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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | 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
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