Semistable I iff
HarderNarasimhan.impl.semistableI_iff
Project documentation
Transport semistability along restriction. This theorem relates: - semistableI μ I, i.e. semistability of the interval I with respect to μ, and - Semistable (Resμ I μ), i.e. global semistability of the restricted function on the interval subtype. API note: this is a key adapter used whenever proofs switch between the “ambient interval” viewpoint a...
Exact Lean statement
theorem semistableI_iff {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ]
{S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S)
(I : {p : ℒ × ℒ // p.1 < p.2}) : semistableI μ I ↔ Semistable (Resμ I μ)Formal artifact
Lean source
theorem semistableI_iff {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ]{S : Type*} [CompleteLattice S](μ : {p :ℒ × ℒ // p.1 < p.2} → S)(I : {p : ℒ × ℒ // p.1 < p.2}) : semistableI μ I ↔ Semistable (Resμ I μ) := by rw [semistable_iff (μ := Resμ I μ)] simp only [semistableI, StI, S₁I, S₂I, TotIntvl, Set.mem_setOf_eq, gt_iff_lt, μA_res_intvl] constructor · rintro ⟨hI, hne, h₁, h₂⟩ refine ⟨in_TotIntvl _, ne_of_lt bot_lt_top, ?_, ?_⟩ · intro y hyI hy exact h₁ y y.prop (fun h => hy <| Subtype.ext h) · intro y hyI hy hy' exact h₂ y y.prop (fun h => hy <| Subtype.ext h) hy' · rintro ⟨hI, hne, h₁, h₂⟩ refine ⟨⟨le_of_lt I.prop, le_rfl⟩, ne_of_lt I.prop, ?_, ?_⟩ · intro y hyI hy exact h₁ ⟨y, hyI⟩ (in_TotIntvl _) (fun h => hy <| congrArg Subtype.val h) · intro y hyI hy hy' exact hyI.2- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/Semistability/Impl.lean:686-705
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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.