openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /EquidimensionalRetainedCut.lean
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeHypersurfaceEquidimension | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.RetainedMinimalPrimeFamily | |
| namespace OAI | |
| namespace PiExponentJets.W22.RetainedPrimeCycle | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| universe u | |
| variable {k σ : Type u} [Field k] [Fintype σ] | |
| theorem retainedChild_dimension | |
| (C : RetainedPrimeCycle.{u,u,u} k σ) (f : MvPolynomial σ k) | |
| (havoid : ∀ a, f ∉ (C.component a).val) (s : ℕ) | |
| (hparent : ∀ a, | |
| (actualHP (C.component a).val (C.component a).homogeneous).natDegree = s+1) | |
| (T : Ideal (MvPolynomial σ k)) (a : C.Index) | |
| (Q : RetainedMinimalPrime (Ideal.span {f} ⊔ (C.component a).val) T) : | |
| ringKrullDim (MvPolynomial σ k ⧸ Q.val) = (s+1 : ℕ) := by | |
| obtain ⟨N, hN⟩ := actualHP_eventually (C.component a).val (C.component a).homogeneous | |
| have hPdim := W24.prime_hilbert_natDegree_add_one_eq_krullDim (C.component a).val | |
| (C.component a).homogeneous (C.component a).coordinate (C.component a).coordinate_not_mem | |
| (actualHP (C.component a).val (C.component a).homogeneous) (N+1) | |
| (fun n hn => (hN n (by omega)).symm) | |
| have hdim : ringKrullDim (MvPolynomial σ k ⧸ (C.component a).val) = (s+2 : ℕ) := by | |
| calc | |
| _ = ((actualHP (C.component a).val (C.component a).homogeneous).natDegree + 1 : | |
| WithBot ℕ∞) := hPdim.symm | |
| _ = ((s+2 : ℕ) : WithBot ℕ∞) := by | |
| rw [← Nat.cast_one, ← Nat.cast_add] | |
| exact congrArg (fun n : ℕ => (n : WithBot ℕ∞)) (by rw [hparent a]) | |
| exact W24.minimal_prime_hypersurface_dimension k σ (C.component a).val Q.val f | |
| (havoid a) (by simpa only [sup_comm] using Q.property.1) s hdim | |
| noncomputable def equidimensionalRetainedCut | |
| (C : RetainedPrimeCycle.{u,u,u} k σ) | |
| (f : MvPolynomial σ k) (d : ℕ) (hd : 0 < d) (hf : f.IsHomogeneous d) | |
| (havoid : ∀ a, f ∉ (C.component a).val) | |
| (s : ℕ) (hparent : ∀ a, | |
| (actualHP (C.component a).val (C.component a).homogeneous).natDegree = s+1) | |
| (T : Ideal (MvPolynomial σ k)) (v : σ) (hv : MvPolynomial.X v ∉ T) : ProperCut C := | |
| canonicalRetainedCut C f d hd hf havoid s hparent T v hv | |
| (retainedChild_dimension C f havoid s hparent T) | |
| theorem equidimensionalRetainedCut_covers | |
| (C : RetainedPrimeCycle.{u,u,u} k σ) | |
| (f : MvPolynomial σ k) (d : ℕ) (hd : 0 < d) (hf : f.IsHomogeneous d) | |
| (havoid : ∀ a, f ∉ (C.component a).val) | |
| (s : ℕ) (hparent : ∀ a, | |
| (actualHP (C.component a).val (C.component a).homogeneous).natDegree = s+1) | |
| (T : Ideal (MvPolynomial σ k)) (v : σ) (hv : MvPolynomial.X v ∉ T) : | |
| (equidimensionalRetainedCut C f d hd hf havoid s hparent T v hv).CoversBelow T := | |
| canonicalRetainedCut_covers C f d hd hf havoid s hparent T v hv | |
| (retainedChild_dimension C f havoid s hparent T) | |
| end PiExponentJets.W22.RetainedPrimeCycle | |
| end OAI | |