| import Mathlib |
| import OAI.RingTheory.Multiplicity.CechReduction |
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|
| namespace OAI |
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| noncomputable section |
| namespace Lech.FiniteModuleCech |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| universe u |
| variable {R : Type u} [CommRing R] {ι : Type} |
| variable {D E G : Diagram R ι} (φ : Map D E) (ψ : Map E G) |
| (hzero : ∀ s,φ.app s ≫ ψ.app s=0) |
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|
| include hzero in |
| lemma Map.cochains_comp_zero (q : ℕ) : ψ.cochains q ∘ₗ φ.cochains q=0 := by |
| apply LinearMap.ext |
| intro x |
| funext s |
| exact congrArg (fun f : D.obj s.val ⟶ G.obj s.val => f.hom (x s)) (hzero s.val) |
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|
| def diagramShortComplex (s : Finset ι) : ShortComplex (ModuleCat.{u} R) := |
| ShortComplex.mk (φ.app s) (ψ.app s) (hzero s) |
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| def cochainsShortComplex (q : ℕ) : ShortComplex (ModuleCat.{u} R) := |
| ShortComplex.mk (ModuleCat.ofHom (φ.cochains q)) (ModuleCat.ofHom (ψ.cochains q)) |
| (ModuleCat.hom_ext (Map.cochains_comp_zero φ ψ hzero q)) |
|
|
| lemma cochains_shortExact (hs : ∀ s,(diagramShortComplex φ ψ hzero s).ShortExact) (q : ℕ) : |
| (cochainsShortComplex φ ψ hzero q).ShortExact where |
| mono_f := (ModuleCat.mono_iff_injective _).mpr (by |
| intro x y hxy |
| funext s |
| exact (hs s.val).moduleCat_injective_f (congrFun hxy s)) |
| epi_g := (ModuleCat.epi_iff_surjective _).mpr (by |
| intro y |
| choose x hx using fun s : Set.powersetCard ι q => (hs s.val).moduleCat_surjective_g (y s) |
| exact ⟨x,funext hx⟩) |
| exact := (ShortComplex.moduleCat_exact_iff _).mpr (by |
| intro y hy |
| have hys (s : Set.powersetCard ι q) : (ψ.app s.val).hom (y s)=0 := congrFun hy s |
| choose x hx using fun s : Set.powersetCard ι q => |
| (ShortComplex.moduleCat_exact_iff _).mp (hs s.val).exact (y s) (hys s) |
| exact ⟨x,funext hx⟩) |
|
|
| variable [Fintype ι] [LinearOrder ι] |
|
|
| def complexShortComplex : ShortComplex (CochainComplex (ModuleCat.{u} R) ℕ) := |
| ShortComplex.mk φ.complex ψ.complex (by |
| apply Hom.ext |
| funext q |
| apply ModuleCat.hom_ext |
| exact Map.cochains_comp_zero φ ψ hzero q) |
|
|
| lemma complex_shortExact (hs : ∀ s,(diagramShortComplex φ ψ hzero s).ShortExact) : |
| (complexShortComplex φ ψ hzero).ShortExact := |
| shortExact_of_degreewise_shortExact _ (cochains_shortExact φ ψ hzero hs) |
|
|
| def positiveShortComplex : ShortComplex (CochainComplex (ModuleCat.{u} R) ℕ) := |
| ShortComplex.mk φ.positiveComplex ψ.positiveComplex (by |
| apply Hom.ext |
| funext q |
| apply ModuleCat.hom_ext |
| exact Map.cochains_comp_zero φ ψ hzero (q+1)) |
|
|
| lemma positive_shortExact (hs : ∀ s,(diagramShortComplex φ ψ hzero s).ShortExact) : |
| (positiveShortComplex φ ψ hzero).ShortExact := |
| shortExact_of_degreewise_shortExact _ (fun q => cochains_shortExact φ ψ hzero hs (q+1)) |
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| end Lech.FiniteModuleCech |
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| end |
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| end OAI |
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