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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 50 51 52 53 54 55 56 57 | 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
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