{"paper":{"title":"Symbolic powers of ideals and their topology over a module","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Adeleh Azari, Reza Naghipour, Simin Mollamahmoudi","submitted_at":"2016-07-26T10:34:45Z","abstract_excerpt":"Let $I$ denote an ideal of a Noetherian ring $R$ and $N$ a non-zero finitely generated $R$-module. In the present paper, some necessary and sufficient conditions are given to determine when the $I$-adic topology on $N$ is equivalent to the $I$-symbolic topology on $N$. Among other things, we shall give a complete solution to the question raised by R. Hartshorne in [{\\it Affine duality and cofiniteness}, Invent. Math. {\\bf9}(1970), 145-164], for a prime ideal $\\frak p$ of dimension one in a local Noetherian ring $R$, by showing that the $\\frak{p}$-adic topology on $N$ is equivalent to the $\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07629","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"}