| import OAI.Algebra.AuslanderReiten.Ext.NatCubicExt |
| import OAI.Algebra.AuslanderReiten.Ext.ExtIsoCongr |
| import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorScalarProducts |
| import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorScalarIso |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| namespace ArExplicit |
| universe u v |
| section General |
| variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{v} C] |
| variable {X X' Y Y' : C} (P : ProjectiveResolution X) (Q : ProjectiveResolution X') |
|
|
| theorem resolutionExt_square (a : X ⟶ X') (b : Y ⟶ Y') |
| (φ : ProjectiveResolution.Hom P Q a) (n : ℕ) |
| (f : P.complex.X n ⟶ Y) (g : Q.complex.X n ⟶ Y') |
| (hf : P.complex.d (n+1) n ≫ f = 0) (hg : Q.complex.d (n+1) n ≫ g = 0) |
| (h : φ.hom.f n ≫ g = f ≫ b) : |
| (Ext.mk₀ a).comp (Q.extMk g (n+1) rfl hg) (zero_add _) = |
| (P.extMk f (n+1) rfl hf).comp (Ext.mk₀ b) (add_zero _) := by |
| rw [ProjectiveResolution.mk₀_comp_extMk g (n+1) rfl hg φ, |
| ProjectiveResolution.extMk_comp_mk₀] |
| congr 1 |
|
|
| theorem selfExtIso_eq_of_square {k : Type v} [CommRing k] [Linear k C] |
| (e : X ≅ X') {n : ℕ} (x : Ext X X n) (y : Ext X' X' n) |
| (h : (Ext.mk₀ e.hom).comp y (zero_add _) = x.comp (Ext.mk₀ e.hom) (add_zero _)) : |
| selfExtIso (k := k) e n x = y := by |
| rw [selfExtIso_apply, Ext.comp_assoc_of_third_deg_zero, ← h, |
| Ext.mk₀_comp_mk₀_assoc, e.inv_hom_id, Ext.mk₀_id_comp] |
| end General |
|
|
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
|
|
