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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 | 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
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