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ae00908 | 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 | 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
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