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udpated def final

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monula95 dutta 2023-06-14 04:49:17 +00:00
parent d30702fb10
commit b6bbf7af8c
1 changed files with 8 additions and 2 deletions
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@ -90,16 +90,17 @@ instance tada3 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGr
-- (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) -- (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
noncomputable def hilbert_function (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] noncomputable def dummyhil_function (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (hilb : ℤ → ℕ∞) := ∀ i, hilb i = (length (𝒜 0) (𝓜 i)) [DirectSum.Gmodule 𝒜 𝓜] (hilb : ℤ → ℕ∞) := ∀ i, hilb i = (length (𝒜 0) (𝓜 i))
lemma hilbertz (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] lemma hilbertz (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] [DirectSum.Gmodule 𝒜 𝓜]
(finlen : ∀ i, (length (𝒜 0) (𝓜 i)) < ⊤ ) : ℤ → ℤ := by (finlen : ∀ i, (length (𝒜 0) (𝓜 i)) < ⊤ ) : ℤ → ℤ := by
intro i intro i
let h := hilbert_function 𝒜 𝓜 let h := dummyhil_function 𝒜 𝓜
simp at h simp at h
let n : ℤ → ℕ := fun i ↦ WithTop.untop _ (finlen i).ne let n : ℤ → ℕ := fun i ↦ WithTop.untop _ (finlen i).ne
have hn : ∀ i, (n i : ℕ∞) = length (𝒜 0) (𝓜 i) := fun i ↦ WithTop.coe_untop _ _ have hn : ∀ i, (n i : ℕ∞) = length (𝒜 0) (𝓜 i) := fun i ↦ WithTop.coe_untop _ _
@ -107,6 +108,11 @@ lemma hilbertz (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGr
exact ((n i) : ℤ ) exact ((n i) : ℤ )
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 dimensionring { A: Type _}
[CommRing A] := krullDim (PrimeSpectrum A) [CommRing A] := krullDim (PrimeSpectrum A)