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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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularCrossHom
import OAI.Algebra.AuslanderReiten.HomComplex.FullHomMap
import OAI.Algebra.AuslanderReiten.Cones.ConeComparison
namespace OAI
noncomputable section
namespace ArExplicit.TriangularInduction
open CategoryTheory CochainComplex.HomComplex
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
variable {P : CochainComplex (ModuleCat.{u} R) ℤ}
variable (v : P ⟶ tensorCochain (k := k) (F := F) P) (n : ℤ)
@[simp] theorem crossCohomologyEquiv_mk
(x : Cocycle P (tensorCochain (k := k) (F := F) P) n) :
crossCohomologyEquiv P P n (CohomologyClass.mk x) =
CohomologyClass.mk ((crossFullHomEquivalence P P).cocycle n x) := rfl
theorem crossCohomology_postcomp (x : CohomologyClass P P n) :
crossCohomologyEquiv P P n (fullHomPostcomp v n x) =
fullHomPostcomp (crossChainMap v) n (mapFullHom (jFunctor k R F) P P n x) := by
obtain ⟨x, rfl⟩ := CohomologyClass.mk_surjective x
rw [fullHomPostcomp_mk, crossCohomologyEquiv_mk, mapFullHom_mk, fullHomPostcomp_mk]
congr 1
apply Cocycle.ext
change crossCochainEquiv P P n (x.1.comp (Cochain.ofHom v) (add_zero n)) =
(x.1.map (jFunctor k R F)).comp (Cochain.ofHom (crossChainMap v)) (add_zero n)
apply Cochain.ext
intro i j hij
rw [crossCochainEquiv_v,
Cochain.comp_v x.1 (Cochain.ofHom v) (add_zero n) i j j hij (add_zero j),
Cochain.comp_v (x.1.map (jFunctor k R F)) (Cochain.ofHom (crossChainMap v))
(add_zero n) i j j hij (add_zero j)]
simp only [Cochain.ofHom_v, Cochain.map_v]
exact crossHomAdd_source _ _
theorem crossCohomology_precomp (x : CohomologyClass P P n) :
crossCohomologyEquiv P P n (fullHomPrecomp v n
(mapFullHom (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)) P P n x)) =
fullHomPrecomp (crossChainMap v) n (mapFullHom (eFunctor k R F) P P n x) := by
obtain ⟨x, rfl⟩ := CohomologyClass.mk_surjective x
rw [mapFullHom_mk, fullHomPrecomp_mk, crossCohomologyEquiv_mk,
mapFullHom_mk, fullHomPrecomp_mk]
congr 1
apply Cocycle.ext
change crossCochainEquiv P P n ((Cochain.ofHom v).comp
(x.1.map (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F))) (zero_add n)) =
(Cochain.ofHom (crossChainMap v)).comp (x.1.map (eFunctor k R F)) (zero_add n)
apply Cochain.ext
intro i j hij
rw [crossCochainEquiv_v,
Cochain.comp_v (Cochain.ofHom v)
(x.1.map (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)))
(zero_add n) i i j (add_zero i) hij,
Cochain.comp_v (Cochain.ofHom (crossChainMap v)) (x.1.map (eFunctor k R F))
(zero_add n) i i j (add_zero i) hij]
simp only [Cochain.ofHom_v, Cochain.map_v]
exact crossHomAdd_target _ _
def tensorComparison : (CohomologyClass P P n × CohomologyClass P P n) →+
CohomologyClass P (tensorCochain (k := k) (F := F) P) n where
toFun x := fullHomPostcomp v n x.2 + (n+1).negOnePow • fullHomPrecomp v n
(mapFullHom (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)) P P n x.1)
map_zero' := by simp
map_add' x y := by simp [smul_add]; abel
theorem cross_tensorComparison (a b : CohomologyClass P P n) :
crossCohomologyEquiv P P n (tensorComparison v n (a,b)) =
coneComparison (crossChainMap v) n
(fullHomCohomologyEquiv (eFunctor k R F) P P n a,
fullHomCohomologyEquiv (jFunctor k R F) P P n b) := by
change crossCohomologyEquiv P P n (fullHomPostcomp v n b +
(n+1).negOnePow • fullHomPrecomp v n
(mapFullHom (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)) P P n a)) = _
rw [map_add]
simp only [Units.smul_def, map_zsmul, crossCohomology_postcomp, crossCohomology_precomp]
rfl
end ArExplicit.TriangularInduction
end
end OAI
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