File size: 3,938 Bytes
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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import Lean.Elab.Tactic.Omega
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.RingTheory.Ideal.Colon
import OAI.NumberTheory.PiExponent.Analysis.HilbertSeries
namespace OAI
namespace PiExponentJets.W64
attribute [local instance] MvPolynomial.gradedAlgebra
variable {k σ : Type*} [Field k] [Finite σ]
omit [Finite σ] in
theorem ideal_le_colon_singleton (I : Ideal (MvPolynomial σ k))
(f : MvPolynomial σ k) : I ≤ I.colon {f} := by
intro p hp
apply Submodule.mem_colon_singleton.mpr
simpa only [smul_eq_mul] using I.mul_mem_right f hp
omit [Finite σ] in
theorem quotientSectionFactor_colon_ker (I : Ideal (MvPolynomial σ k))
{d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) (n : ℕ) :
LinearMap.ker (quotientSectionFactor I (I.colon {f})
(ideal_le_colon_singleton I f) n) =
LinearMap.ker (quotientSectionMultiply I f hf n) := by
ext x
obtain ⟨p, hp, hpx⟩ := Submodule.mem_map.mp x.2
change Ideal.Quotient.mk I p = x.1 at hpx
simp only [LinearMap.mem_ker, Subtype.ext_iff]
change (Ideal.Quotient.factorₐ k (ideal_le_colon_singleton I f)) x.1 = 0 ↔
x.1 * Ideal.Quotient.mk I f = 0
rw [← hpx]
change Ideal.Quotient.mk (I.colon {f}) p = 0 ↔
Ideal.Quotient.mk I (p * f) = 0
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.Quotient.eq_zero_iff_mem,
Submodule.mem_colon_singleton, smul_eq_mul]
theorem quotientSectionMultiply_range_finrank_colon
(I : Ideal (MvPolynomial σ k)) {d : ℕ}
(f : MvPolynomial σ k) (hf : f.IsHomogeneous d) (n : ℕ) :
Module.finrank k (LinearMap.range (quotientSectionMultiply I f hf n)) =
Module.finrank k (quotientSection (I.colon {f}) n) := by
let φ := quotientSectionFactor I (I.colon {f}) (ideal_le_colon_singleton I f) n
have hd := (LinearMap.ker φ).finrank_quotient_add_finrank
let e := φ.quotKerEquivOfSurjective
(quotientSectionFactor_surjective I (I.colon {f}) (ideal_le_colon_singleton I f) n)
rw [e.finrank_eq] at hd
change Module.finrank k (quotientSection (I.colon {f}) n) +
Module.finrank k (LinearMap.ker (quotientSectionFactor I (I.colon {f})
(ideal_le_colon_singleton I f) n)) = _ at hd
rw [quotientSectionFactor_colon_ker I f hf n] at hd
have hm := (quotientSectionMultiply I f hf n).finrank_range_add_finrank_ker
omega
theorem colon_hilbert_step (I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
{d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) (n : ℕ) :
Module.finrank k (quotientSection (Ideal.span {f} ⊔ I) (n+d)) +
Module.finrank k (quotientSection (I.colon {f}) n) =
Module.finrank k (quotientSection I (n+d)) := by
have hd := (LinearMap.range (quotientSectionMultiply I f hf n)).finrank_quotient_add_finrank
rw [quotientSectionMultiply_range_finrank_colon I f hf n,
← quotientSectionFactor_ker_eq I hI f hf n] at hd
let e := (quotientSectionFactor I (Ideal.span {f} ⊔ I) le_sup_right (n+d)).quotKerEquivOfSurjective
(quotientSectionFactor_surjective I (Ideal.span {f} ⊔ I) le_sup_right (n+d))
rw [e.finrank_eq] at hd
exact hd
theorem sectionHilbertSeries_colon_step (I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
{d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) :
sectionHilbertSeries I = sectionHilbertSeries (Ideal.span {f} ⊔ I) +
PowerSeries.X ^ d * sectionHilbertSeries (I.colon {f}) := by
ext n
rw [map_add, PowerSeries.coeff_X_pow_mul']
simp only [coeff_sectionHilbertSeries]
by_cases hdn : d ≤ n
· rw [ite_eq_left hdn]
have h := colon_hilbert_step I hI f hf (n-d)
rw [Nat.sub_add_cancel hdn] at h
exact_mod_cast h.symm
· rw [ite_eq_right hdn, add_zero]
exact congrArg (fun m : ℕ => (m : ℤ))
(hilbert_step_of_degree_lt I hI f hf (Nat.lt_of_not_ge hdn)).symm
end PiExponentJets.W64
end OAI
|