{"paper":{"title":"Quasi-finite modules and asymptotic prime divisors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Daniel Katz, Tony J. Puthenpurakal","submitted_at":"2013-01-29T10:29:14Z","abstract_excerpt":"Let $A$ be a Noetherian ring, $J\\subseteq A$ an ideal and $C$ a finitely generated $A$-module. In this note we would like to prove the following statement. Let $\\{I_n\\}_{n\\geq 0}$ be a collection of ideals satisfying : (i) $I_n\\supseteq J^n$, for all $n$, (ii) $J^s\\cdot I_s \\subseteq I_{r+s}$, for all $r,s\\geq 0$ and (iii) $I_n\\subseteq I_m$, whenever $m\\leq n$. Then $\\Ass_A(I_nC/J^nC)$ is independent of $n$, for $n$ sufficiently large. Note that the set of prime ideals $\\cup_{n\\geq 1} \\Ass_A(I_nC/J^nC)$ is finite, so the issue at hand is the realization that the primes in $\\Ass_A(I_nC/J^nC)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6886","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}