| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensor |
|
|
| namespace OAI |
|
|
| namespace ArExplicit.BalancedTensor |
|
|
| open TensorProduct |
|
|
| variable {k R F M S : Type*} [CommRing k] [Ring R] [Ring S] |
| variable [AddCommGroup F] [AddCommGroup M] [Module k F] [Module k M] |
| variable [Module Rᵐᵒᵖ F] [Module R M] |
| variable [Module S F] [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F] |
|
|
| theorem smul_mem_relations (s : S) {z : F ⊗[k] M} |
| (hz : z ∈ balancingRelations k R F M) : s • z ∈ balancingRelations k R F M := by |
| induction hz using Submodule.span_induction with |
| | mem z hz => |
| obtain ⟨f, r, m, rfl⟩ := hz |
| rw [smul_sub, TensorProduct.smul_tmul', TensorProduct.smul_tmul', smul_comm s] |
| exact Submodule.subset_span ⟨s • f, r, m, rfl⟩ |
| | zero => simp |
| | add x y hx hy ihx ihy => |
| simpa only [smul_add] using (balancingRelations k R F M).add_mem ihx ihy |
| | smul c z hz ih => |
| rw [← smul_comm c s z] |
| exact (balancingRelations k R F M).smul_mem c ih |
|
|
| def relationsAsLeft : Submodule S (F ⊗[k] M) where |
| __ := (balancingRelations k R F M).toAddSubmonoid |
| smul_mem' := fun s _ hz => smul_mem_relations s hz |
| |
| instance (priority := 100) : Module S (BalancedTensor k R F M) := |
| Submodule.Quotient.module (relationsAsLeft (k := k) (R := R) (S := S) (F := F) (M := M)) |
| |
| @[simp] theorem smul_tmul (s : S) (f : F) (m : M) : |
| s • tmul (k := k) (R := R) f m = tmul (s • f) m := rfl |
| |
| variable [Algebra k S] [IsScalarTower k S F] |
| |
| instance : IsScalarTower k S (BalancedTensor k R F M) where |
| smul_assoc c s z := by |
| induction z using Submodule.Quotient.induction_on with |
| | H z => |
| change Submodule.Quotient.mk ((c • s) • z) = |
| (Submodule.Quotient.mk (c • s • z) : BalancedTensor k R F M) |
| rw [smul_assoc] |
| |
| instance : SMulCommClass k S (BalancedTensor k R F M) where |
| smul_comm c s z := by |
| induction z using Submodule.Quotient.induction_on with |
| | H z => |
| change Submodule.Quotient.mk (c • s • z) = |
| (Submodule.Quotient.mk (s • c • z) : BalancedTensor k R F M) |
| rw [smul_comm] |
|
|
| end ArExplicit.BalancedTensor |
|
|
| end OAI |
|
|
