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started the proof for the base case

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monula95 dutta 2023-06-15 23:07:55 +00:00
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CommAlg/final_hil_pol.lean
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@ -4,6 +4,14 @@ import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Ideal.AssociatedPrime
import Mathlib.RingTheory.Artinian
import Mathlib.Order.Height
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.SimpleModule
import CommAlg.krull
#check Ideal.dim_field_eq_zero
#check Ideal.domain_dim_zero.isField
#check Ideal.Quotient.isDomain_iff_prime
-- Setting for "library_search"
@ -67,15 +75,17 @@ instance tada3 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGr
-- Definition of a Hilbert function of a graded module
section
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)) )
[CommRing A] [AddCommGroup M] [Module A M] := Ideal.krullDim (A ⧸ ((⊤ : Submodule A M).annihilator))
lemma equaldim ( A : Type _) [CommRing A] (I : Ideal A): dimensionmodule (A) (A ⧸ I) = Ideal.krullDim (A ⧸ I) := by
sorry
end
@ -121,39 +131,79 @@ def Component_of_graded_as_addsubgroup (𝒜 : ℤ → Type _)
sorry
def graded_morphism (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
def graded_ring_morphism (𝒜 : ℤ → Type _) (ℬ : ℤ → Type _)
[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (ℬ i)]
[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing ℬ] (f : (⨁ i, 𝒜 i) →+* (⨁ i, ℬ i)) := ∀ i, ∀ (r : 𝒜 i), ∀ j, (j ≠ i → f (DirectSum.of _ i r) j = 0)
def graded_module_morphism (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝] (f : (⨁ i, 𝓜 i) → (⨁ i, 𝓝 i)) : ∀ i, ∀ (r : 𝓜 i), ∀ j, (j ≠ i → f (DirectSum.of _ i r) j = 0) ∧ (IsLinearMap (⨁ i, 𝒜 i) f) := by sorry
[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝] (f : (⨁ i, 𝓜 i) → (⨁ i, 𝓝 i)) := ∀ i, ∀ (r : 𝓜 i), ∀ j, (j ≠ i → f (DirectSum.of _ i r) j = 0) ∧ (IsLinearMap (⨁ i, 𝒜 i) f)
def graded_submodule
(𝒜 : ℤ → Type _) (𝓜 : ℤ → Type u) (𝓝 : ℤ → Type u)
def graded_module_isomorphism (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝]
(opn : Submodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) (opnis : opn = (⨁ i, 𝓝 i)) (i : ℤ )
: ∃(piece : Submodule (𝒜 0) (𝓜 i)), piece = 𝓝 i := by
sorry
(f : (⨁ i, 𝓜 i) → (⨁ i, 𝓝 i))
:= (graded_module_morphism 𝒜 𝓜 𝓝 f) ∧ (Function.Bijective f)
def graded_ring_isomorphism (𝒜 : ℤ → Type _) (𝓑 : ℤ → Type _)
[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓑 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓑]
(f : (⨁ i, 𝒜 i) →+* (⨁ i, 𝓑 i))
:= (graded_ring_morphism 𝒜 𝓑 f) ∧ (Function.Bijective f)
def graded_ring_isomorphic (𝒜 : ℤ → Type _) (𝓑 : ℤ → Type _)
[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓑 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓑] := ∃ (f : (⨁ i, 𝒜 i) →+* (⨁ i, 𝓑 i)),graded_ring_isomorphism 𝒜 𝓑 f
-- def graded_submodule
-- (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
-- [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
-- [DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝]
-- (h (⨁ i, 𝓝 i) : Submodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) :
-- Prop :=
-- ∃ (piece : Submodule (𝒜 0) (𝓜 i)), piece = 𝓝 i
end
class DirectSum.GalgebrA
(𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
(𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝓜]
extends DirectSum.Gmodule 𝒜 𝓜
def graded_algebra_morphism (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
(𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝓜] [DirectSum.GalgebrA 𝒜 𝓜]
(𝓝 : ℤ → Type _) [∀ i, AddCommGroup (𝓝 i)] [DirectSum.GCommRing 𝓝] [DirectSum.GalgebrA 𝒜 𝓝]
(f : (⨁ i, 𝓜 i) →+* (⨁ i, 𝓝 i)) := (graded_ring_morphism 𝓜 𝓝 f) ∧ (graded_module_morphism 𝒜 𝓜 𝓝 f)
-- @Quotient of a graded ring R by a graded ideal p is a graded R-alg, preserving each component
-- @Quotient of a graded ring R by a graded ideal p is a graded R-Mod, preserving each component
instance Quotient_of_graded_is_graded
instance Quotient_of_graded_gradedring
(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
: DirectSum.GCommRing (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
sorry
instance Quotient_of_graded_is_gradedalg
(𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
: DirectSum.GalgebrA 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
sorry
lemma Quotient_of_graded_ringiso (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜](p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
(hm : 𝓜 = (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
: Nonempty ((⨁ i, (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) ≃+* ((⨁ i, (𝒜 i))⧸p)) := by
sorry
-- If A_0 is Artinian and local, then A is graded local
lemma Graded_local_if_zero_component_Artinian_and_local (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)) : ∃ ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := by
[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) : ∃! ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := by
sorry
@ -225,20 +275,67 @@ theorem Hilbert_polynomial_d_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [
sorry
-- (reduced version) [BH, 4.1.3] when d = 0
-- If M is a finite graed R-Mod of dimension zero, and M = R⧸ 𝓅 for a graded prime ideal 𝓅, then the Hilbert function H(M, n) = 0 for n >> 0
#check Ideal.dim_field_eq_zero
#check Ideal.domain_dim_zero.isField
--#check Quotient.isDomain_iff_prime
#check DirectSum
-- f (g a) = f (g b)
-- DirectSum _ (fun i => ...) = DirectSum _ (fun i => ...)
theorem Hilbert_polynomial_d_0_reduced
(𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓜]
[DirectSum.GalgebrA 𝒜 𝓜] (st: StandardGraded 𝒜) (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) (Hhilb : hilbert_function 𝒜 𝓜 hilb)
(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
(hm : 𝓜 = (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
: (∃ (N : ℤ), ∀ (n : ℤ), n ≥ N → hilb n = 0) := by
sorry
(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) (hq : HomogeneousPrime 𝒜 p)
(hm : ∀ i, 𝓜 i = ((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
: (∃ (N : ℤ), ∀ (n : ℤ), n ≥ N → hilb n = 0) := by
let 𝓜' := fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)
have h : 𝓜 = 𝓜' := by
ext i
exact hm i
subst h
set R := ⨁ i, 𝒜 i
have : (⨁ i, 𝓜' i )= ⨁ i, ((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
rfl
--have h1 : Nonempty ((⨁ i, 𝓜 i) ≃+* (R⧸p)) := by
-- apply Quotient_of_graded_ringiso 𝒜 p hp
-- have : Ideal.krullDim (R ⧸ p) = 0 := by
-- calc 0 = dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) := by apply findim
-- _ = dimensionmodule (R) (R ⧸ p) := by apply h1
-- _ = Ideal.krullDim (R_mod_p) := by apply equaldim
-- sorry
lemma
-- (reduced version) [BH, 4.1.3] when d = 0
-- If M is a finite graed R-Mod of dimension zero, and M = R⧸ 𝓅 for a graded prime ideal 𝓅, then the Hilbert function H(M, n) = 0 for n >> 0
-- theorem Hilbert_polynomial_d_0_reduced
-- (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
-- [DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓜]
-- [DirectSum.GalgebrA 𝒜 𝓜] (st: StandardGraded 𝒜) (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) (Hhilb : hilbert_function 𝒜 𝓜 hilb)
-- (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) (hq : HomogeneousPrime 𝒜 p)
-- (hm : 𝓜 = (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
-- : (∃ (N : ℤ), ∀ (n : ℤ), n ≥ N → hilb n = 0) := by
-- set R := ⨁ i, 𝒜 i
-- have h := (Ideal.Quotient.isDomain_iff_prime p).mpr hq.1
-- have h1 : Nonempty ((⨁ i, 𝓜 i)) ≃+* (R⧸p)) := by
-- apply Quotient_of_graded_ringiso 𝒜 p hp
-- have : Ideal.krullDim (R ⧸ p) = 0 := by
-- calc 0 = dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) := by apply findim
-- _ = dimensionmodule (R) (R ⧸ p) := by apply h1
-- _ = Ideal.krullDim (R_mod_p) := by apply equaldim
-- sorry