File size: 2,485 Bytes
d7f9a33 | 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 | import OAI.Algebra.AuslanderReiten.Parameters
import OAI.Algebra.AuslanderReiten.BaseChange.ScalarExtension
import Mathlib.Algebra.Category.ModuleCat.Ext.HasExt
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.LinearAlgebra.FiniteDimensional.Defs
import Mathlib.RingTheory.Jacobson.Radical
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.Ideal.Quotient.Operations
namespace OAI
noncomputable section
open CategoryTheory
open scoped TensorProduct
namespace ArExplicit
universe u v w
def GorensteinProjective (R : Type u) [Ring R]
(M : Type v) [AddCommGroup M] [Module R M] : Prop :=
∃ P : ChainComplex (ModuleCat.{max u v} R) ℤ,
(∀ n : ℤ, Module.Finite R (P.X n) ∧ Module.Projective R (P.X n)) ∧
(∀ n : ℤ,
LinearMap.range (P.d (n + 1) n).hom =
LinearMap.ker (P.d n (n - 1)).hom) ∧
(∀ (n : ℤ) (f : P.X n →ₗ[R] R),
f.comp (P.d (n + 1) n).hom = 0 →
∃ g : P.X (n - 1) →ₗ[R] R, f = g.comp (P.d n (n - 1)).hom) ∧
Nonempty (M ≃ₗ[R] (P.X 0 ⧸ LinearMap.range (P.d 1 0).hom))
def PositiveExtVanish (R : Type u) [Ring R]
(M N : Type u) [AddCommGroup M] [Module R M]
[AddCommGroup N] [Module R N] : Prop :=
∀ n : ℕ, 0 < n →
Subsingleton (CategoryTheory.Abelian.Ext (ModuleCat.of R M) (ModuleCat.of R N) n)
def MainConclusions (k : Type u) [Field k]
(R : Type u) [Ring R] [Algebra k R]
(M : Type u) [AddCommGroup M] [Module R M] [Module k M]
[IsScalarTower k R M] : Prop :=
FiniteDimensional k R ∧
FiniteDimensional k M ∧
¬ Module.Projective R M ∧
GorensteinProjective R M ∧
PositiveExtVanish R M M ∧
PositiveExtVanish R M R ∧
Nonempty ((R ⧸ Ring.jacobson R) ≃ₐ[k] (Fin 8 → k)) ∧
(Ring.jacobson R) ^ 4 ≠ ⊥
structure Candidate where
R : Type
[ring : Ring R]
[algebra : Algebra BaseField R]
M : Type
[addCommGroup : AddCommGroup M]
[module : Module R M]
[baseModule : Module BaseField M]
[tower : IsScalarTower BaseField R M]
attribute [instance] Candidate.ring Candidate.algebra Candidate.addCommGroup
Candidate.module Candidate.baseModule Candidate.tower
def FullMainTheorem : Prop :=
∃ c : Candidate,
MainConclusions BaseField c.R c.M ∧
∀ (E : Type v) [Field E] [Algebra BaseField E],
letI := scalarExtensionModule BaseField E c.R c.M
MainConclusions E (E ⊗[BaseField] c.R) (E ⊗[BaseField] c.M)
end ArExplicit
end
end OAI
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