{"paper":{"title":"Regularity over homomorphisms and a Frobenius characterization of Koszul algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Hop D. Nguyen, Thanh Vu","submitted_at":"2013-03-21T04:33:06Z","abstract_excerpt":"Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $\\phi:R\\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^{\\phi}\\!M$ be the abelian group $M$ with the $R$-module structure induced by the Frobenius endomorphism. The $R$-module ${}^{\\phi}\\!M$ has a natural grading given by $\\text{deg} x=j$ if $x\\in M_{jp+i}$ for some $0\\le i \\le p-1$. In this paper, we prove that $R$ is Koszul if and only if there exists a non-zero finitely generated graded $R$-module $M$ such that $\\text{reg}_R\\,{}^{\\phi}\\!M <\\infty$. Thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.5160","kind":"arxiv","version":6},"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"}