Quasi-finite modules and asymptotic prime divisors

2013/01/29 by Daniel Katz, Tony J. Puthenpurakal, Katz, Daniel +1
Mathematics · #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13A17 #Secondary 13A30 #math.AC #msc:13A17 #msc:13A30

paper · pdf · doi:10.48550/arxiv.1301.6886

to appear in Journal of Algebra

arxiv created 2013/01/29 · arxiv updated 2013/01/30

Abstract

Let A be a Noetherian ring, J⊆ A an ideal and C a finitely generated A-module. In this note we would like to prove the following statement. Let \I_n\_{n≥ 0} be a collection of ideals satisfying : (i) I_n⊇ Jⁿ, for all n, (ii) J^s⋅ I_s ⊆ I_r+s, for all r,s≥ 0 and (iii) I_n⊆ I_m, whenever m≤ n. Then \Ass_A(I_nC/JⁿC) is independent of n, for n sufficiently large. Note that the set of prime ideals ∪_{n≥ 1} \Ass_A(I_nC/JⁿC) is finite, so the issue at hand is the realization that the primes in \Ass_A(I_nC/JⁿC) do not behave periodically, as one might have expected, say if \bigoplus _{n≥ 0}I_n were a Noetherian A-algebra generated in degrees greater than one. We also give a multigraded version of our results.

Related

• A convenient category to study asymptotic primes and related questions
• Asymptotic prime divisors over complete intersection rings
• Asymptotic behaviour of integral closures, quintasymptotic primes and ideal topologies
• Asymptotic stability of certain sets of associated prime ideals of local cohomology modules
• A criterion for cofiniteness of modules