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CommAlg/final_hil_pol.lean
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CommAlg/final_hil_pol.lean
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import Mathlib.Order.KrullDimension
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import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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import Mathlib.Algebra.Module.GradedModule
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import Mathlib.RingTheory.Ideal.AssociatedPrime
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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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-- Setting for "library_search"
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set_option maxHeartbeats 0
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macro "ls" : tactic => `(tactic|library_search)
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-- New tactic "obviously"
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macro "obviously" : tactic =>
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`(tactic| (
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first
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| dsimp; simp; done; dbg_trace "it was dsimp simp"
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| simp; done; dbg_trace "it was simp"
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| tauto; done; dbg_trace "it was tauto"
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| simp; tauto; done; dbg_trace "it was simp tauto"
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| rfl; done; dbg_trace "it was rfl"
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| norm_num; done; dbg_trace "it was norm_num"
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| /-change (@Eq ℝ _ _);-/ linarith; done; dbg_trace "it was linarith"
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-- | gcongr; done
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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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open GradedMonoid.GSmul
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open DirectSum
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-- @Definitions (to be classified)
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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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-- Make instance of M_i being an R_0-module
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instance tada1 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
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[DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ) : SMul (𝒜 0) (𝓜 i)
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where smul x y := @Eq.rec ℤ (0+i) (fun a _ => 𝓜 a) (GradedMonoid.GSmul.smul x y) i (zero_add i)
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lemma mylem (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
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[h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ) (a : 𝒜 0) (m : 𝓜 i) :
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of _ _ (a • m) = of _ _ a • of _ _ m := by
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refine' Eq.trans _ (Gmodule.of_smul_of 𝒜 𝓜 a m).symm
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refine' of_eq_of_gradedMonoid_eq _
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exact Sigma.ext (zero_add _).symm <| eq_rec_heq _ _
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instance tada2 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
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[h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ) : SMulWithZero (𝒜 0) (𝓜 i) := by
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letI := SMulWithZero.compHom (⨁ i, 𝓜 i) (of 𝒜 0).toZeroHom
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exact Function.Injective.smulWithZero (of 𝓜 i).toZeroHom Dfinsupp.single_injective (mylem 𝒜 𝓜 i)
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instance tada3 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
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[h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ): Module (𝒜 0) (𝓜 i) := by
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letI := Module.compHom (⨁ j, 𝓜 j) (ofZeroRingHom 𝒜)
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exact Dfinsupp.single_injective.module (𝒜 0) (of 𝓜 i) (mylem 𝒜 𝓜 i)
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-- Definition of a Hilbert function of a graded module
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section
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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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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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noncomputable def hilbert_function (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] (hilb : ℤ → ℤ) := ∀ i, hilb i = (ENat.toNat (length (𝒜 0) (𝓜 i)))
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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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-- Definition of homogeneous ideal
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def Ideal.IsHomogeneous' (𝒜 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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(I : Ideal (⨁ i, 𝒜 i)) := ∀ (i : ℤ )
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⦃r : (⨁ i, 𝒜 i)⦄, r ∈ I → DirectSum.of _ i ( r i : 𝒜 i) ∈ I
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-- Definition of homogeneous prime ideal
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def HomogeneousPrime (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous' 𝒜 I)
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-- Definition of homogeneous maximal ideal
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def HomogeneousMax (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous' 𝒜 I)
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--theorem monotone_stabilizes_iff_noetherian :
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-- (∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by
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-- rw [isNoetherian_iff_wellFounded, WellFounded.monotone_chain_condition]
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instance {𝒜 : ℤ → Type _} [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] :
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Algebra (𝒜 0) (⨁ i, 𝒜 i) :=
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Algebra.ofModule'
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(by
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intro r x
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sorry)
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(by
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intro r x
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sorry)
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class StandardGraded (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] : Prop where
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gen_in_first_piece :
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Algebra.adjoin (𝒜 0) (DirectSum.of _ 1 : 𝒜 1 →+ ⨁ i, 𝒜 i).range = (⊤ : Subalgebra (𝒜 0) (⨁ i, 𝒜 i))
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-- Each component of a graded ring is an additive subgroup
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def Component_of_graded_as_addsubgroup (𝒜 : ℤ → Type _)
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[∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) (i : ℤ) : AddSubgroup (𝒜 i) := by
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sorry
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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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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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/-- `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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[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓑]
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(f : (⨁ i, 𝒜 i) →+* (⨁ i, 𝓑 i))
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:= (graded_ring_morphism 𝒜 𝓑 f) ∧ (Function.Bijective f)
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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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-- def graded_submodule
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-- (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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-- (h (⨁ i, 𝓝 i) : Submodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) :
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-- Prop :=
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-- ∃ (piece : Submodule (𝒜 0) (𝓜 i)), piece = 𝓝 i
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end
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class DirectSum.GalgebrA
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(𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
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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) ∧ (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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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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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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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))⧸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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-- @Existence of a chain of submodules of graded submoduels of a f.g graded R-mod M
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lemma Exist_chain_of_graded_submodules (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]
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(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
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: ∃ (c : List (Submodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))), c.Chain' (· < ·) ∧ ∀ M ∈ c, Ture := by
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sorry
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-- @[BH, 1.5.6 (b)(ii)]
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-- An associated prime of a graded R-Mod M is graded
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lemma Associated_prime_of_graded_is_graded
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(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]
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(p : associatedPrimes (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
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: (Ideal.IsHomogeneous' 𝒜 p) ∧ ((∃ (i : ℤ ), ∃ (x : 𝒜 i), p = (Submodule.span (⨁ i, 𝒜 i) {DirectSum.of _ i x}).annihilator)) := by
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sorry
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-- @[BH, 4.1.3] when d ≥ 1
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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] [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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|
||||||
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|
||||||
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-- (reduced version) [BH, 4.1.3] when d ≥ 1
|
||||||
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-- If M is a finite graed R-Mod of dimension d ≥ 1, and M = R⧸ 𝓅 for a graded prime ideal 𝓅, then the Hilbert function H(M, n) is of polynomial type (d - 1)
|
||||||
|
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)]
|
||||||
|
[DirectSum.GCommRing 𝒜]
|
||||||
|
[DirectSum.Gmodule 𝒜 𝓜] [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) = d)
|
||||||
|
(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)))
|
||||||
|
: PolyType hilb (d - 1) := by
|
||||||
|
sorry
|
||||||
|
|
||||||
|
|
||||||
|
-- @[BH, 4.1.3] when d = 0
|
||||||
|
-- If M is a finite graed R-Mod of dimension zero, then the Hilbert function H(M, n) = 0 for n >> 0
|
||||||
|
theorem Hilbert_polynomial_d_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
|
||||||
|
[DirectSum.GCommRing 𝒜]
|
||||||
|
[DirectSum.Gmodule 𝒜 𝓜] [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)
|
||||||
|
: (∃ (N : ℤ), ∀ (n : ℤ), n ≥ N → hilb n = 0) := by
|
||||||
|
sorry
|
||||||
Loading…
Reference in a new issue