| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomComparison |
| import OAI.Algebra.AuslanderReiten.Algebras.FrobeniusTrace |
| import OAI.Algebra.AuslanderReiten.Complete.SelfInjectiveComplex |
| import Mathlib.Algebra.Homology.Embedding.ExtendHomology |
| import Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence |
| import Mathlib.Algebra.Homology.ShortComplex.ModuleCat |
|
|
| namespace OAI |
|
|
| open CategoryTheory |
|
|
| namespace ArExplicit |
|
|
| universe u |
|
|
| variable {R : Type u} [Ring R] |
|
|
| structure ExactProjectiveCochain (K : CochainComplex (ModuleCat.{u} R) ℤ) : Prop where |
| finite : ∀ i, Module.Finite R (K.X i) |
| projective : ∀ i, Module.Projective R (K.X i) |
| exactAt : ∀ i, K.ExactAt i |
|
|
| theorem chain_exactAt_iff (P : ChainComplex (ModuleCat.{u} R) ℤ) (i : ℤ) : |
| P.ExactAt i ↔ LinearMap.range (P.d (i + 1) i).hom = |
| LinearMap.ker (P.d i (i - 1)).hom := by |
| rw [P.exactAt_iff' (i := i + 1) (j := i) (k := i - 1) |
| (ChainComplex.prev ℤ i) (ChainComplex.next ℤ i), |
| ShortComplex.moduleCat_exact_iff_range_eq_ker] |
| rfl |
|
|
| namespace ExactProjectiveCochain |
|
|
| open CompleteComparison |
|
|
| theorem of_chain {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : ExactProjectiveComplex (chainOfCochain K)) : ExactProjectiveCochain K where |
| finite i := by |
| let := hK.finite (-i) |
| exact Module.Finite.equiv (CategoryTheory.Iso.toLinearEquiv (K.extendXIso negIndexEmbedding (i := i) (i' := -i) rfl)) |
| projective i := by |
| let := hK.projective (-i) |
| exact Module.Projective.of_equiv |
| (CategoryTheory.Iso.toLinearEquiv (K.extendXIso negIndexEmbedding (i := i) (i' := -i) rfl)) |
| exactAt i := |
| (K.extend_exactAt_iff negIndexEmbedding (j := i) (j' := -i) rfl).mp |
| ((chain_exactAt_iff (chainOfCochain K) (-i)).mpr (hK.exactAt (-i))) |
|
|
| theorem to_chain {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : ExactProjectiveCochain K) : ExactProjectiveComplex (chainOfCochain K) where |
| finite i := by |
| let := hK.finite (-i) |
| exact Module.Finite.equiv (CategoryTheory.Iso.toLinearEquiv |
| (K.extendXIso negIndexEmbedding (show negIndexEmbedding.f (-i) = i from neg_neg i))).symm |
| projective i := by |
| let := hK.projective (-i) |
| exact Module.Projective.of_equiv (CategoryTheory.Iso.toLinearEquiv |
| (K.extendXIso negIndexEmbedding (show negIndexEmbedding.f (-i) = i from neg_neg i))).symm |
| exactAt i := (chain_exactAt_iff (chainOfCochain K) i).mp |
| ((K.extend_exactAt_iff negIndexEmbedding (j := -i) (j' := i) (neg_neg i)).mpr |
| (hK.exactAt (-i))) |
|
|
| theorem shift {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : ExactProjectiveCochain K) (n : ℤ) : ExactProjectiveCochain (K⟦n⟧) where |
| finite i := hK.finite (i + n) |
| projective i := hK.projective (i + n) |
| exactAt i := ShortComplex.exact_of_iso |
| ((CochainComplex.shiftShortComplexFunctorIso (ModuleCat.{u} R) n i (n + i) rfl).app K).symm |
| (hK.exactAt (n + i)) |
|
|
| end ExactProjectiveCochain |
|
|
| namespace ExactProjectiveComplex |
|
|
| open CompleteComparison |
|
|
| theorem cochainShift {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : ExactProjectiveComplex (chainOfCochain K)) (n : ℤ) : |
| ExactProjectiveComplex (chainOfCochain (K⟦n⟧)) := |
| ((ExactProjectiveCochain.of_chain hK).shift n).to_chain |
|
|
| end ExactProjectiveComplex |
|
|
| namespace FrobeniusTrace |
|
|
| variable {k : Type u} [Field k] [Algebra k R] [Module.Finite k R] |
| open CompleteComparison |
|
|
| theorem totallyAcyclicShift (τ : FrobeniusTrace k R) |
| {K : CochainComplex (ModuleCat.{u} R) ℤ} |
| (hK : ExactProjectiveComplex (chainOfCochain K)) (n : ℤ) : |
| TotallyAcyclicComplex (chainOfCochain (K⟦n⟧)) := by |
| let := τ.regular_injective |
| exact (hK.cochainShift n).totallyAcyclic |
|
|
| end FrobeniusTrace |
| end ArExplicit |
|
|
| end OAI |
|
|
