| import OAI.Algebra.AuslanderReiten.Triangular.Triangular |
| import OAI.Algebra.AuslanderReiten.Algebras.Radical |
| import Mathlib.RingTheory.Ideal.Quotient.Operations |
|
|
| namespace OAI |
|
|
| namespace ArExplicit.Triangular |
|
|
| variable {R F : Type*} [Ring R] [AddCommGroup F] |
| variable [Module R F] [Module Rᵐᵒᵖ F] [SMulCommClass R Rᵐᵒᵖ F] |
|
|
| theorem diagonal_kernel_square : |
| (RingHom.ker (diagonal (R := R) (F := F))) ^ 2 = ⊥ := by |
| apply le_antisymm _ bot_le |
| rw [Submodule.pow_succ, Submodule.pow_one] |
| apply Ideal.mul_le.mpr |
| intro a ha b hb |
| change (a.top, a.bottom) = (0, 0) at ha |
| change (b.top, b.bottom) = (0, 0) at hb |
| have ha₁ : a.top = 0 := congrArg Prod.fst ha |
| have ha₂ : a.bottom = 0 := congrArg Prod.snd ha |
| have hb₁ : b.top = 0 := congrArg Prod.fst hb |
| have hb₂ : b.bottom = 0 := congrArg Prod.snd hb |
| change a * b = 0 |
| ext <;> simp [ha₁, ha₂, hb₁, hb₂] |
|
|
| theorem radical_eq_comap : |
| Ring.jacobson (Triangular R F) = Ideal.comap diagonal (Ring.jacobson (R × R)) := by |
| apply jacobson_eq_comap_of_nil_kernel _ diagonal_surjective |
| intro a ha |
| refine ⟨2, ?_⟩ |
| have h := Ideal.pow_mem_pow ha 2 |
| simpa only [diagonal_kernel_square, Ideal.mem_bot] using h |
|
|
| theorem mem_jacobson_iff (a : Triangular R F) : |
| a ∈ Ring.jacobson (Triangular R F) ↔ |
| a.top ∈ Ring.jacobson R ∧ a.bottom ∈ Ring.jacobson R := by |
| rw [radical_eq_comap] |
| exact mem_jacobson_prod_iff (a.top, a.bottom) |
|
|
| section Scalars |
|
|
| variable (k : Type*) [CommRing k] [Algebra k R] [Module k F] |
| variable [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] |
| variable [SMulCommClass k R F] [SMulCommClass k Rᵐᵒᵖ F] |
|
|
| def diagonalAlg : Triangular R F →ₐ[k] R × R where |
| __ := diagonal |
| commutes' c := by |
| simp only [Algebra.algebraMap_eq_smul_one] |
| rfl |
|
|
| variable {S : Type*} [Ring S] [Algebra k S] |
|
|
| def quotientMap (f : R →ₐ[k] S) : Triangular R F →ₐ[k] S × S := |
| (f.prodMap f).comp (diagonalAlg k) |
|
|
| theorem quotientMap_surjective (f : R →ₐ[k] S) (hf : Function.Surjective f) : |
| Function.Surjective (quotientMap (F := F) k f) := by |
| rintro ⟨s, t⟩ |
| obtain ⟨r, hr⟩ := hf s |
| obtain ⟨r', hr'⟩ := hf t |
| exact ⟨⟨r, r', 0⟩, Prod.ext hr hr'⟩ |
|
|
| theorem radical_eq_ker_quotientMap (f : R →ₐ[k] S) |
| (hf : Ring.jacobson R = RingHom.ker f) : |
| Ring.jacobson (Triangular R F) = RingHom.ker (quotientMap k f) := by |
| ext a |
| rw [mem_jacobson_iff, hf] |
| change (f a.top = 0 ∧ f a.bottom = 0) ↔ (f a.top, f a.bottom) = (0, 0) |
| simp only [Prod.mk.injEq] |
|
|
| noncomputable def semisimpleQuotientEquiv (f : R →ₐ[k] S) |
| (hsurj : Function.Surjective f) (hker : Ring.jacobson R = RingHom.ker f) : |
| (Triangular R F ⧸ Ring.jacobson (Triangular R F)) ≃ₐ[k] S × S := |
| (Ideal.quotientEquivAlgOfEq k (radical_eq_ker_quotientMap k f hker)).trans |
| (Ideal.quotientKerAlgEquivOfSurjective (quotientMap_surjective k f hsurj)) |
|
|
| end Scalars |
| end ArExplicit.Triangular |
|
|
| end OAI |
|
|
