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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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import OAI.Algebra.AuslanderReiten.Tensor.EnvelopingBalancedRegular
import Mathlib.LinearAlgebra.TensorProduct.Basis
import Mathlib.Algebra.Module.Projective
import Mathlib.LinearAlgebra.Pi
import Mathlib.RingTheory.Finiteness.Cardinality
namespace OAI
noncomputable section
open CategoryTheory
open scoped TensorProduct ModuleCat.Algebra
namespace ArExplicit.EnvelopingBimodule
universe u
variable {k R : Type u} [Field k] [Ring R] [Algebra k R]
variable (X : ModuleCat.{u} R) [Module k X] [IsScalarTower k R X]
private instance finitePiProjective {V : Type u} [AddCommGroup V] [Module R V]
[Module.Projective R V] (n : ℕ) : Module.Projective R (Fin n → V) :=
Module.Projective.of_equiv (DFinsupp.linearEquivFunOnFintype :
(Π₀ _ : Fin n, V) ≃ₗ[R] (Fin n → V))
instance regular_projective :
Module.Projective R (BalancedTensor k R (EnvelopingBimodule k R (regularObject k R)) X) :=
Module.Projective.of_equiv (regularEquiv k R X).symm
variable (M : ModuleCat.{u} (R ⊗[k] Rᵐᵒᵖ))
variable [Module k M] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) M]
abbrev piObject (n : ℕ) : ModuleCat.{u} (R ⊗[k] Rᵐᵒᵖ) :=
ModuleCat.of (R ⊗[k] Rᵐᵒᵖ) (Fin n → M)
def piProjection (n : ℕ) (i : Fin n) : piObject M n ⟶ M :=
ModuleCat.ofHom (LinearMap.proj i)
def piInclusion (n : ℕ) (i : Fin n) : M ⟶ piObject M n :=
ModuleCat.ofHom (LinearMap.single (R ⊗[k] Rᵐᵒᵖ) (fun _ : Fin n => M) i)
def tensorPiCoordinates (n : ℕ) :
BalancedTensor k R (EnvelopingBimodule k R (piObject M n)) X →ₗ[R]
(Fin n → BalancedTensor k R (EnvelopingBimodule k R M) X) :=
LinearMap.pi fun i => tensorMap X (piProjection M n i)
def tensorPiRecombine (n : ℕ) :
(Fin n → BalancedTensor k R (EnvelopingBimodule k R M) X) →ₗ[R]
BalancedTensor k R (EnvelopingBimodule k R (piObject M n)) X :=
∑ i, (tensorMap X (piInclusion M n i)).comp (LinearMap.proj i)
omit [IsScalarTower k R X] in
@[simp] theorem tensorPiRecombine_apply (n : ℕ)
(v : Fin n → BalancedTensor k R (EnvelopingBimodule k R M) X) :
tensorPiRecombine X M n v = ∑ i, tensorMap X (piInclusion M n i) (v i) := by
simp only [tensorPiRecombine, LinearMap.sum_apply, LinearMap.comp_apply, LinearMap.proj_apply]
private theorem pi_reconstruct (n : ℕ) (f : EnvelopingBimodule k R (piObject M n)) :
∑ i, map (piInclusion M n i) (map (piProjection M n i) f) = f := by
apply EnvelopingBimodule.ext
change (scalarEquiv (piObject M n)) (∑ i, map (piInclusion M n i) (map (piProjection M n i) f)) = f.value
rw [_root_.map_sum]
change (∑ i, Pi.single i (f.value i)) = f.value
exact Finset.univ_sum_single f.value
omit [IsScalarTower k R X] in
theorem tensorPi_retraction (n : ℕ) :
(tensorPiRecombine X M n).comp (tensorPiCoordinates X M n) = LinearMap.id := by
apply LinearMap.ext
intro z
change tensorPiRecombine X M n (tensorPiCoordinates X M n z) = z
induction z using BalancedTensor.induction_on with
| zero => simp only [_root_.map_zero]
| add z w hz hw => simp only [_root_.map_add, hz, hw]
| tmul f x =>
rw [tensorPiRecombine_apply]
change (∑ i, tensorMap X (piInclusion M n i)
(tensorMap X (piProjection M n i) (BalancedTensor.tmul f x))) = BalancedTensor.tmul f x
simp only [tensorMap_tmul]
change (∑ i, ((BalancedTensor.mk (k := k) (R := R)).flip x)
(map (piInclusion M n i) (map (piProjection M n i) f))) = _
rw [← _root_.map_sum, pi_reconstruct]
rfl
omit [IsScalarTower k R X] in
theorem tensorPi_projective
[Module.Projective R (BalancedTensor k R (EnvelopingBimodule k R M) X)] (n : ℕ) :
Module.Projective R (BalancedTensor k R (EnvelopingBimodule k R (piObject M n)) X) :=
Module.Projective.of_split (R := R) (tensorPiCoordinates X M n) (tensorPiRecombine X M n)
(tensorPi_retraction X M n)
theorem tensor_projective [Module.Finite (R ⊗[k] Rᵐᵒᵖ) M]
[Module.Projective (R ⊗[k] Rᵐᵒᵖ) M] :
Module.Projective R (BalancedTensor k R (EnvelopingBimodule k R M) X) := by
obtain ⟨n, p, hp⟩ := Module.Finite.exists_fin' (R ⊗[k] Rᵐᵒᵖ) M
obtain ⟨s, hs⟩ := Module.projective_lifting_property p
(LinearMap.id : M →ₗ[R ⊗[k] Rᵐᵒᵖ] M) hp
let := regular_projective (k := k) (R := R) X
let : Module.Projective R (BalancedTensor k R
(EnvelopingBimodule k R (piObject (regularObject k R) n)) X) :=
tensorPi_projective X (regularObject k R) n
apply Module.Projective.of_split (tensorMap X (ModuleCat.ofHom s))
(tensorMap X (ModuleCat.ofHom p))
rw [← tensorMap_comp]
have hsp : ModuleCat.ofHom s ≫ ModuleCat.ofHom p = 𝟙 M := by
apply ModuleCat.hom_ext
exact hs
rw [hsp, tensorMap_id]
end ArExplicit.EnvelopingBimodule
end
end OAI
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