added final_hil_pol

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 _}
+noncomputable def length ( A : Type _) (M : Type _)
- [CommRing A] := krullDim (PrimeSpectrum A)
+ [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 _)
+def graded_ring_morphism (𝒜 : ℤ → Type _) (ℬ : ℤ → Type _)
-[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
+[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (ℬ i)]
-[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝]
+[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)
-(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
+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 _)
+/-- `𝓜 →ᵍₗ[𝒜] 𝓝` denotes the type of graded `𝒜`-linear maps from `𝓜` to `𝓝`. -/
-[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
+notation:25 𝓜 " →ᵍₗ[" 𝒜:25 "] " 𝓝:0 => GradedLinearMap 𝒜 𝓜 𝓝
-[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
 
+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
+def graded_ring_isomorphism (𝒜 : ℤ → Type _) (𝓑 : ℤ → Type _)
-(𝒜 : ℤ → Type _) (𝓜 : ℤ → Type u) (𝓝 : ℤ → Type u)
+[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓑 i)]
-[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
+[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓑]
-[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝]
+(f : (⨁ i, 𝒜 i) →+*  (⨁ i, 𝓑 i))
-(opn : Submodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) (opnis : opn = (⨁ i, 𝓝 i)) (i : ℤ )
+:= (graded_ring_morphism 𝒜 𝓑 f) ∧ (Function.Bijective f)
- : ∃(piece : Submodule (𝒜 0) (𝓜 i)), piece = 𝓝 i := by
+
-  sorry
+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
 
-
+instance Quotient_of_graded_gradedring
-
+    (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
--- @Quotient of a graded ring R by a graded ideal p is a graded R-Mod, preserving each component
+    (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) :
-instance Quotient_of_graded_is_graded
+    DirectSum.GCommRing (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
-(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
 
--- 
+instance Quotient_of_graded_is_gradedalg
-lemma sss
+    (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
-  : true := by
+    (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
+  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
-
-
-
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