From bbfba6a2f722ecd933ed6feb4494885eb28e9675 Mon Sep 17 00:00:00 2001
From: monula95 dutta <monula95@gmail.com>
Date: Wed, 14 Jun 2023 21:50:00 +0000
Subject: [PATCH] added graded submodule
---
CommAlg/monalisa.lean | 25 +++++++++++++++++++++++--
1 file changed, 23 insertions(+), 2 deletions(-)
diff --git a/CommAlg/monalisa.lean b/CommAlg/monalisa.lean
index 68579da..4d5f4d8 100644
--- a/CommAlg/monalisa.lean
+++ b/CommAlg/monalisa.lean
@@ -58,7 +58,7 @@ noncomputable def dimensionring { A: Type _}
noncomputable def dimensionmodule ( A : Type _) (M : Type _)
[CommRing A] [AddCommGroup M] [Module A M] := krullDim (PrimeSpectrum (A ⧸ ((⊤ : Submodule A M).annihilator)) )
--- (∃ (i : ℤ ), ∃ (x : 𝒜 i), p = (Submodule.span (⨁ i, 𝒜 i) {x}).annihilator )
+
-- lemma graded_local (𝒜 : ℤ → Type _) [SetLike (⨁ i, 𝒜 i)] (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
-- [DirectSum.GCommRing 𝒜]
@@ -98,4 +98,25 @@ lemma Associated_prime_of_graded_is_graded
-- def standard_graded {𝒜 : ℤ → Type _} [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (n : ℕ) :
-- Prop :=
--- ∃ J, Ideal.IsHomogeneous' 𝒜 J (J :Nonempty ((⨁ i, 𝒜 i) ≃+* (MvPolynomial (Fin n) (𝒜 0)) ⧸ J)
+
+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
+
+
+
+-- @ 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
+
+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
+ sorry
+
+