| import Mathlib |
|
|
| namespace OAI |
|
|
| /-! |
| # Domain properties of the two concrete auxiliary rings |
|
|
| The determinant equation and the quadric relation are irreducible primitive |
| linear polynomials in one of their variables. This file proves their |
| principal ideals prime, and hence the corresponding affine coordinate rings |
| are domains, over every unique factorization domain. |
| -/ |
|
|
| noncomputable section |
| namespace StableCoordinate.DeterminantDomain |
| open MvPolynomial |
|
|
| variable (k : Type*) [CommRing k] [IsDomain k] [UniqueFactorizationMonoid k] |
|
|
| / |
| def determinantRelation : MvPolynomial (Fin 5) k := |
| X 0 * X 3 - X 2 * X 1 - 1 |
|
|
| / |
| def quadricRelation : MvPolynomial (Fin 4) k := |
| X 0 * X 1 - X 2 * (X 2 + 1) |
|
|
| omit [UniqueFactorizationMonoid k] in |
| theorem determinantRelation_irreducible : Irreducible (determinantRelation k) := by |
| have hrel : IsRelPrime (X (0 : Fin 5) : MvPolynomial (Fin 5) k) |
| (-(X 2 * X 1 + 1)) := by |
| apply (X_prime.irreducible).isRelPrime_iff_not_dvd.mpr |
| intro h |
| have he := map_dvd (eval (fun _ : Fin 5 => (0 : k))) h |
| simp at he |
| have hvars : (3 : Fin 5) ∉ (-(X 2 * X 1 + 1) : MvPolynomial (Fin 5) k).vars := by |
| intro hv |
| rw [vars_neg] at hv |
| have hs := vars_add_subset (X 2 * X 1 : MvPolynomial (Fin 5) k) 1 hv |
| have hm : (3 : Fin 5) ∉ (X 2 * X 1 : MvPolynomial (Fin 5) k).vars := by |
| intro h |
| have h := vars_mul (X 2 : MvPolynomial (Fin 5) k) (X 1) h |
| simp at h |
| simp [hm] at hs |
| have h := irreducible_mul_X_add (X (0 : Fin 5) : MvPolynomial (Fin 5) k) |
| (-(X 2 * X 1 + 1)) 3 (X_ne_zero 0) (by simp) hvars hrel |
| convert h using 1 |
| simp only [determinantRelation] |
| ring |
|
|
| omit [UniqueFactorizationMonoid k] in |
| theorem quadricRelation_irreducible : Irreducible (quadricRelation k) := by |
| have hrel : IsRelPrime (X (0 : Fin 4) : MvPolynomial (Fin 4) k) |
| (-(X 2 * (X 2 + 1))) := by |
| apply (X_prime.irreducible).isRelPrime_iff_not_dvd.mpr |
| intro h |
| have h : (X (0 : Fin 4) : MvPolynomial (Fin 4) k) ∣ X 2 * (X 2 + 1) := by |
| simpa using h |
| rcases X_prime.dvd_mul.mp h with h | h |
| · simp [X_dvd_X] at h |
| · have he := map_dvd (eval (fun _ : Fin 4 => (0 : k))) h |
| simp at he |
| have hvars : (1 : Fin 4) ∉ (-(X 2 * (X 2 + 1)) : MvPolynomial (Fin 4) k).vars := by |
| intro hv |
| rw [vars_neg] at hv |
| have hm := vars_mul (X 2 : MvPolynomial (Fin 4) k) (X 2 + 1) hv |
| have ha : (1 : Fin 4) ∉ (X 2 + 1 : MvPolynomial (Fin 4) k).vars := by |
| intro h |
| have h := vars_add_subset (X 2 : MvPolynomial (Fin 4) k) 1 h |
| simp at h |
| simp [ha] at hm |
| simpa only [quadricRelation, sub_eq_add_neg] using |
| irreducible_mul_X_add (X (0 : Fin 4) : MvPolynomial (Fin 4) k) |
| (-(X 2 * (X 2 + 1))) 1 (X_ne_zero 0) (by simp) hvars hrel |
|
|
| / |
| theorem determinantIdeal_isPrime : |
| (Ideal.span {determinantRelation k}).IsPrime := |
| Ideal.isPrime_span_singleton_of_prime (determinantRelation_irreducible k).prime |
|
|
| / |
| theorem quadricIdeal_isPrime : |
| (Ideal.span {quadricRelation k}).IsPrime := |
| Ideal.isPrime_span_singleton_of_prime (quadricRelation_irreducible k).prime |
|
|
| / |
| abbrev DeterminantRing := |
| MvPolynomial (Fin 5) k ⧸ Ideal.span {determinantRelation k} |
|
|
| / |
| abbrev QuadricRing := |
| MvPolynomial (Fin 4) k ⧸ Ideal.span {quadricRelation k} |
|
|
| instance determinantRing_isDomain : IsDomain (DeterminantRing k) := |
| (Ideal.Quotient.isDomain_iff_prime _).mpr (determinantIdeal_isPrime k) |
|
|
| instance quadricRing_isDomain : IsDomain (QuadricRing k) := |
| (Ideal.Quotient.isDomain_iff_prime _).mpr (quadricIdeal_isPrime k) |
|
|
| end StableCoordinate.DeterminantDomain |
|
|
| end |
|
|
| end OAI |
|
|
