{"paper":{"title":"A Generalized Serre's Condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Brent Holmes","submitted_at":"2017-10-07T03:59:54Z","abstract_excerpt":"Throughout, let $R$ be a commutative Noetherian ring. A ring $R$ satisfies Serre's condition $(S_{\\ell})$ if for all $P \\in \\Spec R,$ $\\depth R_P \\geq \\min \\{ \\ell , \\dim R_P \\}$. Serre's condition has been a topic of expanding interest. In this paper, we examine a generalization of Serre's condition $(S_{\\ell}^j)$. We say a ring satisfies $(S_{\\ell}^j)$ when $\\depth R_P \\geq \\min \\{ \\ell , \\dim R_P -j \\}$ for all $P \\in \\Spec R$. We prove generalizations of results for rings satisfying Serre's condition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.02631","kind":"arxiv","version":4},"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"}