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2c7d975 | 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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCokernel
import OAI.Algebra.AuslanderReiten.Complete.CompleteShiftDual
namespace OAI
noncomputable section
namespace ArExplicit.BalancedTensor
open CategoryTheory
universe u
variable {k R S F : Type u} [Field k] [Ring R] [Ring S] [Algebra k R]
variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F] [Module S F]
variable [IsScalarTower k Rᵐᵒᵖ F]
variable [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F]
variable [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F]
variable [Module.Finite S F] [Module.Projective S F]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
theorem complex_exactProjective (P : CochainComplex (ModuleCat.{u} R) ℤ)
(hP : ExactProjectiveCochain P) :
ExactProjectiveCochain
(((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P) where
finite n := by
let := hP.finite n
exact finite
projective n := by
let := hP.finite n
let := hP.projective n
exact projective
exactAt n := by
rw [HomologicalComplex.exactAt_iff' _ (n-1) n (n+1) (by simp) (by simp),
ShortComplex.moduleCat_exact_iff_range_eq_ker]
have he := hP.exactAt n
rw [HomologicalComplex.exactAt_iff' _ (n-1) n (n+1) (by simp) (by simp),
ShortComplex.moduleCat_exact_iff_range_eq_ker] at he
exact (LinearMap.exact_iff.mp
(mapRight_exact (k := k) (F := F) (S := S) _ _
(LinearMap.exact_iff.mpr he.symm))).symm
theorem complex_totallyAcyclic [Module.Injective S S]
(P : CochainComplex (ModuleCat.{u} R) ℤ) (hP : ExactProjectiveCochain P) :
TotallyAcyclicCochain
(((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P) :=
TotallyAcyclicCochain.of_chain
(complex_exactProjective (k := k) (F := F) (S := S) P hP).to_chain.totallyAcyclic
end ArExplicit.BalancedTensor
end
end OAI
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