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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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularHom
import Mathlib.Algebra.Module.PUnit
namespace OAI
namespace ArExplicit
variable (k R F : Type*) [CommRing k] [Ring R] [Algebra k R]
variable [AddCommGroup F] [Module k F] [Module R F] [Module Rᵐᵒᵖ F]
variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F]
variable (M : Type*) [AddCommGroup M] [Module k M] [Module R M]
abbrev TriangularE := TriangularModule k R F M (BalancedTensor k R F M) LinearMap.id
variable (N : Type*) [AddCommGroup N] [Module R N]
abbrev TriangularJ := TriangularModule k R F PUnit.{1} N 0
namespace TriangularInduction
variable {k R F M N}
variable {M' N' : Type*} [AddCommGroup M'] [Module k M'] [Module R M']
variable [AddCommGroup N'] [Module R N']
variable {ψ : BalancedTensor k R F M' →ₗ[R] N'}
variable [IsScalarTower k R M] [IsScalarTower k R M']
def eHom (a : M →ₗ[R] M') :
TriangularE k R F M →ₗ[Triangular R F] TriangularModule k R F M' N' ψ :=
TriangularModule.hom a (ψ.comp (BalancedTensor.mapRight a)) (by intros; rfl)
def jHom (b : N →ₗ[R] N') :
TriangularJ k R F N →ₗ[Triangular R F] TriangularModule k R F M' N' ψ :=
TriangularModule.hom 0 b (by intros; simp)
theorem eHom_topMap
(g : TriangularE k R F M →ₗ[Triangular R F] TriangularModule k R F M' N' ψ) :
eHom (TriangularModule.topMap g) = g := by
apply TriangularModule.hom_ext
· ext m; rfl
· apply BalancedTensor.leftHom_ext
intro f m
exact (TriangularModule.compatibility g f m).symm
omit [Algebra k R] [IsScalarTower k R M'] in
theorem jHom_bottomMap
(g : TriangularJ k R F N →ₗ[Triangular R F] TriangularModule k R F M' N' ψ) :
jHom (TriangularModule.bottomMap g) = g := by
apply TriangularModule.hom_ext
· apply Subsingleton.elim
· ext n; rfl
variable [Module k N] [Module k N']
variable [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F]
variable [SMulCommClass k Rᵐᵒᵖ F]
variable [IsScalarTower k R N] [IsScalarTower k R N']
variable [SMulCommClass k R M'] [SMulCommClass k R N']
def eHomEquiv :
(TriangularE k R F M →ₗ[Triangular R F] TriangularModule k R F M' N' ψ) ≃ₗ[k]
(M →ₗ[R] M') where
toFun := TriangularModule.topMap
invFun := eHom
left_inv := eHom_topMap
right_inv a := by ext m; rfl
map_add' g h := by ext m; rfl
map_smul' c g := by ext m; rfl
def jHomEquiv :
(TriangularJ k R F N →ₗ[Triangular R F] TriangularModule k R F M' N' ψ) ≃ₗ[k]
(N →ₗ[R] N') where
toFun := TriangularModule.bottomMap
invFun := jHom
left_inv := jHom_bottomMap
right_inv a := by ext n; rfl
map_add' g h := by ext n; rfl
map_smul' c g := by ext n; rfl
theorem hom_E_J_eq_zero (g : TriangularE k R F M →ₗ[Triangular R F] TriangularJ k R F N) :
g = 0 := by
apply eHomEquiv.injective
exact Subsingleton.elim _ _
end TriangularInduction
end ArExplicit
end OAI
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