{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4APCETCWOKG6OY2KVGJ4JUU4WG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c6b95cd5f47bcbd965aafaca4ea74311a3fd474c350e484a9cf5216f8723d138","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-03-21T04:33:06Z","title_canon_sha256":"bb858d8ab681b17eedc163541881927f61417376864d8fb16d6931df0cbee3dc"},"schema_version":"1.0","source":{"id":"1303.5160","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.5160","created_at":"2026-05-18T02:28:12Z"},{"alias_kind":"arxiv_version","alias_value":"1303.5160v6","created_at":"2026-05-18T02:28:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.5160","created_at":"2026-05-18T02:28:12Z"},{"alias_kind":"pith_short_12","alias_value":"4APCETCWOKG6","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"4APCETCWOKG6OY2K","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"4APCETCW","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:cac4a3ffe25cd59698cd7318b8d839bd2c3bc1f764381983dd27c6b121ba2ac0","target":"graph","created_at":"2026-05-18T02:28:12Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Hop D. Nguyen, Thanh Vu","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-03-21T04:33:06Z","title":"Regularity over homomorphisms and a Frobenius characterization of Koszul algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.5160","kind":"arxiv","version":6},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2019250e2eb8acd06ec51c09d7a289bf67d2e02fec30198c80929068f1f62005","target":"record","created_at":"2026-05-18T02:28:12Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c6b95cd5f47bcbd965aafaca4ea74311a3fd474c350e484a9cf5216f8723d138","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-03-21T04:33:06Z","title_canon_sha256":"bb858d8ab681b17eedc163541881927f61417376864d8fb16d6931df0cbee3dc"},"schema_version":"1.0","source":{"id":"1303.5160","kind":"arxiv","version":6}},"canonical_sha256":"e01e224c56728de7634aa993c4d29cb1a2bdd332eee717c23fc5b543ce4ce8f3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e01e224c56728de7634aa993c4d29cb1a2bdd332eee717c23fc5b543ce4ce8f3","first_computed_at":"2026-05-18T02:28:12.266714Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:12.266714Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FN7wwiKs0aSv+j2+/Z/fWFByq6kv8+vf8pX2mXGRwXuJDrjJkk+fdPHjU5P9KVCSzYd5l+vGNm5qzrzUXPypAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:12.267188Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.5160","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2019250e2eb8acd06ec51c09d7a289bf67d2e02fec30198c80929068f1f62005","sha256:cac4a3ffe25cd59698cd7318b8d839bd2c3bc1f764381983dd27c6b121ba2ac0"],"state_sha256":"09affbff89bd038798192a111fa55b73e490547aededd01f9fb7d582e8ae0b4b"}