| import OAI.Algebra.Finitistic.Bimodule.BimoduleTensorUnderlying |
| import OAI.Algebra.Finitistic.Rectification.TotalFinite |
| import OAI.Algebra.Finitistic.Rectification.InducedRectification |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| open scoped TensorProduct ModuleCat.Algebra |
| namespace LittleFinitistic.FullTensorProperties |
| open ExternalTensor.Section4 |
| universe u |
| variable (k B : Type u) [Field k] [Ring B] [Algebra k B] |
|
|
| /-- The tensor input's scalar structure agrees with the original bimodule's |
| scalar restriction through the two algebra homomorphisms. -/ |
| def inputEquiv (M : ModuleCat.{u} (Enveloping k B)) : |
| (forgetK k B).obj M ≃ₗ[k] BimoduleBalancedTensor.Input k B M where |
| toFun := id |
| invFun := id |
| left_inv _ := rfl |
| right_inv _ := rfl |
| map_add' _ _ := rfl |
| map_smul' c x := by |
| change (algebraMap k (Enveloping k B) c) • (show M from x) = |
| (BimoduleVertex.restriction k B (algebraMap k Bᵐᵒᵖ c)) • (show M from x) |
| have h : BimoduleVertex.restriction k B (algebraMap k Bᵐᵒᵖ c) = |
| algebraMap k (Enveloping k B) c := Algebra.TensorProduct.includeRight.commutes c |
| rw [h] |
|
|
| variable (N : Type u) [AddCommGroup N] [Module k N] [Module B N] |
|
|
| theorem finite (M : ModuleCat.{u} (Enveloping k B)) |
| [FiniteDimensional k ((forgetK k B).obj M)] [FiniteDimensional k N] : |
| FiniteDimensional k ((ModuleCat.restrictScalars (algebraMap k B)).obj |
| ((BimoduleBalancedTensor.functor k B N).obj M)) := by |
| have : FiniteDimensional k (BimoduleBalancedTensor.Input k B M) := |
| Module.Finite.of_surjective (inputEquiv k B M).toLinearMap (inputEquiv k B M).surjective |
| have : FiniteDimensional k ((BalancedTensor.tensorFunctor k B N).obj ((forgetRight k B).obj M)) := |
| inferInstanceAs (FiniteDimensional k (BimoduleBalancedTensor.Tensor k B M N)) |
| apply Module.Finite.of_surjective (BimoduleBalancedTensor.underlyingEquiv k B N M).symm.toLinearMap |
| (BimoduleBalancedTensor.underlyingEquiv k B N M).symm.surjective |
|
|
| theorem bounded (P : BiComplex k B) (lo hi : ℤ) |
| (h : ∀ n, n < lo ∨ hi < n → IsZero (P.X n)) : |
| ∀ n, n < lo ∨ hi < n → IsZero |
| ((((BimoduleBalancedTensor.functor k B N).mapHomologicalComplex (.up ℤ)).obj P).X n) := by |
| intro n hn |
| exact (BimoduleBalancedTensor.functor k B N).map_isZero (h n hn) |
| end LittleFinitistic.FullTensorProperties |
|
|
| end |
|
|
| end OAI |
|
|
