| import OAI.Algebra.AuslanderReiten.Stable.StableExt | |
| import OAI.Algebra.AuslanderReiten.Stable.SourceStableShift | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit | |
| open CategoryTheory CompleteComparison | |
| open CochainComplex CochainComplex.HomComplex | |
| universe u | |
| variable {k R : Type u} [CommRing k] [Ring R] [Algebra k R] | |
| variable {K L : CochainComplex (ModuleCat.{u} R) ℤ} | |
| include k | |
| theorem positiveExtVanish_of_fullHom | |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) | |
| (hL : ExactProjectiveComplex (chainOfCochain L)) | |
| (h : ∀ n : ℕ, 0 < n → Subsingleton (CohomologyClass K L (n : ℤ))) : | |
| PositiveExtVanish R (Cokernel (chainOfCochain K) 0) (Cokernel (chainOfCochain L) 0) := by | |
| apply positiveExtVanish_of_stableCokernels hK | |
| intro n | |
| let := h (n + 1) (Nat.zero_lt_succ n) | |
| exact (fullHomSourceStableLinearEquiv (k := k) K L ((n + 1 : ℕ) : ℤ) hK hL).symm.injective.subsingleton | |
| theorem not_projective_of_fullHom_nontrivial | |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) | |
| (h : Nontrivial (CohomologyClass K K 0)) : | |
| ¬ Module.Projective R (Cokernel (chainOfCochain K) 0) := by | |
| intro hp | |
| let := hK.finite 0 | |
| let := hp | |
| have : Subsingleton (StableHom R (Cokernel (chainOfCochain K) 0) | |
| (Cokernel (chainOfCochain K) 0)) := ⟨fun a b => | |
| (StableHom.eq_zero_of_projective_source a).trans (StableHom.eq_zero_of_projective_source b).symm⟩ | |
| have : Subsingleton (CohomologyClass K K 0) := | |
| (fullHomStableZeroLinearEquiv (k := k) K K hK hK.toExactProjectiveComplex).injective.subsingleton | |
| obtain ⟨a, b, hab⟩ := h.exists_pair_ne | |
| exact hab (Subsingleton.elim a b) | |
| end ArExplicit | |
| end | |
| end OAI | |