{"paper":{"title":"Vanishing of relative homology and depth of tensor products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Arash Sadeghi, Li Liang, Olgur Celikbas","submitted_at":"2016-08-25T03:50:46Z","abstract_excerpt":"For finitely generated modules $M$ and $N$ over a Gorenstein local ring $R$, one has $depth M + depth N= depth(M\\otimes N) +depth R$, i.e., the depth formula holds, if $M$ and $N$ are Tor-independent and Tate homology $\\hat{Tor}_{i}^{R}(M,N)$ vanishes for all $i\\in\\mathbb{Z}$. We establish the same conclusion under weaker hypotheses: if $M$ and $N$ are $\\mathcal{G}$-relative Tor-independent, then the vanishing of $\\hat{Tor}_{i}^{R}(M,N)$ for all $i\\le 0$ is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the dep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07011","kind":"arxiv","version":3},"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"}