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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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | 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
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