| import OAI.Algebra.AuslanderReiten.Ext.ExtBridge |
| import OAI.Algebra.AuslanderReiten.Complete.CompleteComparison |
| import OAI.Algebra.AuslanderReiten.Stable.StableHom |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ArExplicit |
|
|
| open CategoryTheory CompleteComparison |
|
|
| universe u |
|
|
| variable {R : Type u} [Ring R] |
| variable {P : ChainComplex (ModuleCat.{u} R) ℤ} |
| variable {N : Type u} [AddCommGroup N] [Module R N] |
|
|
| def completeExtCocycleMap (n : ℤ) (f : P.X n →ₗ[R] N) |
| (hf : f.comp (P.d (n + 1) n).hom = 0) : Cokernel P n →ₗ[R] N := |
| (LinearMap.range (P.d (n + 1) n).hom).liftQ f (LinearMap.range_le_ker_iff.mpr hf) |
|
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| @[simp] theorem completeExtCocycleMap_mk (n : ℤ) (f : P.X n →ₗ[R] N) |
| (hf : f.comp (P.d (n + 1) n).hom = 0) (x : P.X n) : |
| completeExtCocycleMap n f hf ((LinearMap.range (P.d (n + 1) n).hom).mkQ x) = f x := rfl |
|
|
| theorem completeExtCocycleMap_stable_zero_iff (hP : TotallyAcyclicComplex P) |
| (n : ℤ) (f : P.X n →ₗ[R] N) (hf : f.comp (P.d (n + 1) n).hom = 0) : |
| StableHom.classOf R _ N (completeExtCocycleMap n f hf) = 0 ↔ |
| ∃ g : P.X (n - 1) →ₗ[R] N, g.comp (P.d n (n - 1)).hom = f := by |
| rw [StableHom.classOf_eq_zero_iff_projective] |
| constructor |
| · rintro ⟨Q, hQ, hRQ, hproj, hfin, i, p, hip⟩ |
| let π := (LinearMap.range (P.d (n + 1) n).hom).mkQ |
| have hi : (i.comp π).comp (P.d (n + 1) n).hom = 0 := by |
| rw [LinearMap.comp_assoc, LinearMap.range_mkQ_comp, LinearMap.comp_zero] |
| obtain ⟨g, hg⟩ := (hP.dualExact n).projective (i.comp π) hi |
| refine ⟨p.comp g, ?_⟩ |
| rw [LinearMap.comp_assoc, hg, ← LinearMap.comp_assoc, hip] |
| rfl |
| · rintro ⟨g, hg⟩ |
| let := hP.finite (n - 1) |
| let := hP.projective (n - 1) |
| let d : Cokernel P n →ₗ[R] P.X (n - 1) := |
| completeExtCocycleMap n (P.d n (n - 1)).hom (differential_comp P _ _ _) |
| refine ⟨P.X (n - 1), inferInstance, inferInstance, inferInstance, inferInstance, d, g, ?_⟩ |
| apply LinearMap.ext |
| intro x |
| obtain ⟨x, rfl⟩ := Submodule.mkQ_surjective (LinearMap.range (P.d (n + 1) n).hom) x |
| exact DFunLike.congr_fun hg x |
|
|
| theorem positiveExtVanish_of_stableCokernels (hP : TotallyAcyclicComplex P) |
| (h : ∀ n : ℕ, Subsingleton (StableHom R (Cokernel P ((n + 1 : ℕ) : ℤ)) N)) : |
| PositiveExtVanish R (Cokernel P 0) N := by |
| apply (positiveExtVanish_iff_completeResolution R (Cokernel P 0) N P |
| hP.projective hP.exactAt (LinearEquiv.refl R _)).mpr |
| intro n f hf |
| have hi : ((n + 1 : ℕ) : ℤ) + 1 = ((n + 2 : ℕ) : ℤ) := by omega |
| have hf' : f.comp (P.d (((n + 1 : ℕ) : ℤ) + 1) ((n + 1 : ℕ) : ℤ)).hom = 0 := by |
| rw [hi] |
| exact hf |
| have := h n |
| have hz : StableHom.classOf R _ N (completeExtCocycleMap ((n + 1 : ℕ) : ℤ) f hf') = 0 := |
| Subsingleton.elim _ _ |
| have hgex := (completeExtCocycleMap_stable_zero_iff hP _ f hf').mp hz |
| have hn : ((n + 1 : ℕ) : ℤ) - 1 = (n : ℤ) := by omega |
| rw [hn] at hgex |
| obtain ⟨g, hg⟩ := hgex |
| exact ⟨g, hg.symm⟩ |
|
|
| end ArExplicit |
|
|
| end |
|
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| end OAI |
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|
