openAI-math-dataset / lean /OAI /Algebra /Finitistic /Ordinary /OrdinaryBimoduleResolutionProjective.lean
| 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 | |