Proposition 3 8
HarderNarasimhan.proposition_3_8
Plain-language statement
Totality/maximality consequences and the slope decomposition formula. Assuming convexity and an additional admissibility hypothesis (either totality of ≤ on S, or an attainment condition on μ), the internal results show: 1. St μ is totally ordered and, under DCC, admits a greatest element; and 2. for any stable breakpoint x and any y > x, the...
Exact Lean statement
lemma proposition_3_8 {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ] [WellFoundedGT ℒ]
{S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμcvx : Convex μ)
(h : (@Std.Total S (· ≤ ·)) ∨
∀ z : ℒ, (hz : ⊥ ≠ z) → IsAttained μ ⟨(⊥ , z) , lt_of_le_of_ne bot_le hz⟩) :
------------
(
/- `(1)` -/
@Std.Total (St μ) (· ≤ ·) ∧ (μA_DescendingChainCondition μ → ∃ s : ℒ, IsGreatest (St μ) s)
) ∧
/- `(2)` -/
∀ x : ℒ, (hxSt : x ∈ St μ) →
(∀ y : ℒ, (hxy : y > x) →
μA μ ⟨(⊥ , y), lt_of_le_of_lt hxSt.out.choose.1 hxy⟩ = μA μ ⟨(x , y), hxy⟩)
------------Formal artifact
Lean source
lemma proposition_3_8 {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ] [WellFoundedGT ℒ] {S : Type*} [CompleteLattice S] (μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμcvx : Convex μ) (h : (@Std.Total S (· ≤ ·)) ∨ ∀ z : ℒ, (hz : ⊥ ≠ z) → IsAttained μ ⟨(⊥ , z) , lt_of_le_of_ne bot_le hz⟩) :------------ ( /- `(1)` -/ @Std.Total (St μ) (· ≤ ·) ∧ (μA_DescendingChainCondition μ → ∃ s : ℒ, IsGreatest (St μ) s) ) ∧ /- `(2)` -/ ∀ x : ℒ, (hxSt : x ∈ St μ) → (∀ y : ℒ, (hxy : y > x) → μA μ ⟨(⊥ , y), lt_of_le_of_lt hxSt.out.choose.1 hxy⟩ = μA μ ⟨(x , y), hxy⟩)------------ := by rw [← ConvexI_TotIntvl_iff_Convex] at hμcvx constructor · constructor · rcases h with c1 | c2 · exact impl.prop3d8₁ μ TotIntvl hμcvx (Or.inl c1) · exact impl.prop3d8₁ μ TotIntvl hμcvx (Or.inr fun z _ hz ↦ c2 z hz) · intro hμDCC rcases h with c1 | c2 · exact impl.prop3d8₁' μ hμDCC TotIntvl hμcvx (Or.inl c1) · exact impl.prop3d8₁' μ hμDCC TotIntvl hμcvx (Or.inr fun z _ hz ↦ c2 z hz) · intro x hxSt y hxy rcases h with c1 | c2 · exact impl.prop3d8₂ μ TotIntvl hμcvx (Or.inl c1) x hxSt y (in_TotIntvl y) hxy · exact impl.prop3d8₂ μ TotIntvl hμcvx (Or.inr fun z _ hz ↦ c2 z hz) x hxSt y (in_TotIntvl y) hxy- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/Semistability/Results.lean:148-178
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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.