Support quotient mono
HarderNarasimhan.impl.support_quotient_mono
Plain-language statement
Monotonicity of support under enlarging the submodule being quotiented out. If N₁ ≤ N₂, then the support of N₃ / N₂ is contained in the support of N₃ / N₁. This is a standard “support shrinks under quotients” statement.
Exact Lean statement
lemma support_quotient_mono {R : Type*} [CommRing R]
{M : Type*} [AddCommGroup M] [Module R M]
(N₁ N₂ N₃ : Submodule R M) (h : N₁ ≤ N₂) :
Module.support R (N₃⧸ N₂.submoduleOf N₃) ⊆ Module.support R (N₃⧸ N₁.submoduleOf N₃)Formal artifact
Lean source
lemma support_quotient_mono {R : Type*} [CommRing R]{M : Type*} [AddCommGroup M] [Module R M](N₁ N₂ N₃ : Submodule R M) (h : N₁ ≤ N₂) : Module.support R (N₃⧸ N₂.submoduleOf N₃) ⊆ Module.support R (N₃⧸ N₁.submoduleOf N₃) := by intro p hp simp only [Module.mem_support_iff_exists_annihilator] at * rcases hp with ⟨m,hm⟩ use Submodule.Quotient.mk m.out intro z hz have : z • (Submodule.Quotient.mk m.out : ↥N₃ ⧸ N₁.submoduleOf N₃)= 0 := (Submodule.mem_annihilator_span_singleton (Submodule.Quotient.mk (Quotient.out m)) z).mp hz replace : z ∈ (Submodule.span R {m}).annihilator := by rw [Submodule.mem_annihilator_span_singleton] rw [← Submodule.Quotient.mk_smul] at this apply (Submodule.Quotient.mk_eq_zero _).1 at this have this' : z • m = Submodule.Quotient.mk (z • m).out := by unfold Submodule.Quotient.mk Quotient.mk'' rw [Quotient.out_eq] rw [this'] apply (Submodule.Quotient.mk_eq_zero _).2 replace this' : z • m.out - (z • m).out ∈ N₂.submoduleOf N₃ := by apply (Submodule.Quotient.mk_eq_zero _).1 simp only [Submodule.Quotient.mk_sub, Submodule.Quotient.mk_smul] unfold Submodule.Quotient.mk Quotient.mk'' rw [Quotient.out_eq, Quotient.out_eq, sub_self] exact (Submodule.sub_mem_iff_right (N₂.submoduleOf N₃) (h this)).mp this' exact hm this- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/CoprimaryFiltration/Impl.lean:215-241
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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.