| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomProducts |
| import Mathlib.Algebra.Homology.Linear |
|
|
| namespace OAI |
|
|
| open CategoryTheory |
| open CochainComplex CochainComplex.HomComplex |
|
|
| namespace ArExplicit.CompleteComparison |
|
|
| universe u v |
|
|
| section Cochains |
|
|
| variable (k : Type u) [CommRing k] |
| variable {C : Type*} [Category* C] [Preadditive C] [Linear k C] |
| variable (K L : CochainComplex C ℤ) (n : ℤ) |
|
|
| def fullHomCoboundaries : Submodule k (Cocycle K L n) where |
| __ := coboundaries K L n |
| smul_mem' a x hx := by |
| obtain ⟨m, hm, y, hy⟩ := hx |
| refine ⟨m, hm, a • y, ?_⟩ |
| rw [δ_smul, hy] |
| rfl |
| |
| instance fullHomModule : Module k (CohomologyClass K L n) := |
| inferInstanceAs (Module k ((Cocycle K L n) ⧸ fullHomCoboundaries k K L n)) |
| |
| variable {K L n} |
| |
| @[simp] |
| theorem fullHom_mk_smul (a : k) (x : Cocycle K L n) : |
| CohomologyClass.mk (a • x) = a • CohomologyClass.mk x := rfl |
| |
| theorem cocycleHom_smul (a : k) (x : Cocycle K L n) : |
| Cocycle.equivHomShift.symm (a • x) = a • Cocycle.equivHomShift.symm x := by |
| apply HomologicalComplex.Hom.ext |
| funext i |
| simp only [cocycleHom_component, HomologicalComplex.smul_f_apply, |
| Cocycle.coe_smul, Cochain.smul_v] |
| |
| end Cochains |
| |
| variable {k R : Type u} [CommRing k] [Ring R] [Algebra k R] |
| open scoped ModuleCat.Algebra |
| |
| variable {P Q : ChainComplex (ModuleCat.{u} R) ℤ} |
| |
| theorem cokernelMap_smul (a : k) (F : P ⟶ Q) (n : ℤ) : |
| cokernelMap (a • F) n = a • cokernelMap F n := by |
| apply LinearMap.ext |
| intro z |
| obtain ⟨x, rfl⟩ := Submodule.mkQ_surjective (LinearMap.range (P.d (n + 1) n).hom) z |
| change (LinearMap.range (Q.d (n + 1) n).hom).mkQ (a • (F.f n).hom x) = |
| a • (LinearMap.range (Q.d (n + 1) n).hom).mkQ ((F.f n).hom x) |
| exact ((LinearMap.range (Q.d (n + 1) n).hom).mkQ.restrictScalars k).map_smul a _ |
| |
| variable {K L : CochainComplex (ModuleCat.{u} R) ℤ} {n : ℤ} |
| |
| theorem cocycleComparisonMap_smul (a : k) (x : Cocycle K L n) : |
| cocycleComparisonMap (a • x) = a • cocycleComparisonMap x := by |
| unfold cocycleComparisonMap |
| rw [cocycleHom_smul] |
| apply HomologicalComplex.Hom.ext |
| funext i |
| have hi : negIndexEmbedding.f (-i) = i := neg_neg i |
| simp only [HomologicalComplex.extendMap_f _ _ hi, |
| HomologicalComplex.smul_f_apply, Linear.comp_smul, Linear.smul_comp] |
| |
| theorem fullHomStableEquiv_smul (hK : TotallyAcyclicComplex (chainOfCochain K)) |
| (hL : ExactProjectiveComplex (chainOfCochain (L⟦n⟧))) |
| (a : k) (x : CohomologyClass K L n) : |
| fullHomStableEquiv K L n hK hL (a • x) = a • fullHomStableEquiv K L n hK hL x := by |
| obtain ⟨x, rfl⟩ := CohomologyClass.mk_surjective x |
| rw [← fullHom_mk_smul, fullHomStableEquiv_mk, fullHomStableEquiv_mk, |
| cocycleComparisonMap_smul, cokernelMap_smul, StableHom.classOf_smul] |
| |
| noncomputable def fullHomStableLinearEquiv |
| (K L : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ) |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) |
| (hL : ExactProjectiveComplex (chainOfCochain (L⟦n⟧))) : |
| CohomologyClass K L n ≃ₗ[k] |
| StableHom R (Cokernel (chainOfCochain K) 0) (Cokernel (chainOfCochain (L⟦n⟧)) 0) := |
| { fullHomStableEquiv K L n hK hL with |
| map_smul' := fullHomStableEquiv_smul hK hL } |
| |
| end ArExplicit.CompleteComparison |
| |
| end OAI |
| |
