| import OAI.Algebra.Finitistic.Ordinary.OrdinarySourceDerivedComparison | |
| import Mathlib.Logic.Function.Iterate | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| 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 → Limits.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 ordinary square on every native derived object, using essential surjectivity of Q. -/ | |
| def derivedSquareObject (Y : DerivedCategory (ModuleCat.{u} B)) : | |
| (derivedTensor P a l).obj ((derivedTensor P a l).obj | |
| ((BlockModule.derivedFirst B (B ⊗[k] ChainAlgebra k l)).obj Y)) ≅ | |
| (BlockModule.derivedFirst B (B ⊗[k] ChainAlgebra k l)).obj | |
| ((shiftedDerivedTensor P b l).obj Y) := | |
| let e := DerivedCategory.Q.objObjPreimageIso Y | |
| (derivedTensor P a l).mapIso ((derivedTensor P a l).mapIso | |
| ((BlockModule.derivedFirst B (B ⊗[k] ChainAlgebra k l)).mapIso e.symm)) ≪≫ | |
| derivedSquareComparison P a l b hb hP (DerivedCategory.Q.objPreimage Y) ≪≫ | |
| (BlockModule.derivedFirst B (B ⊗[k] ChainAlgebra k l)).mapIso | |
| ((shiftedDerivedTensor P b l).mapIso e) | |
| /-- Every even iterate is the first-block inflation of the corresponding actual P[b] tensor iterate. -/ | |
| def derivedEvenIterate (n : ℕ) (Y : DerivedCategory (ModuleCat.{u} B)) : | |
| ((derivedTensor P a l).obj)^[2*n] | |
| ((BlockModule.derivedFirst B (B ⊗[k] ChainAlgebra k l)).obj Y) ≅ | |
| (BlockModule.derivedFirst B (B ⊗[k] ChainAlgebra k l)).obj | |
| (((shiftedDerivedTensor P b l).obj)^[n] Y) := by | |
| induction n with | |
| | zero => exact Iso.refl _ | |
| | succ n ih => | |
| have hn : 2*(n+1) = (2*n+1)+1 := by omega | |
| rw [hn,Function.iterate_succ_apply',Function.iterate_succ_apply',Function.iterate_succ_apply'] | |
| exact (derivedTensor P a l).mapIso ((derivedTensor P a l).mapIso ih) ≪≫ | |
| derivedSquareObject P a l b hb hP (((shiftedDerivedTensor P b l).obj)^[n] Y) | |
| end LittleFinitistic.OrdinarySimulation.XResolution | |
| end | |
| end OAI | |