import Mathlib.Order.KrullDimension import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic import Mathlib.Algebra.Module.GradedModule import Mathlib.RingTheory.Ideal.AssociatedPrime import Mathlib.RingTheory.Artinian import Mathlib.Order.Height -- Setting for "library_search" set_option maxHeartbeats 0 macro "ls" : tactic => `(tactic|library_search) -- New tactic "obviously" macro "obviously" : tactic => `(tactic| ( first | dsimp; simp; done; dbg_trace "it was dsimp simp" | simp; done; dbg_trace "it was simp" | tauto; done; dbg_trace "it was tauto" | simp; tauto; done; dbg_trace "it was simp tauto" | rfl; done; dbg_trace "it was rfl" | norm_num; done; dbg_trace "it was norm_num" | /-change (@Eq ℝ _ _);-/ linarith; done; dbg_trace "it was linarith" -- | gcongr; done | ring; done; dbg_trace "it was ring" | trivial; done; dbg_trace "it was trivial" -- | nlinarith; done | fail "No, this is not obvious.")) open GradedMonoid.GSmul open DirectSum -- @Definitions (to be classified) 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] [DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ) : SMul (𝒜 0) (𝓜 i) where smul x y := @Eq.rec ℤ (0+i) (fun a _ => 𝓜 a) (GradedMonoid.GSmul.smul x y) i (zero_add i) lemma mylem (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜] [h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ) (a : 𝒜 0) (m : 𝓜 i) : of _ _ (a • m) = of _ _ a • of _ _ m := by refine' Eq.trans _ (Gmodule.of_smul_of 𝒜 𝓜 a m).symm refine' of_eq_of_gradedMonoid_eq _ exact Sigma.ext (zero_add _).symm <| eq_rec_heq _ _ instance tada2 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜] [h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ) : SMulWithZero (𝒜 0) (𝓜 i) := by letI := SMulWithZero.compHom (⨁ i, 𝓜 i) (of 𝒜 0).toZeroHom exact Function.Injective.smulWithZero (of 𝓜 i).toZeroHom Dfinsupp.single_injective (mylem 𝒜 𝓜 i) instance tada3 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜] [h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ): Module (𝒜 0) (𝓜 i) := by letI := Module.compHom (⨁ j, 𝓜 j) (ofZeroRingHom 𝒜) exact Dfinsupp.single_injective.module (𝒜 0) (of 𝓜 i) (mylem 𝒜 𝓜 i) -- Definition of a Hilbert function of a graded module section noncomputable def hilbert_function (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] (hilb : ℤ → ℤ) := ∀ i, hilb i = (ENat.toNat (length (𝒜 0) (𝓜 i))) noncomputable def dimensionring { A: Type _} [CommRing A] := krullDim (PrimeSpectrum A) noncomputable def dimensionmodule ( A : Type _) (M : Type _) [CommRing A] [AddCommGroup M] [Module A M] := krullDim (PrimeSpectrum (A ⧸ ((⊤ : Submodule A M).annihilator)) ) end -- Definition of homogeneous ideal def Ideal.IsHomogeneous' (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)) := ∀ (i : ℤ ) ⦃r : (⨁ i, 𝒜 i)⦄, r ∈ I → DirectSum.of _ i ( r i : 𝒜 i) ∈ I -- Definition of homogeneous prime ideal def HomogeneousPrime (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous' 𝒜 I) -- Definition of homogeneous maximal ideal def HomogeneousMax (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous' 𝒜 I) --theorem monotone_stabilizes_iff_noetherian : -- (∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by -- rw [isNoetherian_iff_wellFounded, WellFounded.monotone_chain_condition] instance {𝒜 : ℤ → Type _} [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] : Algebra (𝒜 0) (⨁ i, 𝒜 i) := Algebra.ofModule' (by intro r x sorry) (by intro r x 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)) -- Each component of a graded ring is an additive subgroup def Component_of_graded_as_addsubgroup (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) (i : ℤ) : AddSubgroup (𝒜 i) := by 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 #check graded_morphism 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 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 end -- @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 sorry -- lemma sss : true := 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 lemma Exist_chain_of_graded_submodules (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] (fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) : ∃ (c : List (Submodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))), c.Chain' (· < ·) ∧ ∀ M ∈ c, Ture := by sorry -- @[BH, 1.5.6 (b)(ii)] -- An associated prime of a graded R-Mod M is graded lemma Associated_prime_of_graded_is_graded (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] (p : associatedPrimes (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) : (Ideal.IsHomogeneous' 𝒜 p) ∧ ((∃ (i : ℤ ), ∃ (x : 𝒜 i), p = (Submodule.span (⨁ i, 𝒜 i) {DirectSum.of _ i x}).annihilator)) := by sorry -- @[BH, 4.1.3] when d ≥ 1 -- 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)) (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 -- (reduced version) [BH, 4.1.3] when d ≥ 1 -- 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 (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)) (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] (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) : (∃ (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