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chelseaandmadrid 2023-06-14 22:10:22 -07:00
parent 1f7a809e2c
commit 5a86902118
1 changed files with 8 additions and 7 deletions
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CommAlg/final_hil_pol.lean
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@ -6,7 +6,6 @@ import Mathlib.RingTheory.Artinian
import Mathlib.Order.Height import Mathlib.Order.Height
-- Setting for "library_search" -- Setting for "library_search"
set_option maxHeartbeats 0 set_option maxHeartbeats 0
macro "ls" : tactic => `(tactic|library_search) macro "ls" : tactic => `(tactic|library_search)
@ -44,7 +43,7 @@ noncomputable def length ( A : Type _) (M : Type _)
[CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < ⊤} [CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < ⊤}
-- Make instance of M_i being an R_0-module -- 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] 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) [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) where smul x y := @Eq.rec ℤ (0+i) (fun a _ => 𝓜 a) (GradedMonoid.GSmul.smul x y) i (zero_add i)
@ -110,7 +109,7 @@ instance {𝒜 : ℤ → Type _} [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GComm
class StandardGraded {𝒜 : ℤ → Type _} [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] : Prop where class StandardGraded (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] : Prop where
gen_in_first_piece : gen_in_first_piece :
Algebra.adjoin (𝒜 0) (DirectSum.of _ 1 : 𝒜 1 →+ ⨁ i, 𝒜 i).range = (⊤ : Subalgebra (𝒜 0) (⨁ i, 𝒜 i)) Algebra.adjoin (𝒜 0) (DirectSum.of _ 1 : 𝒜 1 →+ ⨁ i, 𝒜 i).range = (⊤ : Subalgebra (𝒜 0) (⨁ i, 𝒜 i))
@ -189,10 +188,11 @@ 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) -- 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)] 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.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) [DirectSum.Gmodule 𝒜 𝓜] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) (fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d) (findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d)
(hilb : ℤ → ℤ) (Hhilb: hilbert_function 𝒜 𝓜 hilb) (hilb : ℤ → ℤ) (Hhilb: hilbert_function 𝒜 𝓜 hilb)
: PolyType hilb (d - 1) := by : PolyType hilb (d - 1) := by
sorry sorry
@ -203,7 +203,7 @@ theorem Hilbert_polynomial_d_ge_1_reduced
(d : ℕ) (d1 : 1 ≤ d) (d : ℕ) (d1 : 1 ≤ d)
(𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) [DirectSum.Gmodule 𝒜 𝓜] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) (fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d) (findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d)
(hilb : ℤ → ℤ) (Hhilb: hilbert_function 𝒜 𝓜 hilb) (hilb : ℤ → ℤ) (Hhilb: hilbert_function 𝒜 𝓜 hilb)
@ -217,7 +217,7 @@ 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 -- 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)] theorem Hilbert_polynomial_d_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) [DirectSum.Gmodule 𝒜 𝓜] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) (fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = 0) (findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = 0)
(hilb : ℤ → ℤ) (Hhilb : hilbert_function 𝒜 𝓜 hilb) (hilb : ℤ → ℤ) (Hhilb : hilbert_function 𝒜 𝓜 hilb)
@ -230,7 +230,7 @@ theorem Hilbert_polynomial_d_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [
theorem Hilbert_polynomial_d_0_reduced theorem Hilbert_polynomial_d_0_reduced
(𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) [DirectSum.Gmodule 𝒜 𝓜] (st: StandardGraded 𝒜) (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) (fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = 0) (findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = 0)
(hilb : ℤ → ℤ) (Hhilb : hilbert_function 𝒜 𝓜 hilb) (hilb : ℤ → ℤ) (Hhilb : hilbert_function 𝒜 𝓜 hilb)
@ -256,3 +256,4 @@ theorem Hilbert_polynomial_d_0_reduced