Project documentation
Singleton lower bound for μA: the chosen minimal prime is ≤ every tail _μ. Specializing the previous lemma to the minimal element of a smaller interval, we obtain the order relation needed to show that the singleton {min} is the infimum in the definition of μA.
Exact Lean statement
lemma prop3d12p2 {R : Type*} [CommRing R] [IsNoetherianRing R]
{M : Type*} [Nontrivial M] [AddCommGroup M] [Module R M] [Module.Finite R M]
(I : {z: (ℒ R M) × (ℒ R M) // z.1 < z.2})
(N'' : ℒ R M) (ha1 : InIntvl I N'') (ha2 : N'' ≠ I.val.2) :
@LE.le (S₀ R) Preorder.toLE {(_μ R M I).toFinset.min' <| μ_nonempty I}
(_μ R M ⟨(N'', I.val.2), lt_of_le_of_ne ha1.2 ha2⟩).toFinsetFormal artifact
Lean source
lemma prop3d12p2 {R : Type*} [CommRing R] [IsNoetherianRing R]{M : Type*} [Nontrivial M] [AddCommGroup M] [Module R M] [Module.Finite R M](I : {z: (ℒ R M) × (ℒ R M) // z.1 < z.2})(N'' : ℒ R M) (ha1 : InIntvl I N'') (ha2 : N'' ≠ I.val.2) :@LE.le (S₀ R) Preorder.toLE {(_μ R M I).toFinset.min' <| μ_nonempty I} (_μ R M ⟨(N'', I.val.2), lt_of_le_of_ne ha1.2 ha2⟩).toFinset := by have : @LE.le (S₀ R) Preorder.toLE {(_μ R M I).toFinset.min' <| μ_nonempty I} {(_μ R M ⟨(N'', I.val.2), lt_of_le_of_ne ha1.2 ha2⟩).toFinset.min' <| μ_nonempty _} := by rw [← S₀_order.2] have this' : ((_μ R M ⟨(N'', I.val.2), lt_of_le_of_ne ha1.2 ha2⟩).toFinset.min' <| μ_nonempty _).asIdeal ∈ associatedPrimes R (↥I.val.2 ⧸ Submodule.submoduleOf N'' I.val.2):= by have := ((_μ R M ⟨(N'', I.val.2), lt_of_le_of_ne ha1.2 ha2⟩).toFinset.min'_mem <| μ_nonempty _).out simp only [Finset.mem_val, Set.mem_toFinset, Set.mem_setOf_eq] at this rcases this with ⟨p,⟨hp1,hp2⟩⟩ rwa [← hp2] exact prop3d12p1 I N'' ha1 (((_μ R M ⟨(N'', I.val.2), lt_of_le_of_ne ha1.2 ha2⟩).toFinset.min' <| μ_nonempty _).asIdeal) this' refine le_trans this ?_ apply S₀_order.1 simp only [Set.subset_toFinset, Finset.coe_singleton, Set.singleton_subset_iff] exact Set.mem_toFinset.mp <| (_μ R M ⟨(N'', I.val.2), lt_of_le_of_ne ha1.2 ha2⟩).toFinset.min'_mem <| μ_nonempty _- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/CoprimaryFiltration/Impl.lean:296-318
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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.