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almost finished base case
This commit is contained in:
monula95 dutta
parent
53c4675cb8
commit
96d1b2d83c
2 changed files with 116 additions and 113 deletions
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@ -6,13 +6,11 @@ import Mathlib.RingTheory.Artinian
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import Mathlib.Order.Height
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import Mathlib.RingTheory.Ideal.Quotient
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import Mathlib.RingTheory.SimpleModule
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import Mathlib.Algebra.Module.LinearMap
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import Mathlib.Algebra.Field.Defs
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import CommAlg.krull
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#check Ideal.dim_field_eq_zero
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#check Ideal.domain_dim_zero.isField
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#check Ideal.Quotient.isDomain_iff_prime
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-- Setting for "library_search"
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set_option maxHeartbeats 0
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@ -33,6 +31,7 @@ macro "obviously" : tactic =>
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| ring; done; dbg_trace "it was ring"
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| trivial; done; dbg_trace "it was trivial"
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-- | nlinarith; done
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| aesop; done; dbg_trace "it was aesop"
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| fail "No, this is not obvious."))
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@ -47,6 +46,8 @@ section
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-- Definition of polynomail of type d
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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
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noncomputable def length ( A : Type _) (M : Type _)
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[CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < ⊤}
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@ -84,8 +85,17 @@ noncomputable def dimensionmodule ( A : Type _) (M : Type _)
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[CommRing A] [AddCommGroup M] [Module A M] := Ideal.krullDim (A ⧸ ((⊤ : Submodule A M).annihilator))
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lemma lengthfield ( k : Type _) [Field k] : length (k) (k) = 1 := by
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sorry
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lemma equaldim ( A : Type _) [CommRing A] (I : Ideal A): dimensionmodule (A) (A ⧸ I) = Ideal.krullDim (A ⧸ I) := by
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sorry
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lemma dim_iso ( A : Type _) (M : Type _) (N : Type _) [CommRing A] [AddCommGroup M] [Module A M] [AddCommGroup N] [Module A N] (h : Nonempty (M →ₗ[A] N)) : dimensionmodule A M = dimensionmodule A N := by
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sorry
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end
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@ -135,15 +145,22 @@ def graded_ring_morphism (𝒜 : ℤ → Type _) (ℬ : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (ℬ i)]
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[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)
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def graded_module_morphism (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
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[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)
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structure GradedLinearMap (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
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[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] [DirectSum.Gmodule 𝒜 𝓝]
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extends LinearMap (RingHom.id (⨁ i, 𝒜 i)) (⨁ i, 𝓜 i) (⨁ i, 𝓝 i) where
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respects_grading (i : ℤ) (r : 𝓜 i) (j : ℤ) : j ≠ i → toFun (DirectSum.of _ i r) j = 0
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def graded_module_isomorphism (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
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[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝]
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(f : (⨁ i, 𝓜 i) → (⨁ i, 𝓝 i))
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:= (graded_module_morphism 𝒜 𝓜 𝓝 f) ∧ (Function.Bijective f)
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/-- `𝓜 →ᵍₗ[𝒜] 𝓝` denotes the type of graded `𝒜`-linear maps from `𝓜` to `𝓝`. -/
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notation:25 𝓜 " →ᵍₗ[" 𝒜:25 "] " 𝓝:0 => GradedLinearMap 𝒜 𝓜 𝓝
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structure GradedLinearEquiv (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
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[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝]
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extends (⨁ i, 𝓜 i) ≃ (⨁ i, 𝓝 i), 𝓜 →ᵍₗ[𝒜] 𝓝
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/-- `𝓜 ≃ᵍₗ[𝒜] 𝓝` denotes the type of graded `𝒜`-linear isomorphisms from `(⨁ i, 𝓜 i)` to `(⨁ i, 𝓝 i)`. -/
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notation:25 𝓜 " ≃ᵍₗ[" 𝒜:25 "] " 𝓝:0 => GradedLinearEquiv 𝒜 𝓜 𝓝
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def graded_ring_isomorphism (𝒜 : ℤ → Type _) (𝓑 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓑 i)]
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@ -153,9 +170,7 @@ def graded_ring_isomorphism (𝒜 : ℤ → Type _) (𝓑 : ℤ → Type _)
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def graded_ring_isomorphic (𝒜 : ℤ → Type _) (𝓑 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓑 i)]
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[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓑] := ∃ (f : (⨁ i, 𝒜 i) →+* (⨁ i, 𝓑 i)),graded_ring_isomorphism 𝒜 𝓑 f
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[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓑] := ∃ (f : (⨁ i, 𝒜 i) →+* (⨁ i, 𝓑 i)), graded_ring_isomorphism 𝒜 𝓑 f
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-- def graded_submodule
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-- (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
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@ -173,38 +188,65 @@ class DirectSum.GalgebrA
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(𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝓜]
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extends DirectSum.Gmodule 𝒜 𝓜
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def graded_algebra_morphism (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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(𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝓜] [DirectSum.GalgebrA 𝒜 𝓜]
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(𝓝 : ℤ → Type _) [∀ i, AddCommGroup (𝓝 i)] [DirectSum.GCommRing 𝓝] [DirectSum.GalgebrA 𝒜 𝓝]
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(f : (⨁ i, 𝓜 i) →+* (⨁ i, 𝓝 i)) := (graded_ring_morphism 𝓜 𝓝 f) ∧ (graded_module_morphism 𝒜 𝓜 𝓝 f)
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-- def graded_algebra_morphism (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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-- (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝓜] [DirectSum.GalgebrA 𝒜 𝓜]
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-- (𝓝 : ℤ → Type _) [∀ i, AddCommGroup (𝓝 i)] [DirectSum.GCommRing 𝓝] [DirectSum.GalgebrA 𝒜 𝓝]
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-- (f : (⨁ i, 𝓜 i) → (⨁ i, 𝓝 i)) := (graded_ring_morphism 𝓜 𝓝 f) ∧ (GradedLinearMap 𝒜 𝓜 𝓝 toFun)
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-- @Quotient of a graded ring R by a graded ideal p is a graded R-alg, preserving each component
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instance Quotient_of_graded_gradedring
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(𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
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: DirectSum.GCommRing (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
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(𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) :
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DirectSum.GCommRing (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
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sorry
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instance Quotient_of_graded_is_gradedalg
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(𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
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: DirectSum.GalgebrA 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
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(𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) :
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DirectSum.GalgebrA 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
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sorry
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section
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variable (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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[LocalRing (𝒜 0)] (m : LocalRing.maximalIdeal (𝒜 0))
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-- check if `Pi.Single` or something writes this more elegantly
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def GradedOneComponent (i : ℤ) : Type _ := ite (i = 0) (𝒜 0 ⧸ LocalRing.maximalIdeal (𝒜 0)) PUnit
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instance (i : ℤ) : AddMonoid (GradedOneComponent 𝒜 i) := by
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unfold GradedOneComponent
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sorry -- split into 0 and nonzero cases and then `inferInstance`
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instance : DirectSum.Gmodule 𝒜 (GradedOneComponent 𝒜) := by sorry
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lemma Graded_local [StandardGraded 𝒜] (I : Ideal (⨁ i, 𝒜 i)) (hp : (HomogeneousMax 𝒜 I)) [∀ i, Module (𝒜 0) ((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 I hp.2 i))] (art: IsArtinianRing (𝒜 0)) : (∀ (i : ℤ ), (i ≠ 0 → Nonempty (((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 I hp.2 i)) →ₗ[𝒜 0] (𝒜 i))) ) := by
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sorry
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-- lemma Graded_local [StandardGraded 𝒜] (I : Ideal (⨁ i, 𝒜 i)) (hp : (HomogeneousMax 𝒜 I)) (art: IsArtinianRing (𝒜 0)) : (∀ (i : ℤ ), (i ≠ 0 → (Nonempty (((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 I hp.2 i)) →ₛₗ[𝒜 0] (𝒜 i)))) ∧ (i = 0 → Nonempty (((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 I hp.2 i)) →ₛₗ[𝒜 0] (𝒜 0 ⧸ LocalRing.maximalIdeal (𝒜 0))))) := by
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-- sorry
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end
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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)
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(hm : 𝓜 = (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
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: Nonempty ((⨁ i, (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) ≃+* ((⨁ i, (𝒜 i))⧸p)) := by
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-- (hm : 𝓜 = (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
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: Nonempty (((⨁ i, (𝒜 i))⧸p) →ₗ[(⨁ i, 𝒜 i)] (⨁ i, (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i))) := by
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sorry
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def Is.Graded_local (𝒜 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)][DirectSum.GCommRing 𝒜] := ∃! ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I)
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lemma hilfun_eq (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
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[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝] (iso : GradedLinearEquiv 𝒜 𝓜 𝓝)(hilbm : ℤ → ℤ) (Hhilbm: hilbert_function 𝒜 𝓜 hilbm) (hilbn : ℤ → ℤ) (Hhilbn: hilbert_function 𝒜 𝓝 hilbn) : ∀ (n : ℤ), hilbm n = hilbn n := by
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sorry
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-- If A_0 is Artinian and local, then A is graded local
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lemma Graded_local_if_zero_component_Artinian_and_local (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
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[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) : ∃! ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := by
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sorry
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-- @Existence of a chain of submodules of graded submoduels of a f.g graded R-mod M
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@ -238,11 +280,10 @@ lemma Associated_prime_of_graded_is_graded
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-- 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)
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theorem Hilbert_polynomial_d_ge_1 (d : ℕ) (d1 : 1 ≤ d) (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
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[DirectSum.GCommRing 𝒜]
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[DirectSum.Gmodule 𝒜 𝓜] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
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[DirectSum.Gmodule 𝒜 𝓜] [StandardGraded 𝒜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
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(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
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(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d)
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(hilb : ℤ → ℤ) (Hhilb: hilbert_function 𝒜 𝓜 hilb)
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: PolyType hilb (d - 1) := by
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sorry
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@ -253,7 +294,7 @@ theorem Hilbert_polynomial_d_ge_1_reduced
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(d : ℕ) (d1 : 1 ≤ d)
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(𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
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[DirectSum.GCommRing 𝒜]
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[DirectSum.Gmodule 𝒜 𝓜] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
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[DirectSum.Gmodule 𝒜 𝓜] [StandardGraded 𝒜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
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(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
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(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d)
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(hilb : ℤ → ℤ) (Hhilb: hilbert_function 𝒜 𝓜 hilb)
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@ -267,7 +308,7 @@ theorem Hilbert_polynomial_d_ge_1_reduced
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-- If M is a finite graed R-Mod of dimension zero, then the Hilbert function H(M, n) = 0 for n >> 0
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theorem Hilbert_polynomial_d_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
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[DirectSum.GCommRing 𝒜]
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[DirectSum.Gmodule 𝒜 𝓜] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
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[DirectSum.Gmodule 𝒜 𝓜] [StandardGraded 𝒜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
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(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
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(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = 0)
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(hilb : ℤ → ℤ) (Hhilb : hilbert_function 𝒜 𝓜 hilb)
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@ -275,82 +316,5 @@ theorem Hilbert_polynomial_d_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [
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sorry
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#check Ideal.dim_field_eq_zero
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#check Ideal.domain_dim_zero.isField
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--#check Quotient.isDomain_iff_prime
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#check DirectSum
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-- f (g a) = f (g b)
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-- DirectSum _ (fun i => ...) = DirectSum _ (fun i => ...)
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theorem Hilbert_polynomial_d_0_reduced
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(𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
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[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓜]
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[DirectSum.GalgebrA 𝒜 𝓜] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
|
||||
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
|
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(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = 0)
|
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(hilb : ℤ → ℤ) (Hhilb : hilbert_function 𝒜 𝓜 hilb)
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(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
|
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let 𝓜' := fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)
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have h : 𝓜 = 𝓜' := by
|
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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
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
39
CommAlg/hil_mine.lean
Normal file
39
CommAlg/hil_mine.lean
Normal file
|
|
@ -0,0 +1,39 @@
|
|||
import CommAlg.final_hil_pol
|
||||
import Mathlib.Algebra.Ring.Defs
|
||||
|
||||
set_option maxHeartbeats 0
|
||||
|
||||
|
||||
|
||||
theorem Hilbert_polynomial_d_0_reduced
|
||||
(𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)]
|
||||
[DirectSum.GCommRing 𝒜][LocalRing (𝒜 0)] [StandardGraded 𝒜] (art: IsArtinianRing (𝒜 0)) (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
|
||||
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
|
||||
(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) = 0)
|
||||
(hilb : ℤ → ℤ) (Hhilb : hilbert_function 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) hilb)
|
||||
(hq : HomogeneousPrime 𝒜 p) (n : ℤ) (n_0 : 0 < n)
|
||||
: hilb n = 0 := by
|
||||
have h1 : dimensionmodule (⨁ i, 𝒜 i) ((⨁ i, (𝒜 i))⧸p) = dimensionmodule (⨁ i, 𝒜 i) (⨁ i, ((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i))) := by
|
||||
apply dim_iso (⨁ i, 𝒜 i) ((⨁ i, (𝒜 i))⧸p) (⨁ i, ((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
|
||||
exact Quotient_of_graded_ringiso 𝒜 p hp
|
||||
have h2 : dimensionmodule (⨁ i, 𝒜 i) ((⨁ i, (𝒜 i))⧸p) = Ideal.krullDim ((⨁ i, (𝒜 i))⧸p) := by
|
||||
apply equaldim (⨁ i, 𝒜 i) p
|
||||
have h3 : 0 = Ideal.krullDim ((⨁ i, 𝒜 i) ⧸ p) := by
|
||||
calc 0 = dimensionmodule (⨁ i, 𝒜 i) (⨁ i, ((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i))) := findim.symm
|
||||
_ = dimensionmodule (⨁ i, 𝒜 i) ((⨁ i, (𝒜 i))⧸p) := h1.symm
|
||||
_ = Ideal.krullDim ((⨁ i, (𝒜 i))⧸p) := h2
|
||||
have h4 : IsDomain ((⨁ i, (𝒜 i))⧸p) := (Ideal.Quotient.isDomain_iff_prime p).mpr hq.1
|
||||
have h5 : IsField ((⨁ i, (𝒜 i))⧸p) := Ideal.domain_dim_zero.isField (h3.symm)
|
||||
have h6 : p.IsMaximal := Ideal.Quotient.maximal_of_isField p h5
|
||||
have h7 : HomogeneousMax 𝒜 p := ⟨h6, hq.2⟩
|
||||
-- have h8 : Nonempty ((⨁ i, 𝒜 i)⧸ p →+* (𝒜 0)⧸(LocalRing.maximalIdeal (𝒜 0))) := Graded_local 𝒜 p h7 art
|
||||
-- set m := LocalRing.maximalIdeal (𝒜 0)
|
||||
-- have h0 : m.IsMaximal := LocalRing.maximalIdeal.isMaximal (𝒜 0)
|
||||
-- have h9 : IsField ((𝒜 0)⧸m) := (Ideal.Quotient.maximal_ideal_iff_isField_quotient m).mp h0
|
||||
-- set k := ((𝒜 0)⧸m)
|
||||
have hilb n
|
||||
sorry
|
||||
|
||||
|
||||
|
||||
|
||||
Loading…
Reference in a new issue