File size: 2,767 Bytes
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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorFunctor
import Mathlib.LinearAlgebra.Pi
namespace OAI
noncomputable section
namespace ArExplicit.BalancedTensor
variable {k R S F M : Type*} [CommRing k] [Ring R] [Algebra k R] [Ring S]
variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F]
variable [Module S F] [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F]
variable [AddCommGroup M] [Module k M] [Module R M] [IsScalarTower k R M]
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
omit [Algebra k R] [IsScalarTower k R M] [Fintype ι] [DecidableEq ι] in
theorem tmul_sum (f : F) (s : Finset ι) (m : ι → M) :
tmul (k := k) (R := R) f (∑ i ∈ s, m i) = ∑ i ∈ s, tmul f (m i) := by
exact map_sum (mk (k := k) (R := R) f) m s
def piRightMap : BalancedTensor k R F (ι → M) →ₗ[S] (ι → BalancedTensor k R F M) :=
LinearMap.pi fun i => mapRight (k := k) (S := S) (LinearMap.proj i)
omit [Fintype ι] [DecidableEq ι] in
@[simp] theorem piRightMap_tmul (f : F) (m : ι → M) (i : ι) :
piRightMap (S := S) (tmul f m : BalancedTensor k R F (ι → M)) i = tmul f (m i) := rfl
def piRightInv : (ι → BalancedTensor k R F M) →ₗ[S] BalancedTensor k R F (ι → M) where
toFun z := ∑ i, mapRight (k := k) (S := S) (LinearMap.single R (fun _ : ι => M) i) (z i)
map_add' x y := by simp only [Pi.add_apply, map_add, Finset.sum_add_distrib]
map_smul' s z := by simp only [Pi.smul_apply, map_smul, RingHom.id_apply, Finset.smul_sum]
omit [Fintype ι] in
theorem piRightMap_single (i j : ι) (z : BalancedTensor k R F M) :
piRightMap (S := S) (mapRight (k := k) (S := S) (LinearMap.single R (fun _ : ι => M) j) z) i = if i = j then z else 0 := by
induction z using induction_on with
| zero => simp
| tmul f m =>
simp only [mapRight_tmul, piRightMap_tmul, LinearMap.single_apply]
by_cases h : i = j
· subst i; simp
· simp [h]
| add x y hx hy =>
simp only [map_add, Pi.add_apply, hx, hy]
split_ifs <;> simp
def piRight : BalancedTensor k R F (ι → M) ≃ₗ[S] (ι → BalancedTensor k R F M) where
__ := piRightMap
invFun := piRightInv
right_inv z := by
ext i
change piRightMap (S := S) (∑ j, mapRight (k := k) (S := S) (LinearMap.single R (fun _ : ι => M) j) (z j)) i = z i
simp only [map_sum, Finset.sum_apply, piRightMap_single]
simp
left_inv z := by
change piRightInv (S := S) (piRightMap (S := S) z) = z
induction z using induction_on with
| zero => simp
| tmul f m =>
change (∑ i, tmul (k := k) (R := R) f (Pi.single i (m i))) = tmul f m
rw [← tmul_sum]
congr
exact Finset.univ_sum_single m
| add x y hx hy => simp only [map_add, hx, hy]
end ArExplicit.BalancedTensor
end
end OAI
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