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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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularExact
import OAI.Algebra.AuslanderReiten.Triangular.TriangularDualExact
import OAI.Algebra.AuslanderReiten.BaseChange.ScalarExtensionComplete
import OAI.Algebra.AuslanderReiten.Complete.CompleteComparison
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
private instance moduleCatBaseComm (M : ModuleCat.{u} R) : SMulCommClass k R M where
smul_comm c r m := by
change (algebraMap k R c) • (r • m) = r • ((algebraMap k R c) • m)
rw [← mul_smul, Algebra.commutes c r, mul_smul]
def eComplex (P : ChainComplex (ModuleCat.{u} R) ℤ) :
ChainComplex (ModuleCat.{u} (Triangular R F)) ℤ where
X n := ModuleCat.of (Triangular R F) (TriangularE k R F (P.X n))
d i j := ModuleCat.ofHom (eMap (P.d i j).hom)
shape i j hij := by
rw [P.shape i j hij]
apply ModuleCat.hom_ext
exact eMap_zero
d_comp_d' i j l _ _ := by
apply ModuleCat.hom_ext
change (eMap (P.d j l).hom).comp (eMap (P.d i j).hom) = 0
rw [← eMap_comp, show (P.d j l).hom.comp (P.d i j).hom = 0 from
congrArg ModuleCat.Hom.hom (P.d_comp_d i j l), eMap_zero]
def jComplex (P : ChainComplex (ModuleCat.{u} R) ℤ) :
ChainComplex (ModuleCat.{u} (Triangular R F)) ℤ where
X n := ModuleCat.of (Triangular R F) (TriangularJ k R F (P.X n))
d i j := ModuleCat.ofHom (jMap (P.d i j).hom)
shape i j hij := by
rw [P.shape i j hij]
apply ModuleCat.hom_ext
exact jMap_zero
d_comp_d' i j l _ _ := by
apply ModuleCat.hom_ext
change (jMap (P.d j l).hom).comp (jMap (P.d i j).hom) = 0
rw [← jMap_comp, show (P.d j l).hom.comp (P.d i j).hom = 0 from
congrArg ModuleCat.Hom.hom (P.d_comp_d i j l), jMap_zero]
variable [Module.Finite k R] [Module.Finite k F]
theorem eComplex_totallyAcyclic [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F]
(P : ChainComplex (ModuleCat.{u} R) ℤ) (hP : TotallyAcyclicComplex P) :
TotallyAcyclicComplex (eComplex (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 :=
(LinearMap.exact_iff.mp (eMap_exact (k := k) (F := F) _ _
(LinearMap.exact_iff.mpr (hP.exactAt n).symm))).symm
dualExact n := eHomExact _ _ (hP.dualExact n)
omit [Module.Finite k F] in
theorem jComplex_totallyAcyclic [Module.Finite R F] [Module.Projective R F]
(P : ChainComplex (ModuleCat.{u} R) ℤ) (hP : TotallyAcyclicComplex P) :
TotallyAcyclicComplex (jComplex (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 :=
(LinearMap.exact_iff.mp (jMap_exact (k := k) (F := F) _ _
(LinearMap.exact_iff.mpr (hP.exactAt n).symm))).symm
dualExact n := jHomExact _ _ (hP.dualExact n)
end ArExplicit.TriangularInduction
end
end OAI
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