{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:P5W4D6QN6T3CXM2MUIVY3P7O6C","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":"92f8ca50614de01444d9c80f36f244aca03388d1f7f02c262d885fbe961d054a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2007-09-05T16:47:14Z","title_canon_sha256":"e572913a33cf2e091aa74bcfae86c9772eacf7a2b3c1747b280d2cd9ad7d521b"},"schema_version":"1.0","source":{"id":"0709.0703","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0709.0703","created_at":"2026-05-18T02:58:12Z"},{"alias_kind":"arxiv_version","alias_value":"0709.0703v3","created_at":"2026-05-18T02:58:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0709.0703","created_at":"2026-05-18T02:58:12Z"},{"alias_kind":"pith_short_12","alias_value":"P5W4D6QN6T3C","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"P5W4D6QN6T3CXM2M","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"P5W4D6QN","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:1f1018b91451b988782f95cd9ebef8e9d040292bbae78fc08eeffb0ed594d4be","target":"graph","created_at":"2026-05-18T02:58:12Z","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 the finite group $G$ act linearly on the vector space $V$ over the field $k$ of arbitrary characteristic. If $H<G$ is a subgroup the extension of invariant rings $k[V]^G\\subset k[V]^H$ is studied using modules of covariants.\n  An example of our results is the following. Let $W$ be the subgroup of $G$ generated by the reflections in $G$. A classical theorem due to Serre says that if $k[V]$ is a free $k[V]^G$-module then $G=W$. We generalize this result as follows. If $k[V]^H$ is a free $k[V]^G$-module then $G$ is generated by $H$ and $W$, and the invariant ring $k[V]^{H\\cap W}$ is free over","authors_text":"Abraham Broer, Jianjun Chuai","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2007-09-05T16:47:14Z","title":"Modules of covariants in modular invariant theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.0703","kind":"arxiv","version":3},"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:c00c2c33fe088bfd91a15f4f9f3613c547589bfb274a6671239819d61d3c3b71","target":"record","created_at":"2026-05-18T02:58:12Z","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":"92f8ca50614de01444d9c80f36f244aca03388d1f7f02c262d885fbe961d054a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2007-09-05T16:47:14Z","title_canon_sha256":"e572913a33cf2e091aa74bcfae86c9772eacf7a2b3c1747b280d2cd9ad7d521b"},"schema_version":"1.0","source":{"id":"0709.0703","kind":"arxiv","version":3}},"canonical_sha256":"7f6dc1fa0df4f62bb34ca22b8dbfeef0941019fcef310f962c80f045f8187ddf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7f6dc1fa0df4f62bb34ca22b8dbfeef0941019fcef310f962c80f045f8187ddf","first_computed_at":"2026-05-18T02:58:12.351276Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:12.351276Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+vnxEYdN5dy7TnGVjtBph8mDw++qFoxJ1cYhaZNaF5guVErfjv1IkmooNOfA30sfi/Hie0VWQKcUzV/JsN4WAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:12.352132Z","signed_message":"canonical_sha256_bytes"},"source_id":"0709.0703","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c00c2c33fe088bfd91a15f4f9f3613c547589bfb274a6671239819d61d3c3b71","sha256:1f1018b91451b988782f95cd9ebef8e9d040292bbae78fc08eeffb0ed594d4be"],"state_sha256":"894c28bd2205eee1fc92c34139c80a351c9c490b6e9b1ab622753b571ab67952"}