| import OAI.Algebra.Finitistic.Dimension.PDTensorFunctor |
| import OAI.Algebra.Finitistic.Dimension.PDMinimalResolution |
| import Mathlib.CategoryTheory.Abelian.LeftDerived |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.PDTensor |
| open CategoryTheory CategoryTheory.Limits |
| universe u |
| variable {k A : Type u} [Field k] [Ring A] [Algebra k A] |
| variable (U : Type u) [AddCommGroup U] [Module k U] [Module Aᵐᵒᵖ U] |
|
|
| /-- Tor is the left derived functor of the actual balanced tensor functor. -/ |
| abbrev Tor (M : ModuleCat.{u} A) (n : ℕ) : ModuleCat.{u} k := |
| ((tensorFunctor k A U).leftDerived n).obj M |
|
|
| variable [FiniteDimensional k A] (M : PDMinimal.FiniteModule k A) |
|
|
| def model : ChainComplex (ModuleCat k) ℕ := |
| ((tensorFunctor k A U).mapHomologicalComplex _).obj (PDMinimal.complex M) |
|
|
| /-- The chosen minimal resolution computes this actual derived functor. -/ |
| def torIsoModel (n : ℕ) : Tor (k := k) U M.cat n ≅ (model U M).homology n := |
| (PDMinimal.resolution M).isoLeftDerivedObj (tensorFunctor k A U) n |
|
|
| variable (hU : ∀ a ∈ Ring.jacobson A, ∀ u : U, MulOpposite.op a • u = 0) |
|
|
| include hU in |
| /-- Minimality annihilates every tensor differential, including non-adjacent ones. -/ |
| theorem model_d_zero (i j : ℕ) : (model U M).d i j = 0 := by |
| change (tensorFunctor k A U).map ((PDMinimal.complex M).d i j) = 0 |
| by_cases h : i = j+1 |
| · subst i |
| rw [PDMinimal.complex_d] |
| apply ModuleCat.hom_ext |
| exact @mapRight_radical_zero k A _ _ _ U _ _ _ |
| (ScalarModule k A (PDMinimal.term M (j+1)).cat) |
| (ScalarModule k A (PDMinimal.term M j).cat) |
| _ _ _ (scalarModuleTower k A (PDMinimal.term M (j+1)).cat) |
| _ _ _ (scalarModuleTower k A (PDMinimal.term M j).cat) |
| hU (PDMinimal.differential M j) (PDMinimal.differential_radical M j) |
| · rw [(PDMinimal.complex M).shape i j (by change ¬ j+1=i; exact Ne.symm h)] |
| exact Functor.map_zero _ _ _ |
| |
| /-- Homology of the tensor of a minimal resolution is its term, because all |
| its differentials have been proved zero. -/ |
| def torIsoTerm (n : ℕ) : Tor (k := k) U M.cat n ≅ |
| (tensorFunctor k A U).obj (PDMinimal.term M n).cat := |
| torIsoModel U M n ≪≫ |
| (ShortComplex.LeftHomologyData.ofZeros ((model U M).sc n) |
| (model_d_zero U M hU _ _) (model_d_zero U M hU _ _)).homologyIso |
| end LittleFinitistic.PDTensor |
|
|
| end |
|
|
| end OAI |
|
|
