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d7f9a33 | 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 | import OAI.Algebra.AuslanderReiten.Ext.AExtCoefficientCore
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian
open scoped TensorProduct ModuleCat.Algebra
namespace ArExplicit.Resolution
universe u
variable {k : Type u} [Field k] [CharP k 2] (q : k)
variable [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q))]
variable (hq : q ≠ 0) (hq1 : 1+q ≠ 0) (h : NoPositivePowerOne q)
abbrev aCoefficientFunction (n : ℕ) := fun x : ASelfExt q (3*n) =>
tTensorCoefficientEquiv q hq hq1 h n (aExtKunnethEquiv q (3*n) x)
theorem aCoefficientFunction_injective (n : ℕ) :
Function.Injective (aCoefficientFunction q hq hq1 h n) :=
(tTensorCoefficientEquiv q hq hq1 h n).injective.comp
(aExtKunnethEquiv q (3*n)).injective
theorem aCoefficientFunction_surjective (n : ℕ) :
Function.Surjective (aCoefficientFunction q hq hq1 h n) :=
(tTensorCoefficientEquiv q hq hq1 h n).surjective.comp
(aExtKunnethEquiv q (3*n)).surjective
theorem aCoefficientFunction_add (n : ℕ) (x y : ASelfExt q (3*n)) :
type_of% ((congrArg (tTensorCoefficientEquiv q hq hq1 h n)
((aExtKunnethEquiv q (3*n)).map_add x y)).trans
((tTensorCoefficientEquiv q hq hq1 h n).map_add
(aExtKunnethEquiv q (3*n) x) (aExtKunnethEquiv q (3*n) y))) :=
(congrArg (tTensorCoefficientEquiv q hq hq1 h n)
((aExtKunnethEquiv q (3*n)).map_add x y)).trans
((tTensorCoefficientEquiv q hq hq1 h n).map_add
(aExtKunnethEquiv q (3*n) x) (aExtKunnethEquiv q (3*n) y))
end ArExplicit.Resolution
end
end OAI
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