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d7f9a33 | 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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | import OAI.Algebra.AuslanderReiten.Ext.ExtIsoCongr
import Mathlib.Algebra.Homology.DerivedCategory.Ext.Map
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits
namespace ArExplicit
universe u v w
variable {k : Type u} [CommRing k]
variable {C : Type v} [Category.{u} C] [Abelian C] [HasExt.{u} C] [Linear k C]
variable {D : Type w} [Category.{u} D] [Abelian D] [HasExt.{u} D] [Linear k D]
variable {X Y Z : C}
theorem selfExtIso_trans (e : X ≅ Y) (f : Y ≅ Z) (n : ℕ) (x : Ext X X n) :
selfExtIso (k := k) f n (selfExtIso (k := k) e n x) =
selfExtIso (k := k) (e ≪≫ f) n x := by
simp [selfExtIso_apply]
theorem selfExtIso_symm_apply (e : X ≅ Y) (n : ℕ) (x : Ext Y Y n) :
(selfExtIso (k := k) e n).symm x = selfExtIso (k := k) e.symm n x := by
apply (selfExtIso (k := k) e n).injective
simp [selfExtIso_apply]
theorem selfExtIso_eqToIso (h : X = Y) (n : ℕ) (x : Ext X X n) :
selfExtIso (k := k) (eqToIso h) n x = h ▸ x := by
subst h
simp
variable (F : C ⥤ D) [F.Additive] [PreservesFiniteLimits F] [PreservesFiniteColimits F]
theorem selfExtIso_mapExactFunctor (e : X ≅ Y) (n : ℕ) (x : Ext X X n) :
(selfExtIso (k := k) e n x).mapExactFunctor F =
selfExtIso (k := k) (F.mapIso e) n (x.mapExactFunctor F) := by
simp only [selfExtIso_apply, Ext.mapExactFunctor_comp, Ext.mapExactFunctor_mk₀,
Functor.mapIso_hom, Functor.mapIso_inv]
theorem selfExtIso_conjugated_functor
(G : C ⥤ C) [G.Additive] [PreservesFiniteLimits G] [PreservesFiniteColimits G]
(e : Y ≅ X) (E : Y ≅ G.obj Y) (n : ℕ) (x : Ext X X n) :
selfExtIso (k := k) e n
(selfExtIso (k := k) E.symm n
((selfExtIso (k := k) e.symm n x).mapExactFunctor G)) =
selfExtIso (k := k) (e.symm ≪≫ E ≪≫ G.mapIso e).symm n
(x.mapExactFunctor G) := by
rw [selfExtIso_mapExactFunctor, selfExtIso_trans, selfExtIso_trans]
have heq : G.mapIso e.symm ≪≫ E.symm ≪≫ e =
(e.symm ≪≫ E ≪≫ G.mapIso e).symm := by
apply Iso.ext
simp
rw [heq]
end ArExplicit
end
end OAI
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