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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 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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 | import Mathlib.Algebra.Module.Projective
import Mathlib.RingTheory.Finiteness.Cardinality
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Quotient.Basic
namespace OAI
namespace ArExplicit
universe u v w x y z
variable {R : Type u} [Ring R]
variable {K : Type v} [AddCommGroup K] [Module R K]
variable {L : Type w} [AddCommGroup L] [Module R L]
variable {M : Type x} [AddCommGroup M] [Module R M]
variable {Q : Type y} [AddCommGroup Q] [Module R Q]
def HomExactAt (e : K →ₗ[R] L) (d : L →ₗ[R] M) : Prop :=
∀ f : L →ₗ[R] R, f.comp e = 0 → ∃ g : M →ₗ[R] R, g.comp d = f
namespace HomExactAt
variable {e : K →ₗ[R] L} {d : L →ₗ[R] M}
theorem projective [Module.Finite R Q] [Module.Projective R Q]
(h : HomExactAt e d) (f : L →ₗ[R] Q) (hf : f.comp e = 0) :
∃ g : M →ₗ[R] Q, g.comp d = f := by
obtain ⟨n, π, hπ⟩ := Module.Finite.exists_fin' R Q
obtain ⟨s, hs⟩ := Module.projective_lifting_property π (LinearMap.id : Q →ₗ[R] Q) hπ
have coordinate_exists (i : Fin n) :
∃ g : M →ₗ[R] R, g.comp d = (LinearMap.proj i).comp (s.comp f) := by
apply h
ext z
change (s (f (e z))) i = 0
rw [show f (e z) = 0 from DFunLike.congr_fun hf z, map_zero]
rfl
choose g hg using coordinate_exists
let G : M →ₗ[R] (Fin n → R) := LinearMap.pi g
refine ⟨π.comp G, ?_⟩
ext z
have hz : G (d z) = s (f z) := by
funext i
exact DFunLike.congr_fun (hg i) z
change π (G (d z)) = f z
rw [hz]
exact DFunLike.congr_fun hs (f z)
theorem extend_range [Module.Finite R Q] [Module.Projective R Q]
(h : HomExactAt e d) (hde : d.comp e = 0)
(f : LinearMap.range d →ₗ[R] Q) :
∃ g : M →ₗ[R] Q, g.comp (LinearMap.range d).subtype = f := by
have hf : (f.comp d.rangeRestrict).comp e = 0 := by
ext z
have hz : d.rangeRestrict (e z) = 0 := by
apply Subtype.ext
exact DFunLike.congr_fun hde z
change f (d.rangeRestrict (e z)) = 0
rw [hz, map_zero]
obtain ⟨g, hg⟩ := h.projective (f.comp d.rangeRestrict) hf
refine ⟨g, ?_⟩
apply LinearMap.ext
intro z
obtain ⟨z, rfl⟩ := d.surjective_rangeRestrict z
exact DFunLike.congr_fun hg z
theorem extend_kernel {N : Type z} [AddCommGroup N] [Module R N]
[Module.Finite R Q] [Module.Projective R Q]
(h : HomExactAt e d) (hde : d.comp e = 0)
(d' : M →ₗ[R] N) (hex : LinearMap.range d = LinearMap.ker d')
(f : LinearMap.ker d' →ₗ[R] Q) :
∃ g : M →ₗ[R] Q, g.comp (LinearMap.ker d').subtype = f := by
obtain ⟨g, hg⟩ := h.extend_range hde (f.comp (Submodule.inclusion hex.le))
refine ⟨g, ?_⟩
apply LinearMap.ext
intro t
have ht : (t : M) ∈ LinearMap.range d := by
rw [hex]
exact t.property
exact DFunLike.congr_fun hg ⟨t, ht⟩
end HomExactAt
section ProjectiveLifting
theorem lift_through_exact [Module.Projective R Q]
(e : K →ₗ[R] L) (d : L →ₗ[R] M)
(hex : LinearMap.range e = LinearMap.ker d)
(f : Q →ₗ[R] L) (hf : d.comp f = 0) :
∃ g : Q →ₗ[R] K, e.comp g = f := by
let f' : Q →ₗ[R] LinearMap.range e := f.codRestrict _ (by
intro q
rw [hex]
exact DFunLike.congr_fun hf q)
obtain ⟨g, hg⟩ := Module.projective_lifting_property e.rangeRestrict f'
e.surjective_rangeRestrict
refine ⟨g, ?_⟩
ext q
exact congrArg Subtype.val (DFunLike.congr_fun hg q)
theorem lift_cokernel_map [Module.Projective R K] [Module.Projective R L]
(e : K →ₗ[R] L) (d : M →ₗ[R] Q)
(u : (L ⧸ LinearMap.range e) →ₗ[R] (Q ⧸ LinearMap.range d)) :
∃ (f₀ : L →ₗ[R] Q) (f₁ : K →ₗ[R] M),
(LinearMap.range d).mkQ.comp f₀ = u.comp (LinearMap.range e).mkQ ∧
d.comp f₁ = f₀.comp e := by
obtain ⟨f₀, hf₀⟩ := Module.projective_lifting_property (LinearMap.range d).mkQ
(u.comp (LinearMap.range e).mkQ) (Submodule.mkQ_surjective _)
have hz : (LinearMap.range d).mkQ.comp (f₀.comp e) = 0 := by
ext q
change (LinearMap.range d).mkQ (f₀ (e q)) = 0
calc
(LinearMap.range d).mkQ (f₀ (e q)) =
u ((LinearMap.range e).mkQ (e q)) := DFunLike.congr_fun hf₀ (e q)
_ = 0 := by
rw [show (LinearMap.range e).mkQ (e q) = 0 by
exact (Submodule.Quotient.mk_eq_zero _).mpr ⟨q, rfl⟩, map_zero]
obtain ⟨f₁, hf₁⟩ := lift_through_exact d (LinearMap.range d).mkQ
(Submodule.ker_mkQ _).symm (f₀.comp e) hz
exact ⟨f₀, f₁, hf₀, hf₁⟩
end ProjectiveLifting
section ComparisonSteps
universe a b c
variable {U : Type a} [AddCommGroup U] [Module R U]
variable {V : Type b} [AddCommGroup V] [Module R V]
variable {W : Type c} [AddCommGroup W] [Module R W]
theorem extend_comparison_up [Module.Projective R K]
(e : K →ₗ[R] L) (d : L →ₗ[R] M) (hde : d.comp e = 0)
(e' : U →ₗ[R] V) (d' : V →ₗ[R] W)
(hex : LinearMap.range e' = LinearMap.ker d')
(f : L →ₗ[R] V) (g : M →ₗ[R] W) (hfg : d'.comp f = g.comp d) :
∃ h : K →ₗ[R] U, e'.comp h = f.comp e := by
apply lift_through_exact e' d' hex (f.comp e)
ext t
change d' (f (e t)) = 0
calc
d' (f (e t)) = g (d (e t)) := DFunLike.congr_fun hfg (e t)
_ = 0 := by rw [show d (e t) = 0 from DFunLike.congr_fun hde t, map_zero]
theorem extend_comparison_down [Module.Finite R W] [Module.Projective R W]
(e : K →ₗ[R] L) (d : L →ₗ[R] M) (hdual : HomExactAt e d)
(e' : U →ₗ[R] V) (d' : V →ₗ[R] W) (hde' : d'.comp e' = 0)
(f : K →ₗ[R] U) (g : L →ₗ[R] V) (hfg : e'.comp f = g.comp e) :
∃ h : M →ₗ[R] W, h.comp d = d'.comp g := by
apply hdual.projective (d'.comp g)
ext t
change d' (g (e t)) = 0
calc
d' (g (e t)) = d' (e' (f t)) := congrArg d' (DFunLike.congr_fun hfg t).symm
_ = 0 := DFunLike.congr_fun hde' (f t)
end ComparisonSteps
end ArExplicit
end OAI
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