From 19ef96ef49406abd9d0f53db317961752a8738a4 Mon Sep 17 00:00:00 2001 From: Andre Date: Fri, 16 Jun 2023 18:22:23 -0400 Subject: [PATCH] added final_hil_pol --- CommAlg/final_hil_pol.lean | 207 +++++++++++++++++++++---------------- 1 file changed, 116 insertions(+), 91 deletions(-) diff --git a/CommAlg/final_hil_pol.lean b/CommAlg/final_hil_pol.lean index eff9302..fb79eab 100644 --- a/CommAlg/final_hil_pol.lean +++ b/CommAlg/final_hil_pol.lean @@ -4,6 +4,12 @@ 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 Mathlib.Algebra.Module.LinearMap +import Mathlib.Algebra.Field.Defs +import CommAlg.krull + -- Setting for "library_search" @@ -25,6 +31,7 @@ macro "obviously" : tactic => | ring; done; dbg_trace "it was ring" | trivial; done; dbg_trace "it was trivial" -- | nlinarith; done + | aesop; done; dbg_trace "it was aesop" | fail "No, this is not obvious.")) @@ -39,8 +46,7 @@ section -- Definition of polynomail of type d 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 -noncomputable def length ( A : Type _) (M : Type _) - [CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < ⊤} + -- Make instance of M_i being an R_0-module instance tada1 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜] @@ -67,15 +73,30 @@ 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 length ( A : Type _) (M : Type _) + [CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < ⊤} 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)) + + +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))) + + +lemma lengthfield ( k : Type _) [Field k] : length (k) (k) = 1 := by +sorry + + +lemma equaldim ( A : Type _) [CommRing A] (I : Ideal A): dimensionmodule (A) (A ⧸ I) = Ideal.krullDim (A ⧸ I) := by +sorry + +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 +sorry + + end @@ -108,6 +129,7 @@ instance {𝒜 : ℤ → Type _} [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GComm sorry) + class StandardGraded (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] : Prop where gen_in_first_piece : Algebra.adjoin (𝒜 0) (DirectSum.of _ 1 : 𝒜 1 →+ ⨁ i, 𝒜 i).range = (⊤ : Subalgebra (𝒜 0) (⨁ i, 𝒜 i)) @@ -120,66 +142,109 @@ def Component_of_graded_as_addsubgroup (𝒜 : ℤ → Type _) sorry -def graded_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) →ₗ[(⨁ i, 𝒜 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 +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) -#check graded_morphism +structure GradedLinearMap (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] + extends LinearMap (RingHom.id (⨁ i, 𝒜 i)) (⨁ i, 𝓜 i) (⨁ i, 𝓝 i) where + respects_grading (i : ℤ) (r : 𝓜 i) (j : ℤ) : j ≠ i → toFun (DirectSum.of _ i r) j = 0 -def graded_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] -(f : (⨁ i, 𝓜 i) →ₗ[(⨁ i, 𝒜 i)] (⨁ i, 𝓝 i)) -: IsLinearEquiv f := by - sorry --- f ∈ (⨁ i, 𝓜 i) ≃ₗ[(⨁ i, 𝒜 i)] (⨁ i, 𝓝 i) --- LinearEquivClass f (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) (⨁ i, 𝓝 i) --- #print IsLinearEquiv -#check graded_isomorphism +/-- `𝓜 →ᵍₗ[𝒜] 𝓝` denotes the type of graded `𝒜`-linear maps from `𝓜` to `𝓝`. -/ +notation:25 𝓜 " →ᵍₗ[" 𝒜:25 "] " 𝓝:0 => GradedLinearMap 𝒜 𝓜 𝓝 +structure GradedLinearEquiv (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] + extends (⨁ i, 𝓜 i) ≃ (⨁ i, 𝓝 i), 𝓜 →ᵍₗ[𝒜] 𝓝 +/-- `𝓜 ≃ᵍₗ[𝒜] 𝓝` denotes the type of graded `𝒜`-linear isomorphisms from `(⨁ i, 𝓜 i)` to `(⨁ i, 𝓝 i)`. -/ +notation:25 𝓜 " ≃ᵍₗ[" 𝒜:25 "] " 𝓝:0 => GradedLinearEquiv 𝒜 𝓜 𝓝 -def graded_submodule -(𝒜 : ℤ → Type _) (𝓜 : ℤ → Type u) (𝓝 : ℤ → Type u) -[∀ 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 +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) ∧ (GradedLinearMap 𝒜 𝓜 𝓝 toFun) +-- @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 -(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 +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.GCommRing (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by sorry --- -lemma sss - : true := by +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 +section +variable (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] + [LocalRing (𝒜 0)] (m : LocalRing.maximalIdeal (𝒜 0)) + +-- check if `Pi.Single` or something writes this more elegantly +def GradedOneComponent (i : ℤ) : Type _ := ite (i = 0) (𝒜 0 ⧸ LocalRing.maximalIdeal (𝒜 0)) PUnit + +instance (i : ℤ) : AddMonoid (GradedOneComponent 𝒜 i) := by + unfold GradedOneComponent + sorry -- split into 0 and nonzero cases and then `inferInstance` + +instance : DirectSum.Gmodule 𝒜 (GradedOneComponent 𝒜) := by sorry +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 + sorry + +end + +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))⧸p) →ₗ[(⨁ i, 𝒜 i)] (⨁ i, (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i))) := by + sorry + +def Is.Graded_local (𝒜 : ℤ → Type _) +[∀ i, AddCommGroup (𝒜 i)][DirectSum.GCommRing 𝒜] := ∃! ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) + +lemma hilfun_eq (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] (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 +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 - sorry + -- @Existence of a chain of submodules of graded submoduels of a f.g graded R-mod M @@ -213,11 +278,10 @@ lemma Associated_prime_of_graded_is_graded -- 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_d_ge_1 (d : ℕ) (d1 : 1 ≤ d) (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.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) - : PolyType hilb (d - 1) := by sorry @@ -228,7 +292,7 @@ theorem Hilbert_polynomial_d_ge_1_reduced (d : ℕ) (d1 : 1 ≤ d) (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.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) @@ -242,48 +306,9 @@ theorem Hilbert_polynomial_d_ge_1_reduced -- 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] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) +[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 - - --- (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.Gmodule 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 - - - - - - - - - - - - - - - - - - - - - - - - + sorry \ No newline at end of file