| import Mathlib.Algebra.Category.ModuleCat.Ext.HasExt |
| import Mathlib.LinearAlgebra.FreeModule.Finite.Basic |
| |
| namespace OAI |
|
|
| noncomputable section |
| universe w v u |
|
|
| namespace PiExponentSiegelAux.W31 |
| open Module |
| open CategoryTheory CategoryTheory.Abelian |
| open scoped BigOperators |
| |
| section General |
| variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{w} C] |
|
|
| theorem ext_subsingleton_of_finite_identity_factorization |
| {ι : Type*} [Fintype ι] (X F G : C) (n : ℕ) |
| (p : ι → (F ⟶ G)) (j : ι → (G ⟶ F)) |
| (htotal : (∑ i, p i ≫ j i) = 𝟙 F) |
| (h : Subsingleton (Ext X G n)) : Subsingleton (Ext X F n) := by |
| apply subsingleton_of_forall_eq 0 |
| intro e |
| have he : (∑ i, (e.comp (Ext.mk₀ (p i)) (add_zero n)).comp |
| (Ext.mk₀ (j i)) (add_zero n)) = e := by |
| simp only [Ext.comp_assoc_of_second_deg_zero, Ext.mk₀_comp_mk₀, |
| ← Ext.comp_sum, ← Ext.mk₀_sum, htotal, Ext.comp_mk₀_id] |
| calc |
| e = _ := he.symm |
| _ = 0 := by |
| apply Finset.sum_eq_zero |
| intro i hi |
| rw [h.elim (e.comp (Ext.mk₀ (p i)) (add_zero n)) 0, Ext.zero_comp] |
| |
| end General |
|
|
| variable {R : Type u} [CommRing R] |
| variable {F : Type u} [AddCommGroup F] [Module R F] |
|
|
| theorem ext_subsingleton_of_finite_basis {ι : Type*} [Fintype ι] |
| (b : Basis ι R F) (X : ModuleCat.{u} R) (n : ℕ) |
| (h : Subsingleton (Ext X (ModuleCat.of R R) n)) : |
| Subsingleton (Ext X (ModuleCat.of R F) n) := by |
| apply ext_subsingleton_of_finite_identity_factorization X (ModuleCat.of R F) |
| (ModuleCat.of R R) n |
| (fun i => ModuleCat.ofHom (b.coord i)) |
| (fun i => ModuleCat.ofHom (LinearMap.toSpanSingleton R F (b i))) ?_ h |
| apply ModuleCat.hom_ext |
| ext x |
| simpa only [ModuleCat.hom_sum, ModuleCat.hom_comp, ModuleCat.hom_ofHom, |
| ModuleCat.hom_id, LinearMap.sum_apply, LinearMap.comp_apply, |
| LinearMap.toSpanSingleton_apply, Basis.coord_apply, LinearMap.id_apply] using |
| b.sum_repr x |
|
|
| theorem ext_subsingleton_of_finite_free [Module.Free R F] [Module.Finite R F] |
| (X : ModuleCat.{u} R) (n : ℕ) |
| (h : Subsingleton (Ext X (ModuleCat.of R R) n)) : |
| Subsingleton (Ext X (ModuleCat.of R F) n) := |
| ext_subsingleton_of_finite_basis (Module.Free.chooseBasis R F) X n h |
|
|
| end PiExponentSiegelAux.W31 |
|
|
| end |
|
|
| end OAI |
|
|
