{"paper":{"title":"Necessary conditions for the depth formula over Cohen-Macaulay local rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hailong Dao, Olgur Celikbas","submitted_at":"2010-08-16T05:11:59Z","abstract_excerpt":"Let $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be non-zero finitely generated $R$-modules. We investigate necessary conditions for the depth formula $\\depth(M)+\\depth(N)=\\depth(R)+\\depth(M\\otimes_{R}N)$ to hold. We show that, under certain conditions, $M$ and $N$ satisfy the depth formula if and only if $\\Tor_{i}^{R}(M,N)$ vanishes for all $i\\geq 1$. We also examine the relationship between good depth of $M\\otimes_RN$ and the vanishing of $\\Ext$ modules, with various applications."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2573","kind":"arxiv","version":2},"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"}