| import OAI.AlgebraicGeometry.StableCoordinate.DeterminantPrimeA |
| import OAI.AlgebraicGeometry.StableCoordinate.DeterminantCharts |
| import Mathlib.RingTheory.Ideal.UFD |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| The prime element `a` has a localization which is a polynomial algebra over a |
| localization of a polynomial algebra. Nagata's prime-element criterion then |
| proves unique factorization in the actual determinant quotient. |
| -/ |
|
|
| namespace StableCoordinate.DeterminantRing |
|
|
| variable (k : Type*) [CommRing k] [IsNoetherianRing k] |
|
|
| variable [IsDomain k] [UniqueFactorizationMonoid k] |
| |
| instance C_isNoetherianRing : IsNoetherianRing (C k) := inferInstance |
|
|
| noncomputable instance C_uniqueFactorizationMonoid : UniqueFactorizationMonoid (C k) := by |
| apply (UniqueFactorizationMonoid.iff_localizationAway_of_prime (a_prime k)).mpr |
| exact (chartEquivA k).symm.toMulEquiv.uniqueFactorizationMonoid inferInstance |
|
|
| end StableCoordinate.DeterminantRing |
|
|
| namespace StableCoordinate.Obstruction |
|
|
| variable (k : Type*) [CommRing k] [IsNoetherianRing k] |
| |
| instance C_isNoetherianRing : IsNoetherianRing (C k) := inferInstance |
|
|
| variable [IsDomain k] [UniqueFactorizationMonoid k] |
|
|
| /-- The canonical obstruction presentation is definitionally the determinant ring. -/ |
| noncomputable instance C_uniqueFactorizationMonoid : UniqueFactorizationMonoid (C k) := |
| DeterminantRing.C_uniqueFactorizationMonoid k |
|
|
| end StableCoordinate.Obstruction |
|
|
| end OAI |
|
|
