mirror of
https://github.com/GTBarkley/comm_alg.git
synced 2024-12-26 23:48:36 -06:00
101 lines
5.5 KiB
Text
101 lines
5.5 KiB
Text
import Mathlib.Order.KrullDimension
|
||
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
|
||
import Mathlib.Algebra.Module.GradedModule
|
||
import Mathlib.RingTheory.Ideal.AssociatedPrime
|
||
import Mathlib.RingTheory.Artinian
|
||
import Mathlib.Order.Height
|
||
|
||
noncomputable def length ( A : Type _) (M : Type _)
|
||
[CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < ⊤}
|
||
|
||
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 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
|
||
-- rw [isNoetherian_iff_wellFounded, WellFounded.monotone_chain_condition]
|
||
|
||
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)
|
||
|
||
lemma mylem (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
|
||
[h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ) (a : 𝒜 0) (m : 𝓜 i) :
|
||
of _ _ (a • m) = of _ _ a • of _ _ m := by
|
||
refine' Eq.trans _ (Gmodule.of_smul_of 𝒜 𝓜 a m).symm
|
||
refine' of_eq_of_gradedMonoid_eq _
|
||
exact Sigma.ext (zero_add _).symm <| eq_rec_heq _ _
|
||
|
||
instance tada2 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
|
||
[h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ) : SMulWithZero (𝒜 0) (𝓜 i) := by
|
||
letI := SMulWithZero.compHom (⨁ i, 𝓜 i) (of 𝒜 0).toZeroHom
|
||
exact Function.Injective.smulWithZero (of 𝓜 i).toZeroHom Dfinsupp.single_injective (mylem 𝒜 𝓜 i)
|
||
|
||
instance tada3 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
|
||
[h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ): Module (𝒜 0) (𝓜 i) := by
|
||
letI := Module.compHom (⨁ j, 𝓜 j) (ofZeroRingHom 𝒜)
|
||
exact Dfinsupp.single_injective.module (𝒜 0) (of 𝓜 i) (mylem 𝒜 𝓜 i)
|
||
|
||
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)
|
||
|
||
|
||
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
|
||
|
||
|
||
|
||
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 : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d) (hilb : ℤ → ℤ)
|
||
(Hhilb: hilbert_function 𝒜 𝓜 hilb)
|
||
: PolyType hilb (d - 1) := by
|
||
sorry
|
||
|
||
|
||
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 Associated_prime_of_graded_is_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))
|
||
: (Ideal.IsHomogeneous' 𝒜 p) ∧ ((∃ (i : ℤ ), ∃ (x : 𝒜 i), p = (Submodule.span (⨁ i, 𝒜 i) {DirectSum.of _ i x}).annihilator)) := by
|
||
sorry
|
||
|
||
|
||
-- def standard_graded {𝒜 : ℤ → Type _} [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (n : ℕ) :
|
||
-- Prop :=
|
||
-- ∃ J, Ideal.IsHomogeneous' 𝒜 J (J :Nonempty ((⨁ i, 𝒜 i) ≃+* (MvPolynomial (Fin n) (𝒜 0)) ⧸ J)
|