| import OAI.Algebra.AuslanderReiten.Complete.CompleteShift |
|
|
| namespace OAI |
|
|
| open CategoryTheory |
|
|
| namespace ArExplicit |
|
|
| universe u |
|
|
| variable {R : Type u} [Ring R] |
|
|
| namespace HomExactAt |
|
|
| variable {U V W U' V' W' : Type u} |
| variable [AddCommGroup U] [Module R U] [AddCommGroup V] [Module R V] |
| variable [AddCommGroup W] [Module R W] [AddCommGroup U'] [Module R U'] |
| variable [AddCommGroup V'] [Module R V'] [AddCommGroup W'] [Module R W'] |
|
|
| theorem congr {e : U →ₗ[R] V} {d : V →ₗ[R] W} |
| (h : HomExactAt e d) (e' : U' →ₗ[R] V') (d' : V' →ₗ[R] W') |
| (a : U ≃ₗ[R] U') (b : V ≃ₗ[R] V') (c : W ≃ₗ[R] W') |
| (he : e'.comp a.toLinearMap = b.toLinearMap.comp e) |
| (hd : d'.comp b.toLinearMap = c.toLinearMap.comp d) : HomExactAt e' d' := by |
| intro f hf |
| have hfb : (f.comp b.toLinearMap).comp e = 0 := by |
| rw [LinearMap.comp_assoc, ← he, ← LinearMap.comp_assoc, hf, LinearMap.zero_comp] |
| obtain ⟨g, hg⟩ := h (f.comp b.toLinearMap) hfb |
| refine ⟨g.comp c.symm.toLinearMap, ?_⟩ |
| apply LinearMap.ext |
| intro x' |
| obtain ⟨x, rfl⟩ := b.surjective x' |
| change g (c.symm (d' (b x))) = f (b x) |
| rw [show d' (b x) = c (d x) from DFunLike.congr_fun hd x, c.symm_apply_apply] |
| exact DFunLike.congr_fun hg x |
|
|
| theorem of_shortComplex_iso {S T : ShortComplex (ModuleCat.{u} R)} (E : S ≅ T) |
| (h : HomExactAt S.f.hom S.g.hom) : HomExactAt T.f.hom T.g.hom := by |
| apply h.congr T.f.hom T.g.hom (asIso E.hom.τ₁).toLinearEquiv |
| (asIso E.hom.τ₂).toLinearEquiv (asIso E.hom.τ₃).toLinearEquiv |
| · exact congrArg (fun f : S.X₁ ⟶ T.X₂ => f.hom) E.hom.comm₁₂ |
| · exact congrArg (fun f : S.X₂ ⟶ T.X₃ => f.hom) E.hom.comm₂₃ |
|
|
| end HomExactAt |
|
|
| namespace CompleteComparison |
|
|
| noncomputable def reverseShortComplexIso |
| (K : CochainComplex (ModuleCat.{u} R) ℤ) (i j l i' j' l' : ℤ) |
| (hi : negIndexEmbedding.f i = i') (hj : negIndexEmbedding.f j = j') |
| (hl : negIndexEmbedding.f l = l') : |
| (chainOfCochain K).sc' i' j' l' ≅ K.sc' i j l := |
| ShortComplex.isoMk (K.extendXIso negIndexEmbedding hi) |
| (K.extendXIso negIndexEmbedding hj) (K.extendXIso negIndexEmbedding hl) |
| (by |
| change (K.extendXIso negIndexEmbedding hi).hom ≫ K.d i j = |
| (chainOfCochain K).d i' j' ≫ (K.extendXIso negIndexEmbedding hj).hom |
| rw [K.extend_d_eq negIndexEmbedding hi hj] |
| simp only [Category.assoc, Iso.inv_hom_id, Category.comp_id]) |
| (by |
| change (K.extendXIso negIndexEmbedding hj).hom ≫ K.d j l = |
| (chainOfCochain K).d j' l' ≫ (K.extendXIso negIndexEmbedding hl).hom |
| rw [K.extend_d_eq negIndexEmbedding hj hl] |
| simp only [Category.assoc, Iso.inv_hom_id, Category.comp_id]) |
|
|
| end CompleteComparison |
|
|
| def DualExactCochain (K : CochainComplex (ModuleCat.{u} R) ℤ) : Prop := |
| ∀ i, HomExactAt (K.d (i - 1) i).hom (K.d i (i + 1)).hom |
|
|
| namespace DualExactCochain |
|
|
| open CompleteComparison |
|
|
| theorem of_chain {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) : DualExactCochain K := by |
| intro i |
| let E := reverseShortComplexIso K (i - 1) i (i + 1) (-i + 1) (-i) (-i - 1) |
| (by change -(i - 1) = -i + 1; omega) rfl (by change -(i + 1) = -i - 1; omega) |
| exact HomExactAt.of_shortComplex_iso E (hK.dualExact (-i)) |
| |
| theorem to_chain {K : CochainComplex (ModuleCat.{u} R) ℤ} (hK : DualExactCochain K) : |
| ∀ i, HomExactAt ((chainOfCochain K).d (i + 1) i).hom |
| ((chainOfCochain K).d i (i - 1)).hom := by |
| intro i |
| let E := reverseShortComplexIso K (-i - 1) (-i) (-i + 1) (i + 1) i (i - 1) |
| (by change -(-i - 1) = i + 1; omega) (neg_neg i) |
| (by change -(-i + 1) = i - 1; omega) |
| exact HomExactAt.of_shortComplex_iso E.symm (hK (-i)) |
| |
| theorem shift {K : CochainComplex (ModuleCat.{u} R) ℤ} (hK : DualExactCochain K) (a : ℤ) : |
| DualExactCochain (K⟦a⟧) := by |
| intro i |
| let E := (CochainComplex.shiftShortComplexFunctor' (ModuleCat.{u} R) a (i - 1) i (i + 1) |
| (i + a - 1) (i + a) (i + a + 1) (by omega) (by omega) (by omega)).app K |
| exact HomExactAt.of_shortComplex_iso E.symm (hK (i + a)) |
|
|
| end DualExactCochain |
|
|
| structure TotallyAcyclicCochain (K : CochainComplex (ModuleCat.{u} R) ℤ) : Prop |
| extends ExactProjectiveCochain K where |
| dualExact : DualExactCochain K |
|
|
| namespace TotallyAcyclicCochain |
|
|
| open CompleteComparison |
|
|
| theorem of_chain {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) : TotallyAcyclicCochain K where |
| toExactProjectiveCochain := ExactProjectiveCochain.of_chain hK.toExactProjectiveComplex |
| dualExact := DualExactCochain.of_chain hK |
|
|
| theorem to_chain {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : TotallyAcyclicCochain K) : TotallyAcyclicComplex (chainOfCochain K) where |
| toExactProjectiveComplex := hK.toExactProjectiveCochain.to_chain |
| dualExact := hK.dualExact.to_chain |
|
|
| theorem shift {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : TotallyAcyclicCochain K) (a : ℤ) : TotallyAcyclicCochain (K⟦a⟧) where |
| toExactProjectiveCochain := hK.toExactProjectiveCochain.shift a |
| dualExact := hK.dualExact.shift a |
|
|
| end TotallyAcyclicCochain |
|
|
| namespace TotallyAcyclicComplex |
|
|
| open CompleteComparison |
|
|
| theorem cochainShift {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) (a : ℤ) : |
| TotallyAcyclicComplex (chainOfCochain (K⟦a⟧)) where |
| toExactProjectiveComplex := hK.toExactProjectiveComplex.cochainShift a |
| dualExact := ((DualExactCochain.of_chain hK).shift a).to_chain |
|
|
| end TotallyAcyclicComplex |
| end ArExplicit |
|
|
| end OAI |
|
|
