| import Mathlib.Order.Minimal | |
| import Mathlib.RingTheory.GradedAlgebra.Radical | |
| import Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | |
| import Mathlib.RingTheory.MvPolynomial.Homogeneous | |
| import Mathlib.RingTheory.Polynomial.Basic | |
| import Mathlib.RingTheory.PrincipalIdealDomain | |
| namespace OAI | |
| namespace PiExponentJets.W22 | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {k σ : Type*} [Field k] [Finite σ] | |
| omit [Finite σ] in | |
| theorem homogeneous_colon_singleton | |
| (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) | |
| {d : ℕ} {f : MvPolynomial σ k} (hf : f.IsHomogeneous d) : | |
| (I.colon {f}).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) := by | |
| intro n a ha | |
| apply Submodule.mem_colon_singleton.mpr | |
| have haf : a * f ∈ I := by simpa only [smul_eq_mul] using | |
| (Submodule.mem_colon_singleton.mp ha) | |
| have h := hI (n + d) haf | |
| rw [DirectSum.coe_decompose_mul_add_of_right_mem | |
| (MvPolynomial.homogeneousSubmodule σ k) hf] at h | |
| exact h | |
| theorem exists_homogeneous_prime_colon | |
| (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) | |
| (hproper : I ≠ ⊤) : | |
| ∃ (d : ℕ) (f : MvPolynomial σ k), f.IsHomogeneous d ∧ f ∉ I ∧ | |
| (I.colon {f}).IsPrime := by | |
| classical | |
| let candidates : Ideal (MvPolynomial σ k) → Prop := fun P => | |
| ∃ d f, f.IsHomogeneous d ∧ f ∉ I ∧ P = I.colon {f} | |
| have hnonempty : ∃ P, candidates P := by | |
| refine ⟨I.colon {(1 : MvPolynomial σ k)}, 0, 1, ?_, ?_, rfl⟩ | |
| · exact MvPolynomial.isHomogeneous_one σ k | |
| · exact fun h => hproper (Ideal.eq_top_of_isUnit_mem I h isUnit_one) | |
| obtain ⟨P, hP, hmax⟩ := exists_maximal_of_wellFoundedGT candidates hnonempty | |
| obtain ⟨d, f, hf, hfI, rfl⟩ := hP | |
| refine ⟨d, f, hf, hfI, ?_⟩ | |
| apply (homogeneous_colon_singleton I hI hf).isPrime_of_homogeneous_mem_or_mem | |
| · intro htop | |
| have h : (1 : MvPolynomial σ k) ∈ I.colon {f} := by | |
| rw [htop] | |
| trivial | |
| exact hfI (by simpa only [Submodule.mem_colon_singleton, one_smul] using h) | |
| · intro a b ha hb hab | |
| by_cases hbI : b ∈ I.colon {f} | |
| · exact Or.inr hbI | |
| · left | |
| obtain ⟨e, he⟩ := hb | |
| have hb' : b.IsHomogeneous e := he | |
| have hbf : (b * f).IsHomogeneous (e + d) := hb'.mul hf | |
| have hbfI : b * f ∉ I := by | |
| simpa only [Submodule.mem_colon_singleton, smul_eq_mul] using hbI | |
| have hle : I.colon {f} ≤ I.colon {b * f} := by | |
| intro r hr | |
| apply Submodule.mem_colon_singleton.mpr | |
| have hrf : r * f ∈ I := by | |
| simpa only [smul_eq_mul] using (Submodule.mem_colon_singleton.mp hr) | |
| simpa only [smul_eq_mul, mul_left_comm r b f] using I.mul_mem_left b hrf | |
| have hback : I.colon {b * f} ≤ I.colon {f} := | |
| hmax ⟨e + d, b * f, hbf, hbfI, rfl⟩ hle | |
| apply hback | |
| apply Submodule.mem_colon_singleton.mpr | |
| simpa only [smul_eq_mul, mul_assoc] using (Submodule.mem_colon_singleton.mp hab) | |
| end PiExponentJets.W22 | |
| end OAI | |