Mem associated Primes of mem associated Primes quot ker mk Linear Map of disjoint
HarderNarasimhan.CommutativeAlgebra.mem_associatedPrimes_of_mem_associatedPrimes_quot_ker_mkLinearMap_of_disjoint
Project documentation
If p is an associated prime of the quotient M ⧸ ker(mkLinearMap S M) and p is disjoint from the multiplicative set S, then p is already an associated prime of M. This lemma is used to identify the “disjoint part” of the associated primes of M with the associated primes of the localization quotient.
Exact Lean statement
lemma mem_associatedPrimes_of_mem_associatedPrimes_quot_ker_mkLinearMap_of_disjoint
{R : Type*} [CommRing R] [IsNoetherianRing R]
{M : Type*} [AddCommGroup M] [Module R M]
(S : Submonoid R) {p : Ideal R}
(hp : p ∈ associatedPrimes R (M ⧸ LinearMap.ker (LocalizedModule.mkLinearMap S M)))
(hpDisj : p.carrier ∩ S = ∅) :
p ∈ associatedPrimes R MFormal artifact
Lean source
lemma mem_associatedPrimes_of_mem_associatedPrimes_quot_ker_mkLinearMap_of_disjoint{R : Type*} [CommRing R] [IsNoetherianRing R]{M : Type*} [AddCommGroup M] [Module R M](S : Submonoid R) {p : Ideal R}(hp : p ∈ associatedPrimes R (M ⧸ LinearMap.ker (LocalizedModule.mkLinearMap S M)))(hpDisj : p.carrier ∩ S = ∅) : p ∈ associatedPrimes R M := by haveI : p.IsPrime := hp.1 let K : Submodule R M := LinearMap.ker (LocalizedModule.mkLinearMap S M) have hKloc : K.localized (p := p.primeCompl) = ⊥ := by change Submodule.localized' (Localization p.primeCompl) p.primeCompl (LocalizedModule.mkLinearMap p.primeCompl M) K = ⊥ rw [Submodule.localized'_eq_span] refine Submodule.span_eq_bot.mpr ?_ rintro _ ⟨x, hx, rfl⟩ rcases (LocalizedModule.mem_ker_mkLinearMap_iff (S := S) (m := x)).1 hx with ⟨s, hsS, hsx⟩ have hsP : s ∈ p.primeCompl := fun hsp => Set.notMem_empty s (hpDisj ▸ ⟨hsp, hsS⟩) have hxPker : x ∈ LinearMap.ker (LocalizedModule.mkLinearMap p.primeCompl M) := (LocalizedModule.mem_ker_mkLinearMap_iff (S := p.primeCompl) (m := x)).2 ⟨s, hsP, hsx⟩ simpa [LinearMap.mem_ker] using hxPker let eQ := (localizedQuotientEquiv (p := p.primeCompl) (M' := K)) let eBot := (Submodule.quotEquivOfEqBot (K.localized (p := p.primeCompl)) hKloc) let e : LocalizedModule p.primeCompl (M ⧸ K) ≃ₗ[Localization p.primeCompl] LocalizedModule p.primeCompl M := eQ.symm.trans eBot have hAtPrimeQuot : IsLocalRing.maximalIdeal (Localization.AtPrime p) ∈ associatedPrimes (Localization.AtPrime p) (LocalizedModule.AtPrime p (M ⧸ K)) := by simpa [LocalizedModule.AtPrime, K] using (Module.associatedPrimes.mem_associatedPrimes_atPrime_of_mem_associatedPrimes (R := R) (M := (M ⧸ K)) (p := p) hp) have hAtPrimeM : IsLocalRing.maximalIdeal (Localization.AtPrime p) ∈ associatedPrimes (Localization.AtPrime p) (LocalizedModule.AtPrime p M) := by simpa [LocalizedModule.AtPrime, K] using ((LinearEquiv.AssociatedPrimes.eq (R := Localization.AtPrime p) e) ▸ hAtPrimeQuot) have hComap := associatedPrimes.comap_mem_associatedPrimes_of_mem_associatedPrimes_of_isLocalizedModule_of_fg (S := p.primeCompl) (R' := Localization.AtPrime p) (f := LocalizedModule.mkLinearMap p.primeCompl M) (p := IsLocalRing.maximalIdeal (Localization.AtPrime p)) hAtPrimeM ((isNoetherianRing_iff_ideal_fg R).mp ‹IsNoetherianRing R› _) simpa [Localization.AtPrime.under_maximalIdeal] using hComap- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/CoprimaryFiltration/CommutativeAlgebra.lean:225-266
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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.