{"paper":{"title":"Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Douglas J. Dailey, Srikanth B. Iyengar, Thomas Marley","submitted_at":"2016-12-01T23:10:05Z","abstract_excerpt":"It is proved that a module $M$ over a Noetherian ring $R$ of positive characteristic $p$ has finite flat dimension if there exists an integer $t\\ge 0$ such that $\\operatorname{Tor}_i^R(M, {}^{f^{e}}\\!R)=0$ for $t\\le i\\le t+\\dim R$ and infinitely many $e$. This extends a result of Herzog, who proved it when $M$ is finitely generated, and strengthens a result of the third author and Webb in the case $M$ is arbitrary. It is also proved that when $R$ is a Cohen-Macaulay local ring, it suffices that the Tor vanishing holds for one $e\\ge \\log_{p}e(R)$, where $e(R)$ is the multiplicity of $R$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00509","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"}