File size: 2,254 Bytes
b797a5b | 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 59 60 61 62 63 64 65 66 | 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
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