| import Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear |
| import Mathlib.Algebra.Category.ModuleCat.Algebra |
| import Mathlib.Algebra.Category.ModuleCat.Abelian |
| import Mathlib.Algebra.Category.ModuleCat.Ext.HasExt |
| |
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open scoped ModuleCat.Algebra |
| namespace ArExplicit.ResolutionProducts |
| universe u |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] |
| variable {X : ModuleCat.{u} R} |
|
|
| theorem scalar_end_hom (M : ModuleCat.{u} R) (f : M ⟶ M) (c : k) |
| (hf : ∀ x : M, f.hom x = (algebraMap k R c) • x) : f = c • (𝟙 M) := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| exact hf x |
|
|
| theorem extZero_scalar_of_hom |
| (h : ∀ f : X ⟶ X, ∃ c : k, f = c • 𝟙 X) (x : Ext X X 0) : |
| ∃ c : k, x = c • Ext.mk₀ (𝟙 X) := by |
| obtain ⟨c,hc⟩ := h (Ext.homEquiv₀ x) |
| refine ⟨c, ?_⟩ |
| exact (Ext.mk₀_homEquiv₀_apply x).symm.trans |
| ((congrArg Ext.mk₀ hc).trans (Ext.mk₀_smul (R := k) c (𝟙 X))) |
|
|
| private theorem commute_step (a b : Ext X X 3) (n : ℕ) (x : Ext X X (3*n)) |
| (hab : a.comp b rfl = b.comp a rfl) |
| (hax : a.comp x (by omega : 3+3*n=3*(n+1)) = |
| x.comp a (by omega : 3*n+3=3*(n+1))) : |
| a.comp (b.comp x (by omega : 3+3*n=3*(n+1))) (by omega : 3+3*(n+1)=3*(n+2)) = |
| (b.comp x (by omega : 3+3*n=3*(n+1))).comp a |
| (by omega : 3*(n+1)+3=3*(n+2)) := by |
| rw [← Ext.comp_assoc a b x rfl (by omega : 3+3*n=3*(n+1)) (by omega)] |
| rw [hab] |
| rw [Ext.comp_assoc b a x rfl (by omega : 3+3*n=3*(n+1)) (by omega)] |
| rw [hax] |
| rw [← Ext.comp_assoc b x a (by omega : 3+3*n=3*(n+1)) |
| (by omega : 3*n+3=3*(n+1)) (by omega)] |
|
|
| theorem cubicCentral_of_generation (a b : Ext X X 3) |
| (hab : a.comp b rfl = b.comp a rfl) |
| (hzero : ∀ x : Ext X X 0, |
| a.comp x (by omega : 3+0=3) = x.comp a (by omega : 0+3=3) ∧ |
| b.comp x (by omega : 3+0=3) = x.comp b (by omega : 0+3=3)) |
| (hgen : ∀ (n : ℕ) (x : Ext X X (3*(n+1))), ∃ u v : Ext X X (3*n), |
| x = a.comp u (by omega) + b.comp v (by omega)) : |
| ∀ (n : ℕ) (x : Ext X X (3*n)), |
| a.comp x (by omega : 3+3*n=3*(n+1)) = x.comp a (by omega) ∧ |
| b.comp x (by omega : 3+3*n=3*(n+1)) = x.comp b (by omega) := by |
| intro n |
| induction n with |
| | zero => exact hzero |
| | succ n ih => |
| intro x |
| obtain ⟨u,v,rfl⟩ := hgen n x |
| constructor |
| · rw [Ext.comp_add, Ext.add_comp] |
| exact congrArg₂ (· + ·) (commute_step a a n u rfl (ih u).1) |
| (commute_step a b n v hab (ih v).1) |
| · rw [Ext.comp_add, Ext.add_comp] |
| exact congrArg₂ (· + ·) (commute_step b a n u hab.symm (ih u).2) |
| (commute_step b b n v rfl (ih v).2) |
|
|
| end ArExplicit.ResolutionProducts |
|
|
| end |
|
|
| end OAI |
|
|
