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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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularModule
import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorFunctor
namespace OAI
namespace ArExplicit.TriangularModule
variable {k R F M N M' N' : Type*} [CommRing k] [Ring R]
variable [AddCommGroup F] [Module k F] [Module R F] [Module Rᵐᵒᵖ F]
variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F]
variable [AddCommGroup M] [Module k M] [Module R M]
variable [AddCommGroup N] [Module R N]
variable [AddCommGroup M'] [Module k M'] [Module R M']
variable [AddCommGroup N'] [Module R N']
variable {φ : BalancedTensor k R F M →ₗ[R] N}
variable {ψ : BalancedTensor k R F M' →ₗ[R] N'}
def hom (a : M →ₗ[R] M') (b : N →ₗ[R] N')
(h : ∀ f m, b (φ (BalancedTensor.tmul f m)) = ψ (BalancedTensor.tmul f (a m))) :
TriangularModule k R F M N φ →ₗ[Triangular R F] TriangularModule k R F M' N' ψ where
toFun x := ⟨a x.top, b x.bottom⟩
map_add' x y := by ext <;> simp
map_smul' t x := by
ext
· exact a.map_smul t.top x.top
· simp only [bottom_smul, map_add, h, map_smul, RingHom.id_apply]
@[simp] theorem hom_top (a : M →ₗ[R] M') (b : N →ₗ[R] N')
(h : ∀ f m, b (φ (BalancedTensor.tmul f m)) = ψ (BalancedTensor.tmul f (a m)))
(x : TriangularModule k R F M N φ) : (hom a b h x).top = a x.top := rfl
@[simp] theorem hom_bottom (a : M →ₗ[R] M') (b : N →ₗ[R] N')
(h : ∀ f m, b (φ (BalancedTensor.tmul f m)) = ψ (BalancedTensor.tmul f (a m)))
(x : TriangularModule k R F M N φ) : (hom a b h x).bottom = b x.bottom := rfl
variable (g : TriangularModule k R F M N φ →ₗ[Triangular R F] TriangularModule k R F M' N' ψ)
theorem top_of_map (x : TriangularModule k R F M N φ) :
(g ⟨x.top, 0⟩).top = (g x).top := by
have h := congrArg TriangularModule.top (g.map_smul (⟨1, 0, 0⟩ : Triangular R F) x)
have he : (⟨1, 0, 0⟩ : Triangular R F) • x = ⟨x.top, 0⟩ := by ext <;> simp
rw [he] at h
simpa using h
theorem bottom_of_map (x : TriangularModule k R F M N φ) :
(g ⟨0, x.bottom⟩).bottom = (g x).bottom := by
have h := congrArg TriangularModule.bottom (g.map_smul (⟨0, 1, 0⟩ : Triangular R F) x)
have he : (⟨0, 1, 0⟩ : Triangular R F) • x = ⟨0, x.bottom⟩ := by ext <;> simp
rw [he] at h
simpa using h
def topMap : M →ₗ[R] M' where
toFun m := (g ⟨m, 0⟩).top
map_add' m n := by
have h : (⟨m + n, 0⟩ : TriangularModule k R F M N φ) = ⟨m, 0⟩ + ⟨n, 0⟩ := by
ext <;> simp
rw [h]
rw [g.map_add]
rfl
map_smul' r m := by
have h := congrArg TriangularModule.top
(g.map_smul (⟨r, 0, 0⟩ : Triangular R F) ⟨m, 0⟩)
have he : (⟨r, 0, 0⟩ : Triangular R F) •
(⟨m, 0⟩ : TriangularModule k R F M N φ) = ⟨r • m, 0⟩ := by ext <;> simp
rw [he] at h
exact h
def bottomMap : N →ₗ[R] N' where
toFun n := (g ⟨0, n⟩).bottom
map_add' m n := by
have h : (⟨0, m + n⟩ : TriangularModule k R F M N φ) = ⟨0, m⟩ + ⟨0, n⟩ := by
ext <;> simp
rw [h]
rw [g.map_add]
rfl
map_smul' r n := by
have h := congrArg TriangularModule.bottom
(g.map_smul (⟨0, r, 0⟩ : Triangular R F) ⟨0, n⟩)
have he : (⟨0, r, 0⟩ : Triangular R F) •
(⟨0, n⟩ : TriangularModule k R F M N φ) = ⟨0, r • n⟩ := by ext <;> simp
rw [he] at h
simpa using h
@[simp] theorem topMap_apply (m : M) : topMap g m = (g ⟨m, 0⟩).top := rfl
@[simp] theorem bottomMap_apply (n : N) : bottomMap g n = (g ⟨0, n⟩).bottom := rfl
theorem compatibility (f : F) (m : M) :
bottomMap g (φ (BalancedTensor.tmul f m)) = ψ (BalancedTensor.tmul f (topMap g m)) := by
have h := congrArg TriangularModule.bottom
(g.map_smul (⟨0, 0, f⟩ : Triangular R F) ⟨m, 0⟩)
have he : (⟨0, 0, f⟩ : Triangular R F) •
(⟨m, 0⟩ : TriangularModule k R F M N φ) = ⟨0, φ (BalancedTensor.tmul f m)⟩ := by
ext <;> simp
rw [he] at h
simpa using h
theorem hom_topMap_bottomMap : hom (topMap g) (bottomMap g) (compatibility g) = g := by
apply LinearMap.ext
intro x
ext
· exact top_of_map g x
· exact bottom_of_map g x
@[ext] theorem hom_ext
(g' : TriangularModule k R F M N φ →ₗ[Triangular R F] TriangularModule k R F M' N' ψ)
(ht : topMap g = topMap g') (hb : bottomMap g = bottomMap g') : g = g' := by
apply LinearMap.ext
intro x
ext
· rw [← top_of_map g x, ← top_of_map g' x]
exact congrArg (fun a : M →ₗ[R] M' => a x.top) ht
· rw [← bottom_of_map g x, ← bottom_of_map g' x]
exact congrArg (fun b : N →ₗ[R] N' => b x.bottom) hb
end ArExplicit.TriangularModule
end OAI
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