| import OAI.Algebra.AuslanderReiten.Ext.AExtPolynomial |
| import OAI.Algebra.AuslanderReiten.Stable.TensorStableEnd |
| import OAI.Algebra.AuslanderReiten.Ext.ExtCentralGeneration |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open scoped ModuleCat.Algebra |
| namespace ArExplicit.Resolution |
| universe u |
| variable {k : Type u} [Field k] [CharP k 2] (q : k) (hq1 : 1+q ≠ 0) |
| include hq1 |
|
|
| omit [CharP k 2] in |
| theorem aExtZero_scalar (x : ASelfExt q 0) : |
| ∃ c : k, x = c • Ext.mk₀ (𝟙 (ArExplicit.A.Simple q)) := by |
| apply ResolutionProducts.extZero_scalar_of_hom (k := k) _ x |
| intro f |
| refine ⟨ArExplicit.A.endScalar q f.hom, ?_⟩ |
| apply ResolutionProducts.scalar_end_hom (k := k) (R := ArExplicit.A k q) |
| (ArExplicit.A.Simple q) f |
| intro x |
| have hf := DFunLike.congr_fun (ArExplicit.A.end_eq_scalar_id q hq1 f.hom) x |
| change f.hom x = ArExplicit.A.endScalar q f.hom • x at hf |
| exact hf.trans (IsScalarTower.algebraMap_smul (ArExplicit.A k q) _ _).symm |
|
|
| omit [CharP k 2] in |
| theorem aExtZero_central (n : ℕ) (a : ASelfExt q n) (x : ASelfExt q 0) : |
| a.comp x (by omega : n+0=n) = x.comp a (by omega : 0+n=n) := by |
| obtain ⟨c,rfl⟩ := aExtZero_scalar q hq1 x |
| rw [Ext.comp_smul, Ext.smul_comp, Ext.comp_mk₀_id, Ext.mk₀_id_comp] |
|
|
| end ArExplicit.Resolution |
|
|
| end |
|
|
| end OAI |
|
|
