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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 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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorMap
import Mathlib.Algebra.Module.Projective
namespace OAI
noncomputable section
namespace ArExplicit.BalancedTensor
universe u
variable {k R F G Q X : Type u} [Field k] [Ring R] [Algebra k R]
variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F] [IsScalarTower k Rᵐᵒᵖ F]
variable [AddCommGroup G] [Module k G] [Module Rᵐᵒᵖ G] [IsScalarTower k Rᵐᵒᵖ G]
variable [AddCommGroup Q] [Module k Q] [Module Rᵐᵒᵖ Q] [IsScalarTower k Rᵐᵒᵖ Q]
variable [AddCommGroup X] [Module k X] [Module R X] [IsScalarTower k R X]
def mapFirst (X : Type u) [AddCommGroup X] [Module k X] [Module R X]
(f : F →ₗ[Rᵐᵒᵖ] G) : BalancedTensor k R F X →ₗ[k] BalancedTensor k R G X :=
map (f.restrictScalars k) (LinearMap.id : X →ₗ[k] X)
(fun a x => f.map_smul a x) (fun _ _ => rfl)
omit [IsScalarTower k R X] in
@[simp] theorem mapFirst_tmul (f : F →ₗ[Rᵐᵒᵖ] G) (m : F) (x : X) :
mapFirst (k := k) X f (tmul m x) = tmul (f m) x := rfl
omit [IsScalarTower k R X] in
@[simp] theorem mapFirst_id : mapFirst (k := k) X (LinearMap.id : F →ₗ[Rᵐᵒᵖ] F) = LinearMap.id := by
apply LinearMap.ext
intro z
induction z using induction_on with
| zero => simp
| tmul f x => rfl
| add z w hz hw => simp_all
omit [IsScalarTower k R X] in
@[simp] theorem mapFirst_comp (f : F →ₗ[Rᵐᵒᵖ] G) (g : G →ₗ[Rᵐᵒᵖ] Q) :
mapFirst (k := k) X (g.comp f) = (mapFirst (k := k) X g).comp (mapFirst (k := k) X f) := by
apply LinearMap.ext
intro z
induction z using induction_on with
| zero => simp
| tmul f x => rfl
| add z w hz hw => simp_all
omit [IsScalarTower k R X] in
@[simp] theorem mapFirst_add (f g : F →ₗ[Rᵐᵒᵖ] G) :
mapFirst (k := k) X (f + g) = mapFirst (k := k) X f + mapFirst (k := k) X g := by
apply LinearMap.ext
intro z
induction z using induction_on with
| zero => simp
| tmul f x => simp [BalancedTensor.add_tmul]
| add z w hz hw => simp_all [add_left_comm, add_comm]
omit [IsScalarTower k R X] in
@[simp] theorem mapFirst_zero : mapFirst (k := k) X (0 : F →ₗ[Rᵐᵒᵖ] G) = 0 := by
apply LinearMap.ext
intro z
induction z using induction_on with
| zero => simp
| tmul f x => simp
| add z w hz hw => simp_all
theorem exists_splitting (f : F →ₗ[Rᵐᵒᵖ] G) (g : G →ₗ[Rᵐᵒᵖ] Q)
(hf : Function.Injective f) (hg : Function.Surjective g)
(he : LinearMap.range f = LinearMap.ker g) [Module.Projective Rᵐᵒᵖ Q] :
∃ c : G →ₗ[Rᵐᵒᵖ] F, ∃ s : Q →ₗ[Rᵐᵒᵖ] G,
c.comp f = LinearMap.id ∧ g.comp s = LinearMap.id ∧
f.comp c + s.comp g = LinearMap.id := by
obtain ⟨s, hs⟩ := Module.projective_lifting_property g (LinearMap.id : Q →ₗ[Rᵐᵒᵖ] Q) hg
let e : F ≃ₗ[Rᵐᵒᵖ] LinearMap.range f := LinearEquiv.ofBijective f.rangeRestrict
⟨fun x y h => hf (congrArg Subtype.val h), by
rintro ⟨y, x, rfl⟩
exact ⟨x, rfl⟩⟩
let r : G →ₗ[Rᵐᵒᵖ] LinearMap.range f :=
{ toFun := fun y => ⟨y - s (g y), by
rw [he]
change g (y - s (g y)) = 0
rw [map_sub, show g (s (g y)) = g y from DFunLike.congr_fun hs (g y), sub_self]⟩
map_add' a b := by
apply Subtype.ext
change a + b - s (g (a + b)) = (a - s (g a)) + (b - s (g b))
simp only [map_add, add_sub_add_comm]
map_smul' a b := by
apply Subtype.ext
change a • b - s (g (a • b)) = a • (b - s (g b))
simp only [map_smul, smul_sub] }
let c : G →ₗ[Rᵐᵒᵖ] F := e.symm.toLinearMap.comp r
have hc (y : G) : f (c y) = y - s (g y) := by
exact congrArg Subtype.val (e.apply_symm_apply (r y))
have hgf : g.comp f = 0 := LinearMap.range_le_ker_iff.mp he.le
refine ⟨c, s, ?_, hs, ?_⟩
· apply LinearMap.ext
intro x
apply hf
rw [LinearMap.comp_apply, hc]
have hz : g (f x) = 0 := DFunLike.congr_fun hgf x
simp [hz]
· apply LinearMap.ext
intro y
change f (c y) + s (g y) = y
rw [hc, sub_add_cancel]
omit [IsScalarTower k R X] in
theorem mapFirst_shortExact (f : F →ₗ[Rᵐᵒᵖ] G) (g : G →ₗ[Rᵐᵒᵖ] Q)
(hf : Function.Injective f) (hg : Function.Surjective g)
(he : LinearMap.range f = LinearMap.ker g) [Module.Projective Rᵐᵒᵖ Q] :
Function.Injective (mapFirst (k := k) X f) ∧ Function.Surjective (mapFirst (k := k) X g) ∧
LinearMap.range (mapFirst (k := k) X f) = LinearMap.ker (mapFirst (k := k) X g) := by
obtain ⟨c, s, hcf, hgs, hsum⟩ := exists_splitting f g hf hg he
have hcf' : (mapFirst (k := k) X c).comp (mapFirst (k := k) X f) = LinearMap.id := by
rw [← mapFirst_comp, hcf, mapFirst_id]
have hgs' : (mapFirst (k := k) X g).comp (mapFirst (k := k) X s) = LinearMap.id := by
rw [← mapFirst_comp, hgs, mapFirst_id]
have hsum' : (mapFirst (k := k) X f).comp (mapFirst (k := k) X c) +
(mapFirst (k := k) X s).comp (mapFirst (k := k) X g) = LinearMap.id := by
rw [← mapFirst_comp, ← mapFirst_comp, ← mapFirst_add, hsum, mapFirst_id]
have hgf : g.comp f = 0 := LinearMap.range_le_ker_iff.mp he.le
have hgf' : (mapFirst (k := k) X g).comp (mapFirst (k := k) X f) = 0 := by
rw [← mapFirst_comp, hgf, mapFirst_zero]
refine ⟨?_, ?_, le_antisymm (LinearMap.range_le_ker_iff.mpr hgf') ?_⟩
· intro x y hxy
have hx := DFunLike.congr_fun hcf' x
have hy := DFunLike.congr_fun hcf' y
change mapFirst (k := k) X c (mapFirst (k := k) X f x) = x at hx
change mapFirst (k := k) X c (mapFirst (k := k) X f y) = y at hy
rw [← hx, ← hy, hxy]
· intro z
exact ⟨mapFirst (k := k) X s z, DFunLike.congr_fun hgs' z⟩
· intro z hz
refine ⟨mapFirst (k := k) X c z, ?_⟩
have h := DFunLike.congr_fun hsum' z
change mapFirst (k := k) X f (mapFirst (k := k) X c z) + mapFirst (k := k) X s (mapFirst (k := k) X g z) = z at h
simpa only [show mapFirst (k := k) X g z = 0 from hz, map_zero, add_zero] using h
end ArExplicit.BalancedTensor
end
end OAI
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