| import Mathlib |
| import OAI.RingTheory.Multiplicity.TensorQuotientCommutation |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Lech.TensorIdeal |
| open CategoryTheory HomologicalComplex |
| universe u |
| variable {R : Type u} [CommRing R] (I : Ideal R) |
| variable {ι : Type*} (c : ComplexShape ι) |
|
|
| def complexQuotientNat : 𝟭 (HomologicalComplex (ModuleCat.{u} R) c) ⟶ |
| (quotientFunctor I).mapHomologicalComplex c := |
| (Functor.mapHomologicalComplexIdIso (ModuleCat.{u} R) c).inv ≫ |
| (quotientNat I).mapHomologicalComplex c |
|
|
| lemma complexQuotientNat_f (K : HomologicalComplex (ModuleCat.{u} R) c) (q : ι) : |
| ((complexQuotientNat I c).app K).f q=(quotientNat I).app (K.X q) := by |
| change (𝟙 (K.X q)) ≫ (quotientNat I).app (K.X q)=_ |
| exact Category.id_comp _ |
|
|
| variable {κ : Type*} (d : ComplexShape κ) |
| def bicomplexQuotientNat : |
| 𝟭 (HomologicalComplex (HomologicalComplex (ModuleCat.{u} R) c) d) ⟶ |
| ((quotientFunctor I).mapHomologicalComplex c).mapHomologicalComplex d := |
| (Functor.mapHomologicalComplexIdIso (HomologicalComplex (ModuleCat.{u} R) c) d).inv ≫ |
| (complexQuotientNat I c).mapHomologicalComplex d |
|
|
| lemma bicomplexQuotientNat_f (K : HomologicalComplex (HomologicalComplex (ModuleCat.{u} R) c) d) |
| (p : κ) (q : ι) : |
| (((bicomplexQuotientNat I c d).app K).f p).f q=(quotientNat I).app ((K.X p).X q) := by |
| change ((𝟙 (K.X p)) ≫ (complexQuotientNat I c).app (K.X p)).f q=_ |
| rw [Category.id_comp,complexQuotientNat_f] |
| end Lech.TensorIdeal |
|
|
| end |
|
|
| end OAI |
|
|
