| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorPi |
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorUnit |
| import Mathlib.Algebra.Module.Projective |
| import Mathlib.RingTheory.Finiteness.Cardinality |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ArExplicit.BalancedTensor |
|
|
| private instance finitePiProjective {S P ι : Type*} [Ring S] [AddCommGroup P] |
| [Module S P] [Module.Projective S P] [Fintype ι] : Module.Projective S (ι → P) := |
| Module.Projective.of_equiv (DFinsupp.linearEquivFunOnFintype : |
| (Π₀ _ : ι, P) ≃ₗ[S] (ι → P)) |
|
|
| variable {k R S F M : Type*} [CommRing k] [Ring R] [Ring S] [Algebra k R] |
| variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F] [Module S F] |
| variable [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F] |
| variable [IsScalarTower k Rᵐᵒᵖ F] |
| variable [AddCommGroup M] [Module k M] [Module R M] [IsScalarTower k R M] |
|
|
| theorem projective [Module.Projective S F] [Module.Finite R M] [Module.Projective R M] : |
| Module.Projective S (BalancedTensor k R F M) := by |
| obtain ⟨n, p, hp⟩ := Module.Finite.exists_fin' R M |
| obtain ⟨s, hs⟩ := Module.projective_lifting_property p (LinearMap.id : M →ₗ[R] M) hp |
| let : Module.Projective S (BalancedTensor k R F R) := |
| Module.Projective.of_equiv (runitLeft (k := k) (R := R) (S := S) (F := F)).symm |
| let : Module.Projective S (BalancedTensor k R F (Fin n → R)) := |
| Module.Projective.of_equiv (piRight (k := k) (R := R) (S := S) (F := F) (M := R)).symm |
| apply Module.Projective.of_split (mapRight (k := k) (S := S) s) (mapRight (k := k) (S := S) p) |
| rw [← mapRight_comp, hs, mapRight_id] |
|
|
| theorem finite [Module.Finite S F] [Module.Finite R M] : |
| Module.Finite S (BalancedTensor k R F M) := by |
| obtain ⟨n, p, hp⟩ := Module.Finite.exists_fin' R M |
| let : Module.Finite S (BalancedTensor k R F R) := |
| Module.Finite.equiv (runitLeft (k := k) (R := R) (S := S) (F := F)).symm |
| let : Module.Finite S (BalancedTensor k R F (Fin n → R)) := |
| Module.Finite.equiv (piRight (k := k) (R := R) (S := S) (F := F) (M := R)).symm |
| exact Module.Finite.of_surjective (mapRight (k := k) (S := S) p) (mapRight_surjective p hp) |
|
|
| end ArExplicit.BalancedTensor |
|
|
| end |
|
|
| end OAI |
|
|
