Exists Jordan Holder Series
HarderNarasimhan.exists_JordanHolderSeries
Plain-language statement
Construct a Jordan–Hölder RelSeries from an existing filtration. Given the existence instance for JordanHolderFiltration μ, we build a RelSeries for the relation JordanHolderRel μ whose head is ⊤ and whose last element is ⊥. API note: this is the RelSeries-shaped entry point corresponding to the existence instance.
Exact Lean statement
theorem exists_JordanHolderSeries
{ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ] [WellFoundedGT ℒ]
{S : Type*} [CompleteLinearOrder S]
{μ : {p : ℒ × ℒ // p.1 < p.2} → S} [hftp : FiniteTotalPayoff μ] [hsl : SlopeLike μ]
[hst : Semistable μ] [hwdcc' : StrongDescendingChainCondition' μ] :
∃ s : RelSeries (JordanHolderRel μ), s.head = ⊤ ∧ s.last = ⊥Formal artifact
Lean source
theorem exists_JordanHolderSeries{ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ] [WellFoundedGT ℒ]{S : Type*} [CompleteLinearOrder S]{μ : {p : ℒ × ℒ // p.1 < p.2} → S} [hftp : FiniteTotalPayoff μ] [hsl : SlopeLike μ][hst : Semistable μ] [hwdcc' : StrongDescendingChainCondition' μ] :∃ s : RelSeries (JordanHolderRel μ), s.head = ⊤ ∧ s.last = ⊥:= by have := (inferInstance : Nonempty (JordanHolderFiltration μ)).some let JH : RelSeries (JordanHolderRel μ) := { length := Nat.find this.fin_len, toFun := fun n ↦ this.filtration n.toNat, step := fun n ↦ ⟨this.strict_anti n n.succ (Nat.lt_add_one ↑n) (Fin.is_le n.succ), this.step_cond₁ n n.isLt, this.step_cond₂ n n.isLt⟩ } exact ⟨JH, this.first_eq_top, Nat.find_spec this.fin_len⟩- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/JordanHolderFiltration/Results.lean:105-119
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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.