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added graded morphism def

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monula95 dutta 2023-06-15 04:19:56 +00:00
parent 79a6844707
commit 37cc2c5a3c
1 changed files with 36 additions and 8 deletions
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@ -4,6 +4,18 @@ import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Ideal.AssociatedPrime
import Mathlib.RingTheory.Artinian
import Mathlib.Order.Height
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Module.LinearMap
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)
noncomputable def length ( A : Type _) (M : Type _)
[CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < ⊤}
@ -96,13 +108,28 @@ lemma Associated_prime_of_graded_is_graded
sorry
-- def standard_graded {𝒜 : ℤ → Type _} [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (n : ℕ) :
-- Prop :=
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))
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) := 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) → (⨁ 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_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
-- @ 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
@ -111,12 +138,13 @@ instance Quotient_of_graded_is_graded
: DirectSum.Gmodule 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
sorry
instance 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]
(opn : Submodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) (opnis : opn = (⨁ i, 𝓝 i))
: (𝓝 i : Submodule (𝒜 0) (𝓜 i)) := by
theorem quotient_hilbert_polynomial (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] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) (p : Ideal (⨁ i, 𝒜 i))
(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, ((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) = d) (hilb : ℤ → ℤ)
(Hhilb: hilbert_function 𝒜 𝓜 hilb) (homprime: HomogeneousPrime 𝒜 p)
: PolyType hilb (d - 1) := by
sorry