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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 42 43 44 45 46 47 48 49 | 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
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