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chelseaandmadrid 2023-06-14 14:44:43 -07:00
parent 08b1fd3e7a
commit 900bf12da2
2 changed files with 17 additions and 43 deletions
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38
HilbertFunction.lean
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@ -29,8 +29,6 @@ macro "obviously" : tactic =>
-- @Definitions (to be classified)
section
open GradedMonoid.GSmul
@ -81,15 +79,20 @@ end
-- [DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) : ∃ ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := sorry
-- Definition(s) of homogeneous ideals
-- Definition of homogeneous ideal
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
-- Definition of homogeneous prime ideal
def HomogeneousPrime (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous' 𝒜 I)
-- Definition of homogeneous maximal ideal
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]
@ -99,12 +102,6 @@ end
-- @[BH, 4.1.3] when d ≥ 1
-- If M is a finite graed R-Mod of dimension d ≥ 1, then the Hilbert function H(M, n) is of polynomial type (d - 1)
theorem hilbert_polynomial_ge1 (d : ℕ) (d1 : 1 ≤ d) (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
@ -165,14 +162,7 @@ lemma Associated_prime_of_graded_is_graded
-- sorry
-- instance sdfasdf
-- (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
-- (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
-- : ∀ i, AddCommGroup (p i) := by
-- sorry
-- Each component of a graded ring is an additive subgroup
def Component_of_graded_as_addsubgroup (𝒜 : ℤ → Type _)
[∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) (i : ℤ) : AddSubgroup (𝒜 i) := by
@ -186,10 +176,12 @@ instance Quotient_of_graded_is_graded
: DirectSum.Gmodule 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
sorry
-- @Graded submodule
instance Graded_submodule
(𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
: DirectSum.Gmodule 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
sorry
-- -- @Graded submodule
-- instance Graded_submodule
-- (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
-- (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
-- : DirectSum.Gmodule 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
-- sorry

22
PolyType.lean
View file

@ -1,24 +1,5 @@
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
-- Setting for "library_search"
set_option maxHeartbeats 0
@ -86,6 +67,7 @@ example : Polynomial.eval (100 : ℚ) F = (2 : ℚ) := by
end section
-- @[BH, 4.1.2]
-- All the polynomials are in ℚ[X], all the functions are considered as ℤ → ℤ
noncomputable section
@ -247,6 +229,7 @@ lemma b_to_a (f : ℤ → ℤ) (d : ℕ) : PolyType f d → (∃ (c : ℤ), ∃
end
-- @Additive lemma of length for a SES
-- Given a SES 0 → A → B → C → 0, then length (A) - length (B) + length (C) = 0
section
-- variable {R M N : Type _} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N]
-- (f : M →[R] N)
@ -305,4 +288,3 @@ end section