| import OAI.Algebra.AuslanderReiten.Stable.TargetStableCoordinates | |
| import OAI.Algebra.AuslanderReiten.Complete.TateKoszul | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| open CochainComplex CochainComplex.HomComplex | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.FrobeniusTrace | |
| universe u | |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] [Module.Finite k R] | |
| variable (τ : FrobeniusTrace k R) (hs : ∀ a b : R, τ.trace (a*b)=τ.trace (b*a)) | |
| open CompleteComparison | |
| variable (K : CochainComplex (ModuleCat.{u} R) ℤ) | |
| variable (hK : TotallyAcyclicComplex (chainOfCochain K)) | |
| def fullHomSourceDualityAt (a b : ℤ) (hb : b = -a-1) : | |
| Module.Dual k (CohomologyClass K K a) ≃ₗ[k] CohomologyClass K K b := | |
| (fullHomSourceStableLinearEquiv (k := k) K K a hK | |
| hK.toExactProjectiveComplex).symm.dualMap |>.trans | |
| ((τ.completeStableDualityAt hs (chainOfCochain K) hK.toExactProjectiveComplex a).trans | |
| (fullHomTargetStableLinearEquiv (k := k) (a+1) (by omega) hK | |
| hK.toExactProjectiveComplex).symm) | |
| theorem fullHomSourceDualityAt_symm_apply (a b : ℤ) (hb : b = -a-1) | |
| (v : CohomologyClass K K b) (u : CohomologyClass K K a) : | |
| ((τ.fullHomSourceDualityAt hs K hK a b hb).symm v) u = | |
| ((τ.completeStableDualityAt hs (chainOfCochain K) hK.toExactProjectiveComplex a).symm | |
| (fullHomTargetStableLinearEquiv (k := k) (a+1) (by omega) hK | |
| hK.toExactProjectiveComplex v)) | |
| (fullHomSourceStableLinearEquiv (k := k) K K a hK hK.toExactProjectiveComplex u) := rfl | |
| def fullHomSourceDuality (a : ℤ) : | |
| Module.Dual k (CohomologyClass K K a) ≃ₗ[k] CohomologyClass K K (-a-1) := | |
| τ.fullHomSourceDualityAt hs K hK a (-a-1) rfl | |
| theorem fullHomSourceDualityAt_transport (a b : ℤ) (hb : b = -a-1) | |
| (v : CohomologyClass K K b) : | |
| (τ.fullHomSourceDualityAt hs K hK a b hb).symm v = | |
| (τ.fullHomSourceDuality hs K hK a).symm | |
| (TateKoszul.degreeEquiv (k := k) (CohomologyClass K K) hb v) := by | |
| subst b | |
| rfl | |
| variable [CharP k 2] | |
| theorem fullHomSourceDualityAt_product (a n m : ℤ) (hm : m = -(n+a)-1) | |
| (p : CohomologyClass K K n) (u : CohomologyClass K K a) | |
| (v : CohomologyClass K K m) : | |
| ((τ.fullHomSourceDualityAt hs K hK a (m+n) (by omega)).symm | |
| (fullHomProduct v p)) u = | |
| ((τ.fullHomSourceDualityAt hs K hK (n+a) m hm).symm v) | |
| (fullHomProduct p u) := by | |
| obtain ⟨p,rfl⟩ := CohomologyClass.mk_surjective p | |
| obtain ⟨u,rfl⟩ := CohomologyClass.mk_surjective u | |
| obtain ⟨v,rfl⟩ := CohomologyClass.mk_surjective v | |
| rw [fullHomProduct_mk, fullHomProduct_mk, | |
| fullHomSourceDualityAt_symm_apply, fullHomSourceDualityAt_symm_apply, | |
| fullHomTargetStableLinearEquiv_mk, fullHomTargetStableLinearEquiv_mk, | |
| fullHomSourceStableLinearEquiv_mk, fullHomSourceStableLinearEquiv_mk] | |
| rw [targetCocycleMap_eq_reversed (k := k), targetCocycleMap_eq_reversed (k := k), | |
| sourceCocycleMap_eq_reversed (k := k), sourceCocycleMap_eq_reversed (k := k), | |
| reversedCokernelMap_product (k := k) v p 0 (n+a+1) (a+1) (by omega) (by omega), | |
| reversedCokernelMap_product (k := k) p u (n+a) a 0 (by omega) (by omega)] | |
| exact (τ.completeStableDualityAt_source hs hK.toExactProjectiveComplex | |
| hK.toExactProjectiveComplex (n+a) a | |
| (reversedComponent p.val (n+a) a (by omega)).hom | |
| (reversedCokernelMap (k := k) p (n+a) a (by omega)) | |
| (reversedCokernelMap (k := k) p (n+a+1) (a+1) (by omega)) | |
| (by ext x; rfl) (reversedCokernelMap_syzygy (k := k) p hK.toExactProjectiveComplex | |
| hK.toExactProjectiveComplex (n+a) a (by omega)) | |
| (StableHom.classOf R _ _ (reversedCokernelMap (k := k) u a 0 (by omega))) | |
| (StableHom.classOf R _ _ (reversedCokernelMap (k := k) v 0 (n+a+1) (by omega)))).symm | |
| end ArExplicit.FrobeniusTrace | |
| end | |
| end OAI | |