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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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularFunctor
import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorExact
namespace OAI
namespace ArExplicit.TriangularInduction
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]
variable {M N L : Type*}
variable [AddCommGroup M] [Module k M] [Module R M] [IsScalarTower k R M] [SMulCommClass k R M]
variable [AddCommGroup N] [Module k N] [Module R N] [IsScalarTower k R N] [SMulCommClass k R N]
variable [AddCommGroup L] [Module k L] [Module R L] [IsScalarTower k R L] [SMulCommClass k R L]
omit [IsScalarTower k R F] [SMulCommClass k Rᵐᵒᵖ F] [SMulCommClass k R M] in
theorem eMap_exact [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F]
(e : M →ₗ[R] N) (d : N →ₗ[R] L) (hex : Function.Exact e d) :
Function.Exact (eMap (k := k) (F := F) e) (eMap d) := by
have ht := BalancedTensor.mapRight_exact (k := k) (F := F) (S := R) e d hex
intro z
constructor
· intro hz
obtain ⟨x, hx⟩ := (hex z.top).mp (congrArg TriangularModule.top hz)
obtain ⟨y, hy⟩ := (ht z.bottom).mp (congrArg TriangularModule.bottom hz)
exact ⟨⟨x, y⟩, TriangularModule.ext hx hy⟩
· rintro ⟨w, rfl⟩
apply TriangularModule.ext
· exact (hex (e w.top)).mpr ⟨w.top, rfl⟩
· exact (ht (BalancedTensor.mapRight (S := R) e w.bottom)).mpr ⟨w.bottom, rfl⟩
omit [Algebra k R] [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F]
[SMulCommClass k Rᵐᵒᵖ F] [Module k M] [IsScalarTower k R M] [SMulCommClass k R M]
[Module k N] [IsScalarTower k R N] [SMulCommClass k R N]
[Module k L] [IsScalarTower k R L] [SMulCommClass k R L] in
theorem jMap_exact (e : M →ₗ[R] N) (d : N →ₗ[R] L) (hex : Function.Exact e d) :
Function.Exact (jMap (k := k) (F := F) e) (jMap d) := by
intro z
constructor
· intro hz
obtain ⟨x, hx⟩ := (hex z.bottom).mp (congrArg TriangularModule.bottom hz)
exact ⟨⟨0, x⟩, TriangularModule.ext (Subsingleton.elim _ _) hx⟩
· rintro ⟨w, rfl⟩
exact TriangularModule.ext (Subsingleton.elim _ _) ((hex (e w.bottom)).mpr ⟨w.bottom, rfl⟩)
end ArExplicit.TriangularInduction
end OAI
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