Coprimary Filtration to Harder Narasimhan Filtration
HarderNarasimhan.impl.CoprimaryFiltration.toHarderNarasimhanFiltration
Plain-language statement
Any coprimary filtration underlies a Harder–Narasimhan filtration. We reuse the same filtration function and verify the Harder–Narasimhan axioms: piecewise semistability (via rmk4d14₂ and semistable_res_iff_semistable_quot) and strict decrease of the minimal associated primes.
Exact Lean statement
lemma CoprimaryFiltration.toHarderNarasimhanFiltration {R : Type*} [CommRing R] [IsNoetherianRing R]
{M : Type*} [Nontrivial M] [AddCommGroup M] [Module R M] [Module.Finite R M]
(a : CoprimaryFiltration R M) :
∃ HNFil : HarderNarasimhanFiltration (μ R M), a.filtration = HNFil.filtrationFormal artifact
Lean source
lemma CoprimaryFiltration.toHarderNarasimhanFiltration {R : Type*} [CommRing R] [IsNoetherianRing R]{M : Type*} [Nontrivial M] [AddCommGroup M] [Module R M] [Module.Finite R M](a : CoprimaryFiltration R M) : ∃ HNFil : HarderNarasimhanFiltration (μ R M), a.filtration = HNFil.filtration := by let ahn : HarderNarasimhanFiltration (μ R M) := by refine HarderNarasimhanFiltration.mk a.filtration a.monotone a.first_eq_bot a.fin_len a.strict_mono ?_ ?_ · intro i hi let hstep := a.strict_mono i (i + 1) (Nat.lt_add_one i) hi letI : Nontrivial (↥(a.filtration (i + 1)) ⧸ Submodule.submoduleOf (a.filtration i) (a.filtration (i + 1))) := quot_ntl hstep letI : Nontrivial (ℒ R (↥(a.filtration (i + 1)) ⧸ Submodule.submoduleOf (a.filtration i) (a.filtration (i + 1)))) := quot_ntl' hstep exact (semistable_res_iff_semistable_quot _ _ hstep).mpr <| rmk4d14₂.mpr (a.piecewise_coprimary i hi).coprimary · intro i hi have := a.piecewise_coprimary i (Nat.lt_of_succ_lt hi) repeat rw [prop3d12] simp only [Function.Embedding.toFun_eq_coe, RelEmbedding.coe_toEmbedding, OrderEmbedding.le_iff_le, not_le, gt_iff_lt] apply S₀_order'.1 convert a.strict_mono_associated_prime i hi · replace := (_μ R M ⟨(a.filtration (i + 1), a.filtration (i + 2)), a.strict_mono (i+1) (i+2) (Nat.lt_add_one (i + 1)) hi⟩).toFinset.min'_mem (μ_nonempty _) apply Set.mem_toFinset.mp at this rcases this.out with ⟨p,hp1,hp2⟩ rw [← hp2] simp only [(a.piecewise_coprimary (i+1) hi).coprimary.unique ((a.piecewise_coprimary (i+1) hi).coprimary.exists.choose_spec) hp1] · replace := (_μ R M ⟨(a.filtration i, a.filtration (i + 1)), a.strict_mono i (i+1) (Nat.lt_add_one i) (Nat.le_of_succ_le hi)⟩).toFinset.min'_mem (μ_nonempty _) apply Set.mem_toFinset.mp at this rcases this.out with ⟨p,hp1,hp2⟩ rw [← hp2] simp only [(a.piecewise_coprimary i (Nat.lt_of_succ_lt hi)).coprimary.unique ((a.piecewise_coprimary i (Nat.lt_of_succ_lt hi)).coprimary.exists.choose_spec) hp1] use ahn- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/CoprimaryFiltration/Impl.lean:1095-1131
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Related declarations
Associated Primes ker mk Linear Map eq
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_eq
Project documentation
Associated primes of the kernel of the localization map. This identifies the associated primes of ker (LocalizedModule.mkLinearMap S M) : Submodule R M with the associated primes of M that do meet the multiplicative set S. Equivalently, these are the associated primes of M after removing those disjoint from S. This lemma is used as the “kernel...
Source project: Harder-Narasimhan
Person-level attribution pending.
Associated Primes ker mk Linear Map subset
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_subset
Plain-language statement
One-sided inclusion for associated primes of the kernel of the localization map. If p ∈ associatedPrimes R (ker (mkLinearMap S M)), then p is an associated prime of M and p is not disjoint from the multiplicative set S. This is one direction of associatedPrimes_ker_mkLinearMap_eq.
Source project: Harder-Narasimhan
Person-level attribution pending.
Associated Primes localized Module subset disjoint
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_localizedModule_subset_disjoint
Plain-language statement
Associated primes of a localized module are disjoint from the multiplicative set. More precisely, if p ∈ associatedPrimes R (LocalizedModule S M), then p.carrier ∩ S = ∅. This is a standard fact in commutative algebra: an element of S becomes a unit after localization, so no associated prime of the localized module can contain an element of S.
Source project: Harder-Narasimhan
Person-level attribution pending.