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https://github.com/GTBarkley/comm_alg.git
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ah1112
GitHub
commit
4e38ef0a73
1 changed files with 279 additions and 85 deletions
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import Mathlib
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import Mathlib.Algebra.MonoidAlgebra.Basic
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import Mathlib.Data.Finset.Sort
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import Mathlib.Order.Height
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import Mathlib.Order.KrullDimension
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import Mathlib.Order.JordanHolder
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import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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import Mathlib.Order.Height
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import Mathlib.RingTheory.Ideal.Basic
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import Mathlib.RingTheory.Ideal.Operations
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import Mathlib.LinearAlgebra.Finsupp
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import Mathlib.RingTheory.GradedAlgebra.Basic
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import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
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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.Noetherian
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import Mathlib.RingTheory.Artinian
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import Mathlib.Order.Height
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import Mathlib.Algebra.Module.GradedModule
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import Mathlib.RingTheory.Noetherian
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import Mathlib.RingTheory.Finiteness
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import Mathlib.RingTheory.Ideal.Operations
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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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-- 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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def Ideal.IsHomogeneous' (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)]
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[DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)) := ∀ (i : ℤ ) ⦃r : (⨁ i, 𝒜 i)⦄, r ∈ I → DirectSum.of _ i ( r i : 𝒜 i) ∈ I
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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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| fail "No, this is not obvious."))
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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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-- Testing of Polynomial
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section Polynomial
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variable [Semiring ℕ]
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variable [Semiring ℤ]
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variable [Semiring ℚ]
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noncomputable section
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#check Polynomial
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#check Polynomial (ℚ)
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#check Polynomial.eval
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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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example (f : Polynomial ℚ) (hf : f = Polynomial.C (1 : ℚ)) : Polynomial.eval 2 f = 1 := by
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have : ∀ (q : ℚ), Polynomial.eval q f = 1 := by
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sorry
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obviously
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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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-- example (f : ℤ → ℤ) (hf : ∀ x, f x = x ^ 2) : Polynomial.eval 2 f = 4 := by
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-- sorry
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open GradedMonoid.GSmul
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-- degree of a constant function is ⊥ (is this same as -1 ???)
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#print Polynomial.degree_zero
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open DirectSum
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def F : Polynomial ℚ := Polynomial.C (2 : ℚ)
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#print F
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#check F
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#check Polynomial.degree F
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#check Polynomial.degree 0
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#check WithBot ℕ
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-- #eval Polynomial.degree F
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#check Polynomial.eval 1 F
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example : Polynomial.eval (100 : ℚ) F = (2 : ℚ) := by
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refine Iff.mpr (Rat.ext_iff (Polynomial.eval 100 F) 2) ?_
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simp only [Rat.ofNat_num, Rat.ofNat_den]
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rw [F]
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simp
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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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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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noncomputable def dimensionring { A: Type _}
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[CommRing A] := krullDim (PrimeSpectrum A)
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noncomputable def dimensionmodule ( A : Type _) (M : Type _)
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[CommRing A] [AddCommGroup M] [Module A M] := krullDim (PrimeSpectrum (A ⧸ ((⊤ : Submodule A M).annihilator)) )
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-- (∃ (i : ℤ ), ∃ (x : 𝒜 i), p = (Submodule.span (⨁ i, 𝒜 i) {x}).annihilator )
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-- lemma graded_local (𝒜 : ℤ → Type _) [SetLike (⨁ i, 𝒜 i)] (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] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) : ∃ ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := sorry
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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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-- Treat polynomial f ∈ ℚ[X] as a function f : ℚ → ℚ
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#check CoeFun
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theorem hilbert_polynomial (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] (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) (hilb : ℤ → ℤ)
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(Hhilb: hilbert_function 𝒜 𝓜 hilb)
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: PolyType hilb (d - 1) := by
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end section
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-- @[BH, 4.1.2]
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-- All the polynomials are in ℚ[X], all the functions are considered as ℤ → ℤ
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noncomputable section
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-- Polynomial type of degree d
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@[simp]
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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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section
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-- structure PolyType (f : ℤ → ℤ) where
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-- Poly : Polynomial ℤ
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-- d :
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-- N : ℤ
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-- Poly_equal : ∀ n ∈ ℤ → f n = Polynomial.eval n : ℤ Poly
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#check PolyType
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example (f : ℤ → ℤ) (hf : ∀ x, f x = x ^ 2) : PolyType f 2 := by
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unfold PolyType
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sorry
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-- use Polynomial.monomial (2 : ℤ) (1 : ℤ)
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-- have' := hf 0; ring_nf at this
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-- exact this
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end section
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-- Δ operator (of d times)
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@[simp]
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def Δ : (ℤ → ℤ) → ℕ → (ℤ → ℤ)
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| f, 0 => f
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| f, d + 1 => fun (n : ℤ) ↦ (Δ f d) (n + 1) - (Δ f d) (n)
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section
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-- def Δ (f : ℤ → ℤ) (d : ℕ) := fun (n : ℤ) ↦ f (n + 1) - f n
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-- def add' : ℕ → ℕ → ℕ
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-- | 0, m => m
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-- | n+1, m => (add' n m) + 1
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-- #eval add' 5 10
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#check Δ
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def f (n : ℤ) := n
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#eval (Δ f 1) 100
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-- #check (by (show_term unfold Δ) : Δ f 0=0)
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end section
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-- (NO need to prove) Constant polynomial function = constant function
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lemma Poly_constant (F : Polynomial ℚ) (c : ℚ) :
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(F = Polynomial.C c) ↔ (∀ r : ℚ, (Polynomial.eval r F) = c) := by
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constructor
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· intro h
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rintro r
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refine Iff.mpr (Rat.ext_iff (Polynomial.eval r F) c) ?_
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simp only [Rat.ofNat_num, Rat.ofNat_den]
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rw [h]
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simp
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· sorry
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-- PolyType 0 = constant function
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lemma PolyType_0 (f : ℤ → ℤ) : (PolyType f 0) ↔ (∃ (c : ℤ), ∃ (N : ℤ), ∀ (n : ℤ), (N ≤ n → f n = c) ∧ c ≠ 0) := by
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constructor
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· intro h
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rcases h with ⟨Poly, hN⟩
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rcases hN with ⟨N, hh⟩
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have H1 := λ n hn => (hh n hn).left
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have H2 := λ n hn => (hh n hn).right
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clear hh
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specialize H2 (N + 1)
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have this1 : Polynomial.degree Poly = 0 := by
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have : N ≤ N + 1 := by
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dsimp
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simp
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tauto
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have this2 : ∃ (c : ℤ), Poly = Polynomial.C (c : ℚ) := by
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have HH : ∃ (c : ℚ), Poly = Polynomial.C (c : ℚ) := by
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use Poly.coeff 0
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apply Polynomial.eq_C_of_degree_eq_zero
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exact this1
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cases' HH with c HHH
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have HHHH : ∃ (d : ℤ), d = c := by
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sorry
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cases' HHHH with d H5
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use d
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rw [H5]
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exact HHH
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clear this1
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rcases this2 with ⟨c, hthis2⟩
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use c
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use N
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intro n
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specialize H1 n
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constructor
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· intro HH1
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have this3 : f n = Polynomial.eval (n : ℚ) Poly := by
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tauto
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have this4 : Polynomial.eval (n : ℚ) Poly = c := by
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rw [hthis2]
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dsimp
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simp
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have this5 : f n = (c : ℚ) := by
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rw [←this4, this3]
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exact Iff.mp (Rat.coe_int_inj (f n) c) this5
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· sorry
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· intro h
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rcases h with ⟨c, N, aaa⟩
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let (Poly : Polynomial ℚ) := Polynomial.C (c : ℚ)
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use Poly
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use N
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intro n Nn
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specialize aaa n
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have this1 : c ≠ 0 → f n = c := by
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tauto
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constructor
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· sorry
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· sorry
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-- apply Polynomial.degree_C c
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-- Δ of d times maps polynomial of degree d to polynomial of degree 0
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lemma Δ_PolyType_d_to_PolyType_0 (f : ℤ → ℤ) (d : ℕ): PolyType f d → PolyType (Δ f d) 0 := by
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intro h
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rcases h with ⟨Poly, hN⟩
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rcases hN with ⟨N, hh⟩
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have H1 := λ n hn => (hh n hn).left
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have H2 := λ n hn => (hh n hn).right
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clear hh
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have HH2 : d = Polynomial.degree Poly := by
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sorry
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induction' d with d hd
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· rw [PolyType_0]
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sorry
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· sorry
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-- [BH, 4.1.2] (a) => (b)
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-- Δ^d f (n) = c for some nonzero integer c for n >> 0 → f is of polynomial type d
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lemma a_to_b (f : ℤ → ℤ) (d : ℕ) : (∃ (c : ℤ), ∃ (N : ℤ), ∀ (n : ℤ), ((N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0)) → PolyType f d := by
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intro h
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rcases h with ⟨c, N, hh⟩
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have H1 := λ n => (hh n).left
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have H2 := λ n => (hh n).right
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clear hh
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have H2 : c ≠ 0 := by
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tauto
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induction' d with d hd
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· rw [PolyType_0]
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use c
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use N
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tauto
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· sorry
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-- [BH, 4.1.2] (a) <= (b)
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-- f is of polynomial type d → Δ^d f (n) = c for some nonzero integer c for n >> 0
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lemma b_to_a (f : ℤ → ℤ) (d : ℕ) : PolyType f d → (∃ (c : ℤ), ∃ (N : ℤ), ∀ (n : ℤ), ((N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0)) := by
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intro h
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have : PolyType (Δ f d) 0 := by
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apply Δ_PolyType_d_to_PolyType_0
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exact h
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have this1 : (∃ (c : ℤ), ∃ (N : ℤ), ∀ (n : ℤ), ((N ≤ n → (Δ f d) n = c) ∧ c ≠ 0)) := by
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rw [←PolyType_0]
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exact this
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exact this1
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end
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-- @Additive lemma of length for a SES
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section
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-- variable {R M N : Type _} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N]
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-- (f : M →[R] N)
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open LinearMap
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-- variable {R M : Type _} [CommRing R] [AddCommGroup M] [Module R M]
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-- noncomputable def length := Set.chainHeight {M' : Submodule R M | M' < ⊤}
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-- Definitiion of the length of a module
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noncomputable def length (R M : Type _) [CommRing R] [AddCommGroup M] [Module R M] := Set.chainHeight {M' : Submodule R M | M' < ⊤}
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#check length ℤ ℤ
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-- #eval length ℤ ℤ
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-- @[ext]
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-- structure SES (R : Type _) [CommRing R] where
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-- A : Type _
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-- B : Type _
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-- C : Type _
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-- f : A →ₗ[R] B
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-- g : B →ₗ[R] C
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-- left_exact : LinearMap.ker f = 0
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-- middle_exact : LinearMap.range f = LinearMap.ker g
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-- right_exact : LinearMap.range g = C
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-- Definition of a SES (Short Exact Sequence)
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-- @[ext]
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structure SES {R A B C : Type _} [CommRing R] [AddCommGroup A] [AddCommGroup B]
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[AddCommGroup C] [Module R A] [Module R B] [Module R C]
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(f : A →ₗ[R] B) (g : B →ₗ[R] C)
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where
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left_exact : LinearMap.ker f = ⊥
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middle_exact : LinearMap.range f = LinearMap.ker g
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right_exact : LinearMap.range g = ⊤
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#check SES.right_exact
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#check SES
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-- Additive lemma
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lemma length_Additive (R A B C : Type _) [CommRing R] [AddCommGroup A] [AddCommGroup B] [AddCommGroup C] [Module R A] [Module R B] [Module R C]
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(f : A →ₗ[R] B) (g : B →ₗ[R] C)
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: (SES f g) → ((length R A) + (length R C) = (length R B)) := by
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intro h
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rcases h with ⟨left_exact, middle_exact, right_exact⟩
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sorry
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theorem hilbert_polynomial_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] (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) (hilb : ℤ → ℤ)
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: true := by
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sorry
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lemma ass_graded (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)) : (HomogeneousMax 𝒜 p) := by
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sorry
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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 x i}).annihilator)) := by
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sorry
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def standard_graded (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)) := (⨁ i, 𝒜 i)
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end section
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Reference in a new issue