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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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularCategory
import OAI.Algebra.AuslanderReiten.Complete.CompleteShiftDual
namespace OAI
noncomputable section
namespace ArExplicit.TriangularInduction
open CategoryTheory
universe u
variable {k R F : Type u} [Field 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]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
abbrev eCochain (P : CochainComplex (ModuleCat.{u} R) ℤ) :=
((eFunctor k R F).mapHomologicalComplex _).obj P
abbrev jCochain (P : CochainComplex (ModuleCat.{u} R) ℤ) :=
((jFunctor k R F).mapHomologicalComplex _).obj P
variable [Module.Finite k R] [Module.Finite k F]
theorem eCochain_totallyAcyclic [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F]
(P : CochainComplex (ModuleCat.{u} R) ℤ) (hP : TotallyAcyclicCochain P) :
TotallyAcyclicCochain (eCochain (k := k) (F := F) P) where
finite n := by
let := hP.finite n
let : Module.Finite k (P.X n) := Module.Finite.trans R (P.X n)
exact Module.Finite.of_restrictScalars_finite k (Triangular R F) (TriangularE k R F (P.X n))
projective n := by
let := hP.finite n
let := hP.projective n
exact eProjective
exactAt n := by
rw [HomologicalComplex.exactAt_iff' _ (n - 1) n (n + 1) (by simp) (by simp),
ShortComplex.moduleCat_exact_iff_range_eq_ker]
have h := hP.exactAt n
rw [HomologicalComplex.exactAt_iff' _ (n - 1) n (n + 1) (by simp) (by simp),
ShortComplex.moduleCat_exact_iff_range_eq_ker] at h
exact (LinearMap.exact_iff.mp (eMap_exact (k := k) (F := F) _ _
(LinearMap.exact_iff.mpr h.symm))).symm
dualExact n := eHomExact _ _ (hP.dualExact n)
omit [Module.Finite k F] in
theorem jCochain_totallyAcyclic [Module.Finite R F] [Module.Projective R F]
(P : CochainComplex (ModuleCat.{u} R) ℤ) (hP : TotallyAcyclicCochain P) :
TotallyAcyclicCochain (jCochain (k := k) (F := F) P) where
finite n := by
let := hP.finite n
let : Module.Finite k (P.X n) := Module.Finite.trans R (P.X n)
exact Module.Finite.of_restrictScalars_finite k (Triangular R F) (TriangularJ k R F (P.X n))
projective n := by
let := hP.finite n
let := hP.projective n
exact jProjective
exactAt n := by
rw [HomologicalComplex.exactAt_iff' _ (n - 1) n (n + 1) (by simp) (by simp),
ShortComplex.moduleCat_exact_iff_range_eq_ker]
have h := hP.exactAt n
rw [HomologicalComplex.exactAt_iff' _ (n - 1) n (n + 1) (by simp) (by simp),
ShortComplex.moduleCat_exact_iff_range_eq_ker] at h
exact (LinearMap.exact_iff.mp (jMap_exact (k := k) (F := F) _ _
(LinearMap.exact_iff.mpr h.symm))).symm
dualExact n := jHomExact _ _ (hP.dualExact n)
end ArExplicit.TriangularInduction
end
end OAI
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