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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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularFunctor
import Mathlib.Algebra.Module.Projective
import Mathlib.RingTheory.Finiteness.Cardinality
namespace OAI
noncomputable section
namespace ArExplicit.TriangularInduction
private instance finitePiProjective {S P ι : Type*} [Ring S] [AddCommGroup P]
[Module S P] [Module.Projective S P] [Fintype ι] : Module.Projective S (ι → P) :=
Module.Projective.of_equiv (DFinsupp.linearEquivFunOnFintype :
(Π₀ _ : ι, P) ≃ₗ[S] (ι → P))
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 [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F]
def eRegularInclusion : TriangularE k R F R →ₗ[Triangular R F] Triangular R F where
toFun x := ⟨x.top, 0, BalancedTensor.runitLeft (S := R) x.bottom⟩
map_add' x y := by ext <;> simp
map_smul' a x := by
ext
· rfl
· simp
· change BalancedTensor.runitLeft (S := R)
(BalancedTensor.tmul a.off x.top + a.bottom • x.bottom) =
a.bottom • BalancedTensor.runitLeft (S := R) x.bottom + MulOpposite.op x.top • a.off
rw [map_add, map_smul]
exact add_comm _ _
def eRegularProjection : Triangular R F →ₗ[Triangular R F] TriangularE k R F R where
toFun a := ⟨a.top, (BalancedTensor.runitLeft (k := k) (R := R) (S := R)).symm a.off⟩
map_add' a b := by ext <;> simp
map_smul' a b := by
ext
· rfl
· change (BalancedTensor.runitLeft (S := R)).symm
(a.bottom • b.off + MulOpposite.op b.top • a.off) =
BalancedTensor.tmul a.off b.top +
a.bottom • (BalancedTensor.runitLeft (S := R)).symm b.off
apply (BalancedTensor.runitLeft (S := R)).injective
rw [LinearEquiv.apply_symm_apply, map_add, map_smul, LinearEquiv.apply_symm_apply]
exact add_comm _ _
omit [IsScalarTower k R F] [SMulCommClass k Rᵐᵒᵖ F] in
theorem eRegular_split :
(eRegularProjection (k := k) (R := R) (F := F)).comp eRegularInclusion = LinearMap.id := by
apply LinearMap.ext
intro x
ext
· rfl
· exact (BalancedTensor.runitLeft (S := R)).symm_apply_apply x.bottom
instance eRegular_projective : Module.Projective (Triangular R F) (TriangularE k R F R) :=
Module.Projective.of_split eRegularInclusion eRegularProjection eRegular_split
def jRegularInclusion : TriangularJ k R F R →ₗ[Triangular R F] Triangular R F where
toFun x := ⟨0, x.bottom, 0⟩
map_add' x y := by ext <;> simp
map_smul' a x := by ext <;> simp
def jRegularProjection : Triangular R F →ₗ[Triangular R F] TriangularJ k R F R where
toFun a := ⟨0, a.bottom⟩
map_add' a b := by ext; simp
map_smul' a b := by ext; simp
omit [Algebra k R] [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F]
[SMulCommClass k Rᵐᵒᵖ F] in
theorem jRegular_split :
(jRegularProjection (k := k) (R := R) (F := F)).comp jRegularInclusion = LinearMap.id := by
apply LinearMap.ext
intro x
ext
exact rfl
instance jRegular_projective : Module.Projective (Triangular R F) (TriangularJ k R F R) :=
Module.Projective.of_split jRegularInclusion jRegularProjection jRegular_split
variable {M : Type*} [AddCommGroup M] [Module k M] [Module R M]
variable [IsScalarTower k R M] [SMulCommClass k R M]
omit [SMulCommClass k R M] in
theorem eProjective [Module.Finite R M] [Module.Projective R M] :
Module.Projective (Triangular R F) (TriangularE k R F M) := by
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
let : Module.Projective (Triangular R F) (TriangularE k R F (Fin n → R)) :=
Module.Projective.of_equiv (ePi (k := k) (R := R) (F := F) (M := R) (Fin n)).symm
apply Module.Projective.of_split (eMap s) (eMap p)
rw [← eMap_comp, hs, eMap_id]
omit [SMulCommClass k R M] in
theorem jProjective [Module.Finite R M] [Module.Projective R M] :
Module.Projective (Triangular R F) (TriangularJ k R F M) := by
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
let : Module.Projective (Triangular R F) (TriangularJ k R F (Fin n → R)) :=
Module.Projective.of_equiv (jPi (k := k) (R := R) (F := F) (M := R) (Fin n)).symm
apply Module.Projective.of_split (jMap s) (jMap p)
rw [← jMap_comp, hs, jMap_id]
end ArExplicit.TriangularInduction
end
end OAI
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