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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1303.5160","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-03-21T04:33:06Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"bb858d8ab681b17eedc163541881927f61417376864d8fb16d6931df0cbee3dc","abstract_canon_sha256":"c6b95cd5f47bcbd965aafaca4ea74311a3fd474c350e484a9cf5216f8723d138"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:12.267188Z","signature_b64":"FN7wwiKs0aSv+j2+/Z/fWFByq6kv8+vf8pX2mXGRwXuJDrjJkk+fdPHjU5P9KVCSzYd5l+vGNm5qzrzUXPypAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e01e224c56728de7634aa993c4d29cb1a2bdd332eee717c23fc5b543ce4ce8f3","last_reissued_at":"2026-05-18T02:28:12.266714Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:12.266714Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\text{reg}_R\\,{}^{\\phi}\\!M <\\infty$. 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