| import OAI.Algebra.AuslanderReiten.Complete.CompleteResolution |
| import Mathlib.Algebra.Module.Injective |
|
|
| namespace OAI |
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| noncomputable section |
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|
| namespace ArExplicit |
|
|
| universe u |
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|
| variable {R M N : Type u} [Ring R] [AddCommGroup M] [Module R M] |
| variable [AddCommGroup N] [Module R N] |
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|
| theorem finiteProjective_injective [Module.Injective R R] |
| [Module.Finite R M] [Module.Projective R M] : Module.Injective R M where |
| out := by |
| intro X Y _ _ _ _ f hf g |
| obtain ⟨n, p, hp⟩ := Module.Finite.exists_fin' R M |
| obtain ⟨s, hs⟩ := Module.projective_lifting_property p (LinearMap.id : M →ₗ[R] M) hp |
| obtain ⟨g', hg'⟩ := Module.Injective.extension_property R (Fin n → R) X Y f hf (s.comp g) |
| refine ⟨p.comp g', ?_⟩ |
| intro x |
| change p (g' (f x)) = g x |
| rw [show g' (f x) = s (g x) from DFunLike.congr_fun hg' x] |
| exact DFunLike.congr_fun hs (g x) |
|
|
| def splitKernelProjection (f : M →ₗ[R] N) (s : N →ₗ[R] M) (hs : f.comp s = LinearMap.id) : |
| M →ₗ[R] LinearMap.ker f := |
| (LinearMap.id - s.comp f).codRestrict _ (by |
| intro x |
| change f (x - s (f x)) = 0 |
| rw [map_sub, show f (s (f x)) = f x from DFunLike.congr_fun hs (f x)] |
| exact sub_self _) |
|
|
| @[simp] theorem splitKernelProjection_subtype (f : M →ₗ[R] N) (s : N →ₗ[R] M) |
| (hs : f.comp s = LinearMap.id) : |
| (splitKernelProjection f s hs).comp (LinearMap.ker f).subtype = LinearMap.id := by |
| apply LinearMap.ext |
| intro x |
| apply Subtype.ext |
| simp [splitKernelProjection, show f x.val = 0 from x.property] |
|
|
| theorem projective_kernel_of_surjective [Module.Projective R M] [Module.Projective R N] |
| (f : M →ₗ[R] N) (hf : Function.Surjective f) : Module.Projective R (LinearMap.ker f) := by |
| obtain ⟨s, hs⟩ := Module.projective_lifting_property f (LinearMap.id : N →ₗ[R] N) hf |
| exact Module.Projective.of_split (LinearMap.ker f).subtype (splitKernelProjection f s hs) |
| (splitKernelProjection_subtype f s hs) |
|
|
| theorem finite_kernel_of_surjective [Module.Finite R M] [Module.Projective R N] |
| (f : M →ₗ[R] N) (hf : Function.Surjective f) : Module.Finite R (LinearMap.ker f) := by |
| obtain ⟨s, hs⟩ := Module.projective_lifting_property f (LinearMap.id : N →ₗ[R] N) hf |
| apply Module.Finite.of_surjective (splitKernelProjection f s hs) |
| intro x |
| exact ⟨x.val, DFunLike.congr_fun (splitKernelProjection_subtype f s hs) x⟩ |
|
|
| def splitCokernelSection (i : M →ₗ[R] N) (r : N →ₗ[R] M) (hr : r.comp i = LinearMap.id) : |
| (N ⧸ LinearMap.range i) →ₗ[R] N := |
| (LinearMap.range i).liftQ (LinearMap.id - i.comp r) (by |
| rintro x ⟨m, rfl⟩ |
| change i m - i (r (i m)) = 0 |
| rw [show r (i m) = m from DFunLike.congr_fun hr m] |
| exact sub_self _) |
|
|
| @[simp] theorem splitCokernelSection_mkQ (i : M →ₗ[R] N) (r : N →ₗ[R] M) |
| (hr : r.comp i = LinearMap.id) : |
| (LinearMap.range i).mkQ.comp (splitCokernelSection i r hr) = LinearMap.id := by |
| apply LinearMap.ext |
| intro x |
| obtain ⟨x, rfl⟩ := Submodule.mkQ_surjective (LinearMap.range i) x |
| change (LinearMap.range i).mkQ (x - i (r x)) = (LinearMap.range i).mkQ x |
| rw [map_sub] |
| have hi : (LinearMap.range i).mkQ (i (r x)) = 0 := |
| DFunLike.congr_fun (LinearMap.range_mkQ_comp i) (r x) |
| rw [hi, sub_zero] |
|
|
| theorem projective_cokernel_of_injective [Module.Injective R M] [Module.Projective R N] |
| (i : M →ₗ[R] N) (hi : Function.Injective i) : Module.Projective R (N ⧸ LinearMap.range i) := by |
| obtain ⟨r, hr⟩ := Module.Injective.extension_property R M M N i hi (LinearMap.id : M →ₗ[R] M) |
| exact Module.Projective.of_split (splitCokernelSection i r hr) (LinearMap.range i).mkQ |
| (splitCokernelSection_mkQ i r hr) |
|
|
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
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|
