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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 $\\Ass_A(I_nC/J^nC)$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1301.6886","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-01-29T10:29:14Z","cross_cats_sorted":[],"title_canon_sha256":"68bd4464b5a5ce45bf23c740883ef06bc7aaf067143d555bf66eee128d8eeecc","abstract_canon_sha256":"6553de55cf4ec5485a3861bd57c93d68685dc36409068113ec9ff670aa25bd24"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:12.032987Z","signature_b64":"xBH2J6EJHUEX/8ObJoi/HlRiXnOA+XXnUAePEDIHTFDJSIOoG9ix41cF9u64K20hTfVLBBZSgr8rRhfsmax6AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86a40ad147719cb6a5a06efb69b45300b8130296b20fb0e74d6b6e7fd1b5aa88","last_reissued_at":"2026-05-18T03:35:12.032040Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:12.032040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1301.6886","created_at":"2026-05-18T03:35:12.032170+00:00"},{"alias_kind":"arxiv_version","alias_value":"1301.6886v1","created_at":"2026-05-18T03:35:12.032170+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.6886","created_at":"2026-05-18T03:35:12.032170+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q2SAVUKHOGOL","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q2SAVUKHOGOLNJNA","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q2SAVUKH","created_at":"2026-05-18T12:27:57.521954+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Q2SAVUKHOGOLNJNAN35WTNCTAC","json":"https://pith.science/pith/Q2SAVUKHOGOLNJNAN35WTNCTAC.json","graph_json":"https://pith.science/api/pith-number/Q2SAVUKHOGOLNJNAN35WTNCTAC/graph.json","events_json":"https://pith.science/api/pith-number/Q2SAVUKHOGOLNJNAN35WTNCTAC/events.json","paper":"https://pith.science/paper/Q2SAVUKH"},"agent_actions":{"view_html":"https://pith.science/pith/Q2SAVUKHOGOLNJNAN35WTNCTAC","download_json":"https://pith.science/pith/Q2SAVUKHOGOLNJNAN35WTNCTAC.json","view_paper":"https://pith.science/paper/Q2SAVUKH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1301.6886&json=true","fetch_graph":"https://pith.science/api/pith-number/Q2SAVUKHOGOLNJNAN35WTNCTAC/graph.json","fetch_events":"https://pith.science/api/pith-number/Q2SAVUKHOGOLNJNAN35WTNCTAC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q2SAVUKHOGOLNJNAN35WTNCTAC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q2SAVUKHOGOLNJNAN35WTNCTAC/action/storage_attestation","attest_author":"https://pith.science/pith/Q2SAVUKHOGOLNJNAN35WTNCTAC/action/author_attestation","sign_citation":"https://pith.science/pith/Q2SAVUKHOGOLNJNAN35WTNCTAC/action/citation_signature","submit_replication":"https://pith.science/pith/Q2SAVUKHOGOLNJNAN35WTNCTAC/action/replication_record"}},"created_at":"2026-05-18T03:35:12.032170+00:00","updated_at":"2026-05-18T03:35:12.032170+00:00"}