| import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal |
| import Mathlib.RingTheory.Ideal.Span |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.QuotientSections |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W64 |
|
|
| attribute [local instance] MvPolynomial.gradedAlgebra |
|
|
| variable {k σ : Type*} [Field k] |
|
|
| theorem homogeneous_mem_step_decomposition |
| (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| {d n : ℕ} (f p : MvPolynomial σ k) |
| (hf : f.IsHomogeneous d) (hp : p.IsHomogeneous (n + d)) |
| (hmem : p ∈ Ideal.span {f} ⊔ I) : |
| ∃ a : MvPolynomial σ k, a.IsHomogeneous n ∧ p - a * f ∈ I := by |
| obtain ⟨a, b, hb, hab⟩ := Ideal.mem_span_singleton_sup.mp hmem |
| let aₙ := DirectSum.decompose (MvPolynomial.homogeneousSubmodule σ k) a n |
| refine ⟨aₙ.1, aₙ.2, ?_⟩ |
| have hproj := congrArg |
| (GradedRing.proj (MvPolynomial.homogeneousSubmodule σ k) (n + d)) hab |
| rw [map_add, GradedRing.proj_apply, GradedRing.proj_apply, |
| GradedRing.proj_apply, |
| DirectSum.coe_decompose_mul_add_of_right_mem |
| (MvPolynomial.homogeneousSubmodule σ k) hf, |
| DirectSum.decompose_of_mem_same (MvPolynomial.homogeneousSubmodule σ k) hp] at hproj |
| have hb' := (Ideal.IsHomogeneous.mem_iff |
| (MvPolynomial.homogeneousSubmodule σ k) hI).mp hb (n + d) |
| have heq : p - aₙ.1 * f = |
| (DirectSum.decompose (MvPolynomial.homogeneousSubmodule σ k) b (n + d) : |
| MvPolynomial σ k) := by |
| apply sub_eq_iff_eq_add.mpr |
| simpa only [aₙ, add_comm] using hproj.symm |
| rw [heq] |
| exact hb' |
|
|
| theorem homogeneous_mem_step_of_degree_lt |
| (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| {d n : ℕ} (f p : MvPolynomial σ k) |
| (hf : f.IsHomogeneous d) (hp : p.IsHomogeneous n) (hnd : n < d) |
| (hmem : p ∈ Ideal.span {f} ⊔ I) : p ∈ I := by |
| obtain ⟨a, b, hb, hab⟩ := Ideal.mem_span_singleton_sup.mp hmem |
| have hproj := congrArg |
| (GradedRing.proj (MvPolynomial.homogeneousSubmodule σ k) n) hab |
| rw [map_add, GradedRing.proj_apply, GradedRing.proj_apply, |
| GradedRing.proj_apply, |
| DirectSum.coe_decompose_mul_of_right_mem_of_not_le |
| (MvPolynomial.homogeneousSubmodule σ k) hf (Nat.not_le_of_lt hnd), |
| zero_add, |
| DirectSum.decompose_of_mem_same (MvPolynomial.homogeneousSubmodule σ k) hp] at hproj |
| rw [← hproj] |
| exact (Ideal.IsHomogeneous.mem_iff |
| (MvPolynomial.homogeneousSubmodule σ k) hI).mp hb n |
|
|
| end PiExponentJets.W64 |
|
|
| end OAI |
|
|
