openAI-math-dataset / lean /OAI /Algebra /Finitistic /Ordinary /OrdinaryDerivedSquareComparison.lean
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryXBlockComparison | |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBlockDerived | |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryShiftDerivedTensor | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory Limits | |
| open scoped TensorProduct | |
| namespace LittleFinitistic.OrdinarySimulation.XResolution | |
| universe u | |
| variable {k B : Type u} [Field k] [Ring B] [Algebra k B] | |
| variable (P : CochainComplex (ModuleCat.{u} (Enveloping k B)) ℤ) (a : ℤ) (l : ℕ) | |
| variable (b : ℤ) (hb : a+(l:ℤ)=b) | |
| variable (hP : ∀ n : ℤ, n < a ∨ b < n → IsZero (P.X n)) | |
| variable [∀ n, Module.Projective Bᵐᵒᵖ ((underlyingRightComplex P).X n)] | |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} B)] | |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} (AlgebraD k l B))] | |
| /-- The actual native derived square comparison on every complex Z, with first-block support. | |
| Both tensor functors are the localization lifts of their verified bounded projective models. -/ | |
| def derivedSquareComparison (Z : CochainComplex (ModuleCat.{u} B) ℤ) : | |
| (derivedTensor P a l).obj ((derivedTensor P a l).obj | |
| ((BlockModule.derivedFirst B (B ⊗[k] ChainAlgebra k l)).obj (DerivedCategory.Q.obj Z))) ≅ | |
| (BlockModule.derivedFirst B (B ⊗[k] ChainAlgebra k l)).obj | |
| ((shiftedDerivedTensor P b l).obj (DerivedCategory.Q.obj Z)) := | |
| (derivedTensor P a l).mapIso ((derivedTensor P a l).mapIso | |
| ((BlockModule.derivedFirstComparison B (B ⊗[k] ChainAlgebra k l)).app Z)) ≪≫ | |
| (derivedTensor P a l).mapIso ((derivedTensorComparison P a l).app | |
| (BlockModule.firstComplex B (B ⊗[k] ChainAlgebra k l) Z)) ≪≫ | |
| (derivedTensorComparison P a l).app | |
| ((tensorFunctor P a l).obj (BlockModule.firstComplex B (B ⊗[k] ChainAlgebra k l) Z)) ≪≫ | |
| DerivedCategory.Q.mapIso (squareComparison P a l b hb hP Z) ≪≫ | |
| ((BlockModule.derivedFirstComparison B (B ⊗[k] ChainAlgebra k l)).app | |
| (BimoduleComplexTensor.total k B B (P⟦b⟧) Z | |
| (TensorResolution.finiteDegrees l) (TensorResolution.finiteDegrees_strictMono l))).symm ≪≫ | |
| (BlockModule.derivedFirst B (B ⊗[k] ChainAlgebra k l)).mapIso | |
| ((shiftedDerivedTensorComparison P b l).app Z).symm | |
| end LittleFinitistic.OrdinarySimulation.XResolution | |
| end | |
| end OAI | |