{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:VRLV3YKO23IY6V7USSC7RRQFIM","short_pith_number":"pith:VRLV3YKO","canonical_record":{"source":{"id":"1806.11024","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-06-28T15:04:35Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"4479e4b4833e0911863d694959390a3680e7a67a3451a54c5e868f3c833e612a","abstract_canon_sha256":"f23e02f8bbd04e200f698c45bc641d441c5af7509579a156f1272a398e59fc21"},"schema_version":"1.0"},"canonical_sha256":"ac575de14ed6d18f57f49485f8c6054328da45a36c75c70fad31272a80bcebbd","source":{"kind":"arxiv","id":"1806.11024","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.11024","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"arxiv_version","alias_value":"1806.11024v3","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.11024","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"pith_short_12","alias_value":"VRLV3YKO23IY","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VRLV3YKO23IY6V7U","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VRLV3YKO","created_at":"2026-05-18T12:32:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:VRLV3YKO23IY6V7USSC7RRQFIM","target":"record","payload":{"canonical_record":{"source":{"id":"1806.11024","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-06-28T15:04:35Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"4479e4b4833e0911863d694959390a3680e7a67a3451a54c5e868f3c833e612a","abstract_canon_sha256":"f23e02f8bbd04e200f698c45bc641d441c5af7509579a156f1272a398e59fc21"},"schema_version":"1.0"},"canonical_sha256":"ac575de14ed6d18f57f49485f8c6054328da45a36c75c70fad31272a80bcebbd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:01.271635Z","signature_b64":"Sa+0+aDqbjnzgZKRU5qWevhr33a/y2XzlLRXsSKX/OsCR858IxX1R9N3KS7KLLsdqACS0wK4gnNfw8DEDzxeDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac575de14ed6d18f57f49485f8c6054328da45a36c75c70fad31272a80bcebbd","last_reissued_at":"2026-05-17T23:41:01.271020Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:01.271020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.11024","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:41:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Oqpaxp+QaFU7kf6NQMn6Nq37+hPz/L3tNyxmBDecCj+cEnt1Nypspw7Kv8yFPe6Fdr9Mkne0gnZ30I0ejsELBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T03:53:55.956634Z"},"content_sha256":"927e0dfc42603fa9d6ba92c6f884391825ef44027b6ca3fb5d330a2c0fa3f310","schema_version":"1.0","event_id":"sha256:927e0dfc42603fa9d6ba92c6f884391825ef44027b6ca3fb5d330a2c0fa3f310"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:VRLV3YKO23IY6V7USSC7RRQFIM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modular Covariants of Cyclic Groups of Order p","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.AC","authors_text":"Jonathan Elmer","submitted_at":"2018-06-28T15:04:35Z","abstract_excerpt":"Let $G$ be a cyclic group of order $p$, let $k$ be a field of characteristic $p$, and let $V, W$ be $kG$-modules. We study the modules of covariants $k[V,W]^G = (S(V^*) \\otimes W)^G$. For $V$ indecomposable with dimension 2, and $W$ an arbitrary indecomposable module, we show $k[V,W]^G$ is a free $k[V]^G$-module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating $k[V,W]^G$ freely over $k[V]^G$. For $V$ indecomposable with dimension 3 and $W$ an indecomposable module with dimension at most 5, we show that $k[V,W]^G$ is a Cohen-Macaulay $k[V]^G$-module "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.11024","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:41:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bHycsn2bgIgzdteaM7mAryl+Oxz0J+r8/LaA0TuE8hhujlL1YA0ouJuUtQ0lNNshTF61W7MkzuP+d8GmIG7gAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T03:53:55.957290Z"},"content_sha256":"858b47451815432d979311d952c9f6b962a4948420c5c70c7c60cfd1ded188ea","schema_version":"1.0","event_id":"sha256:858b47451815432d979311d952c9f6b962a4948420c5c70c7c60cfd1ded188ea"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VRLV3YKO23IY6V7USSC7RRQFIM/bundle.json","state_url":"https://pith.science/pith/VRLV3YKO23IY6V7USSC7RRQFIM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VRLV3YKO23IY6V7USSC7RRQFIM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T03:53:55Z","links":{"resolver":"https://pith.science/pith/VRLV3YKO23IY6V7USSC7RRQFIM","bundle":"https://pith.science/pith/VRLV3YKO23IY6V7USSC7RRQFIM/bundle.json","state":"https://pith.science/pith/VRLV3YKO23IY6V7USSC7RRQFIM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VRLV3YKO23IY6V7USSC7RRQFIM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:VRLV3YKO23IY6V7USSC7RRQFIM","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":"f23e02f8bbd04e200f698c45bc641d441c5af7509579a156f1272a398e59fc21","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-06-28T15:04:35Z","title_canon_sha256":"4479e4b4833e0911863d694959390a3680e7a67a3451a54c5e868f3c833e612a"},"schema_version":"1.0","source":{"id":"1806.11024","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.11024","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"arxiv_version","alias_value":"1806.11024v3","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.11024","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"pith_short_12","alias_value":"VRLV3YKO23IY","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VRLV3YKO23IY6V7U","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VRLV3YKO","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:858b47451815432d979311d952c9f6b962a4948420c5c70c7c60cfd1ded188ea","target":"graph","created_at":"2026-05-17T23:41:01Z","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 $G$ be a cyclic group of order $p$, let $k$ be a field of characteristic $p$, and let $V, W$ be $kG$-modules. We study the modules of covariants $k[V,W]^G = (S(V^*) \\otimes W)^G$. For $V$ indecomposable with dimension 2, and $W$ an arbitrary indecomposable module, we show $k[V,W]^G$ is a free $k[V]^G$-module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating $k[V,W]^G$ freely over $k[V]^G$. For $V$ indecomposable with dimension 3 and $W$ an indecomposable module with dimension at most 5, we show that $k[V,W]^G$ is a Cohen-Macaulay $k[V]^G$-module ","authors_text":"Jonathan Elmer","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-06-28T15:04:35Z","title":"Modular Covariants of Cyclic Groups of Order p"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.11024","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":"sha256:927e0dfc42603fa9d6ba92c6f884391825ef44027b6ca3fb5d330a2c0fa3f310","target":"record","created_at":"2026-05-17T23:41:01Z","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":"f23e02f8bbd04e200f698c45bc641d441c5af7509579a156f1272a398e59fc21","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-06-28T15:04:35Z","title_canon_sha256":"4479e4b4833e0911863d694959390a3680e7a67a3451a54c5e868f3c833e612a"},"schema_version":"1.0","source":{"id":"1806.11024","kind":"arxiv","version":3}},"canonical_sha256":"ac575de14ed6d18f57f49485f8c6054328da45a36c75c70fad31272a80bcebbd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac575de14ed6d18f57f49485f8c6054328da45a36c75c70fad31272a80bcebbd","first_computed_at":"2026-05-17T23:41:01.271020Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:01.271020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Sa+0+aDqbjnzgZKRU5qWevhr33a/y2XzlLRXsSKX/OsCR858IxX1R9N3KS7KLLsdqACS0wK4gnNfw8DEDzxeDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:01.271635Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.11024","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:927e0dfc42603fa9d6ba92c6f884391825ef44027b6ca3fb5d330a2c0fa3f310","sha256:858b47451815432d979311d952c9f6b962a4948420c5c70c7c60cfd1ded188ea"],"state_sha256":"db652041b6b338696a17e48b6b0fb2339585591c0a7525626e4cf5a977dd3e6f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7g/nksGTGxLMVjxlwh0JsGaoVIMlZOWkgNqrtwFlrtW2ILmgF0bNE6CKtwmhsGQzdZcM4EmdTo2c+SGRAzxsBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T03:53:55.961184Z","bundle_sha256":"b417ee3a5ced6dedad378524384318d037e5e66e3a72c34eb589a1a19ee1b98b"}}