{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:7LX6MH5YGVUBFGFVFR2XNKKPDF","short_pith_number":"pith:7LX6MH5Y","schema_version":"1.0","canonical_sha256":"faefe61fb835681298b52c7576a94f197bd38d057b611e0cb5d315de4542eccc","source":{"kind":"arxiv","id":"1101.3834","version":2},"attestation_state":"computed","paper":{"title":"Productive elements in group cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Ergun Yalcin","submitted_at":"2011-01-20T07:54:20Z","abstract_excerpt":"Let $G$ be a finite group and $k$ be a field of characteristic $p>0$. A cohomology class $\\zeta \\in H^n(G,k)$ is called productive if it annihilates $\\Ext^*_{kG}(L_{\\zeta},L_{\\zeta})$. We consider the chain complex $\\bPz$ of projective $kG$-modules which has the homology of an $(n-1)$-sphere and whose $k$-invariant is $\\zeta$ under a certain polarization. We show that $\\zeta$ is productive if and only if there is a chain map $\\Delta: \\bPz \\to \\bPz \\otimes \\bPz$ such that $(\\id \\otimes \\epsilon)\\Delta\\simeq \\id$ and $(\\epsilon \\otimes \\id)\\Delta \\simeq \\id$. Using the Postnikov decomposition of"},"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":"1101.3834","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-01-20T07:54:20Z","cross_cats_sorted":[],"title_canon_sha256":"59ec75cbdf38c9733ac4ecb1964bf0017ece44f88dcafc88a551310d24c9b2ab","abstract_canon_sha256":"f509e3d8e00e85b643a000bc4bfdbd131fe9ea4a78981c7363aeae1f43bdcc51"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:58.235003Z","signature_b64":"0FoiKw/wLaSYMPpe+I1rk2PXauiovubIdmdkIhxi8v9fpiMXWTe1868cr6wiomWF8ArCbvMJNiML5i5TG9d4DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"faefe61fb835681298b52c7576a94f197bd38d057b611e0cb5d315de4542eccc","last_reissued_at":"2026-05-18T03:56:58.234256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:58.234256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Productive elements in group cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Ergun Yalcin","submitted_at":"2011-01-20T07:54:20Z","abstract_excerpt":"Let $G$ be a finite group and $k$ be a field of characteristic $p>0$. A cohomology class $\\zeta \\in H^n(G,k)$ is called productive if it annihilates $\\Ext^*_{kG}(L_{\\zeta},L_{\\zeta})$. We consider the chain complex $\\bPz$ of projective $kG$-modules which has the homology of an $(n-1)$-sphere and whose $k$-invariant is $\\zeta$ under a certain polarization. We show that $\\zeta$ is productive if and only if there is a chain map $\\Delta: \\bPz \\to \\bPz \\otimes \\bPz$ such that $(\\id \\otimes \\epsilon)\\Delta\\simeq \\id$ and $(\\epsilon \\otimes \\id)\\Delta \\simeq \\id$. Using the Postnikov decomposition of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3834","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1101.3834","created_at":"2026-05-18T03:56:58.234384+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.3834v2","created_at":"2026-05-18T03:56:58.234384+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3834","created_at":"2026-05-18T03:56:58.234384+00:00"},{"alias_kind":"pith_short_12","alias_value":"7LX6MH5YGVUB","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"7LX6MH5YGVUBFGFV","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"7LX6MH5Y","created_at":"2026-05-18T12:26:22.705136+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF","json":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF.json","graph_json":"https://pith.science/api/pith-number/7LX6MH5YGVUBFGFVFR2XNKKPDF/graph.json","events_json":"https://pith.science/api/pith-number/7LX6MH5YGVUBFGFVFR2XNKKPDF/events.json","paper":"https://pith.science/paper/7LX6MH5Y"},"agent_actions":{"view_html":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF","download_json":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF.json","view_paper":"https://pith.science/paper/7LX6MH5Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.3834&json=true","fetch_graph":"https://pith.science/api/pith-number/7LX6MH5YGVUBFGFVFR2XNKKPDF/graph.json","fetch_events":"https://pith.science/api/pith-number/7LX6MH5YGVUBFGFVFR2XNKKPDF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/action/storage_attestation","attest_author":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/action/author_attestation","sign_citation":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/action/citation_signature","submit_replication":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/action/replication_record"}},"created_at":"2026-05-18T03:56:58.234384+00:00","updated_at":"2026-05-18T03:56:58.234384+00:00"}