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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 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 90 91 92 93 94 95 96 97 | import Mathlib.Algebra.BigOperators.Fin
import OAI.NumberTheory.PiExponent.Geometry.NormalizationGrowth
import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousQuotientDimension
namespace OAI
noncomputable section
namespace PiExponentJets.W25
open MvPolynomial DirectSum
open PiExponentJets.W24 PiExponentJets.W64
attribute [local instance] MvPolynomial.gradedAlgebra
variable {k σ : Type*} [Field k]
theorem restrictTotalDegree_eq_iSup_homogeneous (n : ℕ) :
restrictTotalDegree σ k n =
⨆ j : Fin (n + 1), homogeneousSubmodule σ k j.val := by
classical
apply le_antisymm
· intro p hp
have hpdegree : p.totalDegree ≤ n := (mem_restrictTotalDegree σ _ _).mp hp
have hsum : (∑ j ∈ Finset.range (p.totalDegree + 1), homogeneousComponent j p) ∈
⨆ j : Fin (n + 1), homogeneousSubmodule σ k j.val := by
apply Submodule.sum_mem
intro j hj
have hjdegree : j ≤ n :=
(Nat.le_of_lt_succ (Finset.mem_range.mp hj)).trans hpdegree
exact (le_iSup (fun a : Fin (n + 1) => homogeneousSubmodule σ k a.val)
⟨j, Nat.lt_succ_of_le hjdegree⟩) (homogeneousComponent_mem j p)
simpa only [sum_homogeneousComponent] using hsum
· apply iSup_le
intro j p hp
apply (mem_restrictTotalDegree σ _ _).mpr
exact (show p.IsHomogeneous j.val from hp).totalDegree_le.trans
(Nat.le_of_lt_succ j.isLt)
theorem quotientDegreeFiltration_eq_iSup_sections
(I : Ideal (MvPolynomial σ k)) (n : ℕ) :
quotientDegreeFiltration I n =
⨆ j : Fin (n + 1), quotientSection I j.val := by
unfold quotientDegreeFiltration quotientSection
rw [restrictTotalDegree_eq_iSup_homogeneous, Submodule.map_iSup]
theorem cumulativeSectionMap_injective (I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ) :
Function.Injective
(DirectSum.coeLinearMap (fun j : Fin (n + 1) => quotientSection I j.val)) :=
((quotientSection_iSupIndep I hI).comp Fin.val_injective).dfinsupp_lsum_injective
theorem cumulativeSectionMap_range (I : Ideal (MvPolynomial σ k)) (n : ℕ) :
LinearMap.range
(DirectSum.coeLinearMap (fun j : Fin (n + 1) => quotientSection I j.val)) =
quotientDegreeFiltration I n := by
rw [DirectSum.range_coeLinearMap, quotientDegreeFiltration_eq_iSup_sections]
def cumulativeHomogeneousQuotientEquiv (I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ) :
(⨁ j : Fin (n + 1), quotientSection I j.val) ≃ₗ[k]
quotientDegreeFiltration I n :=
(LinearEquiv.ofInjective
(DirectSum.coeLinearMap (fun j : Fin (n + 1) => quotientSection I j.val))
(cumulativeSectionMap_injective I hI n)).trans
(LinearEquiv.ofEq _ _ (cumulativeSectionMap_range I n))
@[simp] theorem cumulativeHomogeneousQuotientEquiv_apply
(I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ)
(x : ⨁ j : Fin (n + 1), quotientSection I j.val) :
(cumulativeHomogeneousQuotientEquiv I hI n x : MvPolynomial σ k ⧸ I) =
DirectSum.coeLinearMap (fun j : Fin (n + 1) => quotientSection I j.val) x := rfl
variable [Finite σ]
theorem quotientDegreeFiltration_finrank_eq_sum_fin
(I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ) :
Module.finrank k (quotientDegreeFiltration I n) =
∑ j : Fin (n + 1), Module.finrank k (quotientSection I j.val) := by
rw [← (cumulativeHomogeneousQuotientEquiv I hI n).finrank_eq, Module.finrank_directSum]
theorem quotientDegreeFiltration_finrank_eq_sum
(I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ) :
Module.finrank k (quotientDegreeFiltration I n) =
∑ j ∈ Finset.range (n + 1), Module.finrank k (quotientSection I j) := by
rw [quotientDegreeFiltration_finrank_eq_sum_fin I hI n]
exact Fin.sum_univ_eq_sum_range
(fun j : ℕ => Module.finrank k (quotientSection I j)) (n + 1)
end PiExponentJets.W25
end
end OAI
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