0 => M |
| | n + 1 => (syzygies M n).syzygy |
|
|
| noncomputable def envelopeRank (M : FiniteLeftModule k R) : ℕ := |
| Module.finrank k M.carrier |
|
|
| noncomputable def envelopeMap (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| M.carrier →ₗ[R] (Fin M.envelopeRank → R) := τ.finiteFreeEmbedding |
|
|
| theorem envelopeMap_injective (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| Function.Injective (M.envelopeMap τ) := τ.finiteFreeEmbedding_injective |
|
|
| noncomputable def cosyzygy (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| FiniteLeftModule k R where |
| carrier := ModuleCat.of R ((Fin M.envelopeRank → R) ⧸ LinearMap.range (M.envelopeMap τ)) |
| scalar := inferInstance |
| tower := inferInstance |
| finite := inferInstance |
|
|
| noncomputable def cosyzygies (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| ℕ → FiniteLeftModule k R |
| | 0 => M |
| | n + 1 => (cosyzygies τ M n).cosyzygy τ |
|
|
| noncomputable abbrev positiveTerm (M : FiniteLeftModule k R) (n : ℕ) := |
| Fin (M.syzygies n).cover.rank → R |
|
|
| noncomputable def positiveDifferential (M : FiniteLeftModule k R) (n : ℕ) : |
| M.positiveTerm (n + 1) →ₗ[R] M.positiveTerm n := |
| (LinearMap.ker (M.syzygies n).cover.map).subtype.comp (M.syzygies (n + 1)).cover.map |
|
|
| theorem positiveDifferential_range (M : FiniteLeftModule k R) (n : ℕ) : |
| LinearMap.range (M.positiveDifferential n) = LinearMap.ker (M.syzygies n).cover.map := by |
| ext x |
| constructor |
| · rintro ⟨y, rfl⟩ |
| exact ((M.syzygies (n + 1)).cover.map y).property |
| · intro hx |
| obtain ⟨y, hy⟩ := (M.syzygies (n + 1)).cover.surjective ⟨x, hx⟩ |
| exact ⟨y, congrArg Subtype.val hy⟩ |
|
|
| theorem positiveDifferential_kernel (M : FiniteLeftModule k R) (n : ℕ) : |
| LinearMap.ker (M.positiveDifferential n) = LinearMap.ker (M.syzygies (n + 1)).cover.map := by |
| ext x |
| change ((M.syzygies (n + 1)).cover.map x).val = 0 ↔ (M.syzygies (n + 1)).cover.map x = 0 |
| exact ⟨fun h => Subtype.ext h, fun h => congrArg Subtype.val h⟩ |
|
|
| theorem positiveDifferential_exact (M : FiniteLeftModule k R) (n : ℕ) : |
| LinearMap.range (M.positiveDifferential (n + 1)) = |
| LinearMap.ker (M.positiveDifferential n) := by |
| rw [positiveDifferential_range, positiveDifferential_kernel] |
|
|
| noncomputable abbrev negativeTerm (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) (n : ℕ) := |
| Fin (M.cosyzygies τ n).envelopeRank → R |
|
|
| noncomputable def negativeDifferential (τ : FrobeniusTrace k R) |
| (M : FiniteLeftModule k R) (n : ℕ) : |
| M.negativeTerm τ n →ₗ[R] M.negativeTerm τ (n + 1) := |
| ((M.cosyzygies τ (n + 1)).envelopeMap τ).comp |
| (LinearMap.range ((M.cosyzygies τ n).envelopeMap τ)).mkQ |
|
|
| theorem negativeDifferential_range (τ : FrobeniusTrace k R) |
| (M : FiniteLeftModule k R) (n : ℕ) : |
| LinearMap.range (M.negativeDifferential τ n) = |
| LinearMap.range ((M.cosyzygies τ (n + 1)).envelopeMap τ) := by |
| ext x |
| constructor |
| · rintro ⟨y, rfl⟩ |
| exact ⟨_, rfl⟩ |
| · rintro ⟨y, rfl⟩ |
| obtain ⟨z, rfl⟩ := Submodule.mkQ_surjective |
| (LinearMap.range ((M.cosyzygies τ n).envelopeMap τ)) y |
| exact ⟨z, rfl⟩ |
|
|
| theorem negativeDifferential_kernel (τ : FrobeniusTrace k R) |
| (M : FiniteLeftModule k R) (n : ℕ) : |
| LinearMap.ker (M.negativeDifferential τ n) = |
| LinearMap.range ((M.cosyzygies τ n).envelopeMap τ) := by |
| ext x |
| change ((M.cosyzygies τ (n + 1)).envelopeMap τ) |
| ((LinearMap.range ((M.cosyzygies τ n).envelopeMap τ)).mkQ x) = 0 ↔ _ |
| exact (map_eq_zero_iff _ ((M.cosyzygies τ (n + 1)).envelopeMap_injective τ)).trans |
| (Submodule.Quotient.mk_eq_zero _) |
|
|
| theorem negativeDifferential_exact (τ : FrobeniusTrace k R) |
| (M : FiniteLeftModule k R) (n : ℕ) : |
| LinearMap.range (M.negativeDifferential τ n) = |
| LinearMap.ker (M.negativeDifferential τ (n + 1)) := by |
| rw [negativeDifferential_range, negativeDifferential_kernel] |
|
|
| noncomputable def seam (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| M.positiveTerm 0 →ₗ[R] M.negativeTerm τ 0 := |
| (M.envelopeMap τ).comp M.cover.map |
|
|
| theorem seam_range (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| LinearMap.range (M.seam τ) = LinearMap.range (M.envelopeMap τ) := by |
| ext x |
| constructor |
| · rintro ⟨y, rfl⟩ |
| exact ⟨_, rfl⟩ |
| · rintro ⟨y, rfl⟩ |
| obtain ⟨z, rfl⟩ := M.cover.surjective y |
| exact ⟨z, rfl⟩ |
|
|
| theorem seam_kernel (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| LinearMap.ker (M.seam τ) = LinearMap.ker M.cover.map := by |
| ext x |
| exact map_eq_zero_iff _ (M.envelopeMap_injective τ) |
|
|
| noncomputable def term (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : ℤ → ModuleCat.{u} R |
| | Int.ofNat n => ModuleCat.of R (M.positiveTerm n) |
| | Int.negSucc n => ModuleCat.of R (M.negativeTerm τ n) |
|
|
| noncomputable def differential (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| (i : ℤ) → M.term τ (i + 1) ⟶ M.term τ i |
| | Int.ofNat n => ModuleCat.ofHom (M.positiveDifferential n) |
| | Int.negSucc 0 => ModuleCat.ofHom (M.seam τ) |
| | Int.negSucc (n + 1) => ModuleCat.ofHom (M.negativeDifferential τ n) |
|
|
| theorem adjacent_exact (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) (i : ℤ) : |
| LinearMap.range (M.differential τ (i + 1)).hom = |
| LinearMap.ker (M.differential τ i).hom := by |
| cases i with |
| | ofNat n => exact M.positiveDifferential_exact n |
| | negSucc n => |
| cases n with |
| | zero => |
| change LinearMap.range (M.positiveDifferential 0) = LinearMap.ker (M.seam τ) |
| rw [positiveDifferential_range, seam_kernel] |
| rfl |
| | succ n => |
| cases n with |
| | zero => |
| change LinearMap.range (M.seam τ) = LinearMap.ker (M.negativeDifferential τ 0) |
| rw [seam_range, negativeDifferential_kernel] |
| rfl |
| | succ n => exact M.negativeDifferential_exact τ n |
|
|
| theorem differential_square (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) (i : ℤ) : |
| M.differential τ (i + 1) ≫ M.differential τ i = 0 := by |
| apply ModuleCat.hom_ext |
| exact LinearMap.range_le_ker_iff.mp (M.adjacent_exact τ i).le |
|
|
| noncomputable def completeComplex (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| ChainComplex (ModuleCat.{u} R) ℤ := |
| ChainComplex.of (M.term τ) (M.differential τ) (M.differential_square τ) |
|
|
| theorem completeComplex_exact (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) (n : ℤ) : |
| LinearMap.range ((M.completeComplex τ).d (n + 1) n).hom = |
| LinearMap.ker ((M.completeComplex τ).d n (n - 1)).hom := by |
| have aux (i j : ℤ) (hi : i = j + 1) : |
| LinearMap.range ((M.completeComplex τ).d (i + 1) i).hom = |
| LinearMap.ker ((M.completeComplex τ).d i j).hom := by |
| subst i |
| simp only [completeComplex, ChainComplex.of_d] |
| exact M.adjacent_exact τ j |
| exact aux n (n - 1) (by omega) |
|
|
| theorem completeComplex_exactProjective (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| ExactProjectiveComplex (M.completeComplex τ) where |
| finite n := by |
| cases n with |
| | ofNat n => exact inferInstanceAs (Module.Finite R (M.positiveTerm n)) |
| | negSucc n => exact inferInstanceAs (Module.Finite R (M.negativeTerm τ n)) |
| projective n := by |
| cases n with |
| | ofNat n => exact inferInstanceAs (Module.Projective R (M.positiveTerm n)) |
| | negSucc n => exact inferInstanceAs (Module.Projective R (M.negativeTerm τ n)) |
| exactAt := M.completeComplex_exact τ |
|
|
| theorem completeComplex_totallyAcyclic (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| TotallyAcyclicComplex (M.completeComplex τ) := by |
| let := τ.regular_injective |
| exact (M.completeComplex_exactProjective τ).totallyAcyclic |
|
|
| noncomputable def completeCokernelEquiv (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) : |
| CompleteComparison.Cokernel (M.completeComplex τ) 0 ≃ₗ[R] M.carrier := by |
| change (M.positiveTerm 0 ⧸ LinearMap.range (M.positiveDifferential 0)) ≃ₗ[R] M.carrier |
| exact (Submodule.quotEquivOfEq _ _ (M.positiveDifferential_range 0)).trans |
| (M.cover.map.quotKerEquivOfSurjective M.cover.surjective) |
|
|
| end FiniteLeftModule |
|
|
| theorem exists_finite_complete_resolution [Module.Finite k R] |
| (τ : FrobeniusTrace k R) (M : Type u) [AddCommGroup M] [Module R M] |
| [Module k M] [IsScalarTower k R M] [Module.Finite k M] : |
| ∃ P : ChainComplex (ModuleCat.{u} R) ℤ, |
| TotallyAcyclicComplex P ∧ Nonempty (M ≃ₗ[R] CompleteComparison.Cokernel P 0) := by |
| let X : FiniteLeftModule k R := { carrier := ModuleCat.of R M } |
| exact ⟨X.completeComplex τ, X.completeComplex_totallyAcyclic τ, |
| ⟨(X.completeCokernelEquiv τ).symm⟩⟩ |
|
|
| end ArExplicit |
|
|
| end OAI |
|
|
