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monula95 dutta 2023-06-14 18:20:41 +00:00
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CommAlg/monalisa.lean
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@ -1,8 +1,5 @@
import Mathlib.Order.KrullDimension
import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.Height
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
@ -11,11 +8,6 @@ import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Ideal.AssociatedPrime
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Finiteness
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.FiniteType
import Mathlib.Order.Height
import Mathlib.RingTheory.PrincipalIdealDomain
@ -25,27 +17,12 @@ import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Algebra.DirectSum.Ring
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib
--import Mathlib
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
import Mathlib.Order.Height
import Mathlib.Order.KrullDimension
import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.Height
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Ideal.AssociatedPrime
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Finiteness
import Mathlib.RingTheory.Ideal.Operations
@ -53,11 +30,14 @@ import Mathlib.RingTheory.Ideal.Operations
noncomputable def length ( A : Type _) (M : Type _)
[CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < ⊤}
def HomogeneousPrime { A σ : Type _} [CommRing A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ℤ → σ) [GradedRing 𝒜] (I : Ideal A):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous 𝒜 I)
def Ideal.IsHomogeneous' (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)]
[DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)) := ∀ (i : ℤ ) ⦃r : (⨁ i, 𝒜 i)⦄, r ∈ I → DirectSum.of _ i ( r i : 𝒜 i) ∈ I
def HomogeneousMax { A σ : Type _} [CommRing A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ℤ → σ) [GradedRing 𝒜] (I : Ideal A):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous 𝒜 I)
def HomogeneousPrime (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous' 𝒜 I)
def HomogeneousMax (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous' 𝒜 I)
--theorem monotone_stabilizes_iff_noetherian :
-- (∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by
@ -67,6 +47,7 @@ open GradedMonoid.GSmul
open DirectSum
instance tada1 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ) : SMul (𝒜 0) (𝓜 i)
where smul x y := @Eq.rec ℤ (0+i) (fun a _ => 𝓜 a) (GradedMonoid.GSmul.smul x y) i (zero_add i)
@ -88,32 +69,10 @@ instance tada3 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGr
letI := Module.compHom (⨁ j, 𝓜 j) (ofZeroRingHom 𝒜)
exact Dfinsupp.single_injective.module (𝒜 0) (of 𝓜 i) (mylem 𝒜 𝓜 i)
-- (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
noncomputable def dummyhil_function (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (hilb : ℤ → ℕ∞) := ∀ i, hilb i = (length (𝒜 0) (𝓜 i))
lemma hilbertz (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜]
(finlen : ∀ i, (length (𝒜 0) (𝓜 i)) < ⊤ ) : ℤ → ℤ := by
intro i
let h := dummyhil_function 𝒜 𝓜
simp at h
let n : ℤ → ℕ := fun i ↦ WithTop.untop _ (finlen i).ne
have hn : ∀ i, (n i : ℕ∞) = length (𝒜 0) (𝓜 i) := fun i ↦ WithTop.coe_untop _ _
have' := hn i
exact ((n i) : ℤ )
noncomputable def hilbert_function (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (hilb : ℤ → ℤ) := ∀ i, hilb i = (ENat.toNat (length (𝒜 0) (𝓜 i)))
noncomputable def dimensionring { A: Type _}
[CommRing A] := krullDim (PrimeSpectrum A)
@ -121,20 +80,38 @@ noncomputable def dimensionring { A: Type _}
noncomputable def dimensionmodule ( A : Type _) (M : Type _)
[CommRing A] [AddCommGroup M] [Module A M] := krullDim (PrimeSpectrum (A ⧸ ((⊤ : Submodule A M).annihilator)) )
-- (∃ (i : ℤ ), ∃ (x : 𝒜 i), p = (Submodule.span (⨁ i, 𝒜 i) {x}).annihilator )
-- lemma graded_local (𝒜 : ℤ → Type _) [SetLike (⨁ i, 𝒜 i)] (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
-- [DirectSum.GCommRing 𝒜]
-- [DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) : ∃ ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := sorry
def PolyType (f : ℤ → ℤ) (d : ℕ) := ∃ Poly : Polynomial ℚ, ∃ (N : ℤ), ∀ (n : ℤ), N ≤ n → f n = Polynomial.eval (n : ℚ) Poly ∧ d = Polynomial.degree Poly
def PolyType (f : ℤ → ℤ) (d : ℕ ) := ∃ Poly : Polynomial ℚ, ∃ (N : ℤ), ∀ (n : ℤ), N ≤ n → f n = Polynomial.eval (n : ℚ) Poly ∧ d = Polynomial.degree Poly
theorem hilbert_polynomial (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
theorem hilbert_polynomial (d : ℕ) (d1 : 1 ≤ d) (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) (fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim : ∃ d : ℕ , dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d):True := sorry
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d) (hilb : ℤ → ℤ)
(Hhilb: hilbert_function 𝒜 𝓜 hilb)
: PolyType hilb (d - 1) := by
sorry
-- Semiring A]
-- variable [SetLike σ A]
theorem hilbert_polynomial_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = 0) (hilb : ℤ → ℤ)
: true := by
sorry
lemma ass_graded (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _)
[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜]
(p : associatedPrimes (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) : (HomogeneousMax 𝒜 p) := by
sorry