File size: 1,503 Bytes
e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | 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
|