File size: 3,005 Bytes
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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularComparison
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)
def comparisonSourceEquiv :
(CohomologyClass P P n × CohomologyClass P P n) ≃+
(CohomologyClass (eCochain (k := k) (F := F) P)
(eCochain (k := k) (F := F) P) n ×
CohomologyClass (jCochain (k := k) (F := F) P)
(jCochain (k := k) (F := F) P) n) :=
AddEquiv.prodCongr (fullHomCohomologyEquiv (eFunctor k R F) P P n)
(fullHomCohomologyEquiv (jFunctor k R F) P P n)
@[simp] theorem comparisonSourceEquiv_apply (a b : CohomologyClass P P n) :
comparisonSourceEquiv (F := F) (P := P) n (a,b) =
(fullHomCohomologyEquiv (eFunctor k R F) P P n a,
fullHomCohomologyEquiv (jFunctor k R F) P P n b) := rfl
theorem coneComparison_injective_of_tensor
(h : Function.Injective (tensorComparison v n)) :
Function.Injective (coneComparison (crossChainMap v) n) := by
intro x y hxy
obtain ⟨a,rfl⟩ := (comparisonSourceEquiv (F := F) (P := P) n).surjective x
obtain ⟨b,rfl⟩ := (comparisonSourceEquiv (F := F) (P := P) n).surjective y
apply congrArg
apply h
apply (crossCohomologyEquiv P P n).injective
rw [cross_tensorComparison, cross_tensorComparison]
exact hxy
theorem coneComparison_surjective_of_tensor
(h : Function.Surjective (tensorComparison v n)) :
Function.Surjective (coneComparison (crossChainMap v) n) := by
intro y
obtain ⟨x,hx⟩ := h ((crossCohomologyEquiv P P n).symm y)
refine ⟨comparisonSourceEquiv (F := F) (P := P) n x, ?_⟩
rcases x with ⟨a,b⟩
rw [comparisonSourceEquiv_apply, ← cross_tensorComparison, hx, AddEquiv.apply_symm_apply]
theorem coneComparison_nonzero_kernel_of_tensor
(h : ∃ x : CohomologyClass P P n × CohomologyClass P P n,
x ≠ 0 ∧ tensorComparison v n x = 0) :
∃ x : CohomologyClass (eCochain (k := k) (F := F) P)
(eCochain (k := k) (F := F) P) n ×
CohomologyClass (jCochain (k := k) (F := F) P)
(jCochain (k := k) (F := F) P) n,
x ≠ 0 ∧ coneComparison (crossChainMap v) n x = 0 := by
obtain ⟨x,hx,h0⟩ := h
refine ⟨comparisonSourceEquiv (F := F) (P := P) n x, ?_, ?_⟩
· exact fun he => hx ((comparisonSourceEquiv (F := F) (P := P) n).map_eq_zero_iff.mp he)
· rcases x with ⟨a,b⟩
rw [comparisonSourceEquiv_apply, ← cross_tensorComparison, h0, map_zero]
end ArExplicit.TriangularInduction
end
end OAI
|