| 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 |
|
|
