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2c7d975 | 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import OAI.Algebra.AuslanderReiten.Complete.CompleteResolution
import Mathlib.Algebra.Module.Injective
namespace OAI
noncomputable section
namespace ArExplicit
universe u
variable {R M N : Type u} [Ring R] [AddCommGroup M] [Module R M]
variable [AddCommGroup N] [Module R N]
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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