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.
Exact Lean statement
lemma associatedPrimes_ker_mkLinearMap_subset
{R : Type*} [CommRing R]
{M : Type*} [AddCommGroup M] [Module R M]
(S : Submonoid R) :
associatedPrimes R (LinearMap.ker (LocalizedModule.mkLinearMap S M)) ⊆
(associatedPrimes R M) \ { p ∈ associatedPrimes R M | p.carrier ∩ S = ∅ }Formal artifact
Lean source
lemma associatedPrimes_ker_mkLinearMap_subset{R : Type*} [CommRing R]{M : Type*} [AddCommGroup M] [Module R M](S : Submonoid R) : associatedPrimes R (LinearMap.ker (LocalizedModule.mkLinearMap S M)) ⊆ (associatedPrimes R M) \ { p ∈ associatedPrimes R M | p.carrier ∩ S = ∅ } := by intro p hp refine ⟨?_, ?_⟩ · exact associatedPrimes.subset_of_injective (R := R) (f := (LinearMap.ker (LocalizedModule.mkLinearMap S M)).subtype) (Submodule.injective_subtype _) hp · intro hpempty rcases hp with ⟨hpPrime, x, hx⟩ rcases (LocalizedModule.mem_ker_mkLinearMap_iff (S := S) (m := x.1)).1 x.2 with ⟨r, hrS, hrx⟩ have hrx' : r • x = 0 := Subtype.ext (by simpa using hrx) have hrp : r ∈ p := by rw [hx] rw [Ideal.mem_radical_iff] refine ⟨1, ?_⟩ rw [pow_one, Submodule.mem_colon_singleton] simpa using hrx' have hnonempty : (p.carrier ∩ S).Nonempty := ⟨r, hrp, hrS⟩ exact hnonempty.ne_empty hpempty.2- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/CoprimaryFiltration/CommutativeAlgebra.lean:37-59
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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 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.
Associated Primes quot ker mk Linear Map subset disjoint
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_quot_ker_mkLinearMap_subset_disjoint
Plain-language statement
Associated primes of the localization quotient are disjoint from the multiplicative set. More precisely, any p ∈ associatedPrimes R (M ⧸ ker(mkLinearMap S M)) satisfies p.carrier ∩ S = ∅. In this file this is used as a “black box” input in the Bourbaki-style splitting of associated primes under localization.
Source project: Harder-Narasimhan
Person-level attribution pending.