{"paper":{"title":"Remarks on Auslander's depth formula for quasi-projective dimension","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Paulo Martins, Victor D. Mendoza-Rubio, Victor H. Jorge-P\\'erez","submitted_at":"2024-09-13T17:22:04Z","abstract_excerpt":"For nonzero finitely generated $R$-modules $M$ and $N$ over a Noetherian local ring $R$, Auslander's depth formula is the equality $$ \\operatorname{depth} M + \\operatorname{depth} N = \\operatorname{depth} R + \\operatorname{depth}(\\operatorname{Tor}_q^R(M,N)) - q, $$ where $ q := \\sup\\{ i \\ge 0 \\mid \\operatorname{Tor}_i^R(M,N) \\neq 0 \\}$. Gheibi, Jorgensen, and Takahashi introduced a homological invariant called quasi-projective dimension, which generalizes projective dimension, and proved that Auslander's depth formula holds when $M$ has finite quasi-projective dimension and $q=0$. In this pap"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.08996","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.08996/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}