Plain-language statement
Remark 2.5 (paper-facing form). Under convexity, this states: - μmax μ is convex, and - μmax is idempotent and leaves μA unchanged on every interval. API note: the second component is universally quantified over intervals to facilitate rewriting in later files.
Exact Lean statement
lemma remark_2_5 {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ]
{S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμcvx : Convex μ) :
------------
Convex (μmax μ) ∧
∀ I : {p : ℒ × ℒ // p.1 < p.2},
μmax μ I = μmax (μmax μ) I ∧ μA μ I = μA (μmax μ) I
------------Formal artifact
Lean source
lemma remark_2_5 {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ] {S : Type*} [CompleteLattice S] (μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμcvx : Convex μ) :------------ Convex (μmax μ) ∧ ∀ I : {p : ℒ × ℒ // p.1 < p.2}, μmax μ I = μmax (μmax μ) I ∧ μA μ I = μA (μmax μ) I------------ := by apply (ConvexI_TotIntvl_iff_Convex _).2 at hμcvx rw [← ConvexI_TotIntvl_iff_Convex] exact ⟨impl.rmk2d5₁ TotIntvl μ hμcvx,fun I ↦ ⟨impl.rmk2d5₂ I μ (Convex_of_Convex_large TotIntvl I ⟨bot_le,le_top⟩ μ hμcvx), impl.rmk2d5₃ I μ (Convex_of_Convex_large TotIntvl I ⟨bot_le,le_top⟩ μ hμcvx)⟩⟩- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/Convexity/Results.lean:84-97
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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.