{"paper":{"title":"The Acyclicity of the Frobenius Functor for Modules of Finite Flat Dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Marcus Webb, Thomas Marley","submitted_at":"2015-01-02T02:09:06Z","abstract_excerpt":"Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\\to R$ the Frobenius ring homomorphism. For $e\\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and Herzog state that for finitely generated modules $M$, $M$ has finite projective dimension if and only if $\\operatorname{Tor}_i^R(R^{(e)},M)=0$ for all $i>0$ and all (equivalently, infinitely many) $e\\ge 1$. We prove this statement holds for arbitrary modules using the theory of flat covers and minimal flat resolutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00336","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"}