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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 | 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
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