{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7LX6MH5YGVUBFGFVFR2XNKKPDF","short_pith_number":"pith:7LX6MH5Y","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"},"canonical_sha256":"faefe61fb835681298b52c7576a94f197bd38d057b611e0cb5d315de4542eccc","source":{"kind":"arxiv","id":"1101.3834","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.3834","created_at":"2026-05-18T03:56:58Z"},{"alias_kind":"arxiv_version","alias_value":"1101.3834v2","created_at":"2026-05-18T03:56:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3834","created_at":"2026-05-18T03:56:58Z"},{"alias_kind":"pith_short_12","alias_value":"7LX6MH5YGVUB","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7LX6MH5YGVUBFGFV","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7LX6MH5Y","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7LX6MH5YGVUBFGFVFR2XNKKPDF","target":"record","payload":{"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"},"canonical_sha256":"faefe61fb835681298b52c7576a94f197bd38d057b611e0cb5d315de4542eccc","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"},"source_kind":"arxiv","source_id":"1101.3834","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:56:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DrcC6QcAQXWultBdqCN6EZ461TDQAK2WnHwpMNZIq8tFwYjES/3X0Yi7J/ibDL4RnnEiVb51gLdDOw518EWlBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T11:01:59.611331Z"},"content_sha256":"a98da8a712f3af217cd01c6f361dc2bd12eb73587ee051c3d053e1baacfdedec","schema_version":"1.0","event_id":"sha256:a98da8a712f3af217cd01c6f361dc2bd12eb73587ee051c3d053e1baacfdedec"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7LX6MH5YGVUBFGFVFR2XNKKPDF","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:56:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"upc1sKQdMPXe01RpCpIU8eSIun5rJqK3FhKvdcjU3LaAT2bTIZktQipz4VxKh/5CfFQGQnU0l/fxGYA+Q/5wDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T11:01:59.611991Z"},"content_sha256":"cdd1a9c3100a43ef14dc4e782f8c2b214460ca31f38b713a373f793f3dfa16e0","schema_version":"1.0","event_id":"sha256:cdd1a9c3100a43ef14dc4e782f8c2b214460ca31f38b713a373f793f3dfa16e0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/bundle.json","state_url":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/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":"2026-05-25T11:01:59Z","links":{"resolver":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF","bundle":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/bundle.json","state":"https://pith.science/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7LX6MH5YGVUBFGFVFR2XNKKPDF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7LX6MH5YGVUBFGFVFR2XNKKPDF","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":"f509e3d8e00e85b643a000bc4bfdbd131fe9ea4a78981c7363aeae1f43bdcc51","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-01-20T07:54:20Z","title_canon_sha256":"59ec75cbdf38c9733ac4ecb1964bf0017ece44f88dcafc88a551310d24c9b2ab"},"schema_version":"1.0","source":{"id":"1101.3834","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.3834","created_at":"2026-05-18T03:56:58Z"},{"alias_kind":"arxiv_version","alias_value":"1101.3834v2","created_at":"2026-05-18T03:56:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3834","created_at":"2026-05-18T03:56:58Z"},{"alias_kind":"pith_short_12","alias_value":"7LX6MH5YGVUB","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7LX6MH5YGVUBFGFV","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7LX6MH5Y","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:cdd1a9c3100a43ef14dc4e782f8c2b214460ca31f38b713a373f793f3dfa16e0","target":"graph","created_at":"2026-05-18T03:56:58Z","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 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","authors_text":"Ergun Yalcin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-01-20T07:54:20Z","title":"Productive elements in group cohomology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3834","kind":"arxiv","version":2},"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:a98da8a712f3af217cd01c6f361dc2bd12eb73587ee051c3d053e1baacfdedec","target":"record","created_at":"2026-05-18T03:56:58Z","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":"f509e3d8e00e85b643a000bc4bfdbd131fe9ea4a78981c7363aeae1f43bdcc51","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-01-20T07:54:20Z","title_canon_sha256":"59ec75cbdf38c9733ac4ecb1964bf0017ece44f88dcafc88a551310d24c9b2ab"},"schema_version":"1.0","source":{"id":"1101.3834","kind":"arxiv","version":2}},"canonical_sha256":"faefe61fb835681298b52c7576a94f197bd38d057b611e0cb5d315de4542eccc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"faefe61fb835681298b52c7576a94f197bd38d057b611e0cb5d315de4542eccc","first_computed_at":"2026-05-18T03:56:58.234256Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:56:58.234256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0FoiKw/wLaSYMPpe+I1rk2PXauiovubIdmdkIhxi8v9fpiMXWTe1868cr6wiomWF8ArCbvMJNiML5i5TG9d4DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:56:58.235003Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.3834","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a98da8a712f3af217cd01c6f361dc2bd12eb73587ee051c3d053e1baacfdedec","sha256:cdd1a9c3100a43ef14dc4e782f8c2b214460ca31f38b713a373f793f3dfa16e0"],"state_sha256":"31303c27413d565c7da14ac6f06f12589763faec29adb47cafb388816d1e3c53"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gEbD9BKEaz9DqHzmaK61+WrN1Akv3fil9XTs0ItojYgpKQqr0M3Pstl+gVjwKbA4YA1d9PI0sSj23UKnmjDZCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T11:01:59.615252Z","bundle_sha256":"66c364375de852fa93acfc126945aa016fc26e4d76110a09913c9d3ce04486cd"}}