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216fb31 | 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 | import OAI.Algebra.Finitistic.Ordinary.OrdinaryTensorBimoduleResolution
import OAI.Algebra.Finitistic.Ordinary.OrdinaryBimoduleTensorTerms
import OAI.Algebra.Finitistic.Ordinary.OrdinaryTensorTotalAcyclic
namespace OAI
noncomputable section
open CategoryTheory Limits
open scoped TensorProduct
namespace LittleFinitistic.OrdinarySimulation.TensorResolution
universe u
variable (k B : Type u) [Field k] [Ring B] [Algebra k B] (l : ℕ)
/-- The actual genuine bimodule resolution is right-projective in every integer degree. -/
instance integerBimoduleRight_projective (n : ℤ) :
Module.Projective (B ⊗[k] ChainAlgebra k l)ᵐᵒᵖ
(ComplexTensor.rightTerm k (B ⊗[k] ChainAlgebra k l)
(BimoduleComplexTensor.rightComplex k B (B ⊗[k] ChainAlgebra k l) (integerBimoduleComplex k B l)) n) := by
let e := (HomologicalComplex.eval _ _ n).mapIso (forgetIntegerBimoduleComplex k B l)
let : Projective ((((forgetBimodule k B l).mapHomologicalComplex (.up ℤ)).obj
(integerBimoduleComplex k B l)).X n) := Projective.of_iso e.symm (integerTerm_projective k B l n)
exact ModuleCat.projective_of_module_projective
((((forgetBimodule k B l).mapHomologicalComplex (.up ℤ)).obj (integerBimoduleComplex k B l)).X n)
theorem integerBimodule_lower (n : ℤ) (hn : n < -(l:ℤ)) : IsZero ((integerBimoduleComplex k B l).X n) := by
apply (bimoduleComplex k B l).isZero_extend_X (degreeEmbedding l) n
intro j hj
simp only [degreeEmbedding_apply] at hj
omega
theorem integerBimodule_upper (n : ℤ) (hn : 0 < n) : IsZero ((integerBimoduleComplex k B l).X n) := by
have h := (integerTerm_upper k B l n hn).of_iso
((HomologicalComplex.eval _ _ n).mapIso (forgetIntegerBimoduleComplex k B l))
apply ModuleCat.isZero_iff_subsingleton.mpr
have hs := ModuleCat.isZero_iff_subsingleton.mp h
exact hs
end LittleFinitistic.OrdinarySimulation.TensorResolution
end
end OAI
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