{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:Y25RCQR47TILLEU2XIC43IFEXP","short_pith_number":"pith:Y25RCQR4","canonical_record":{"source":{"id":"1011.1183","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-04T15:09:34Z","cross_cats_sorted":["math.RA","math.RT"],"title_canon_sha256":"57d3435b4449047307d47c3a5d9cd6dddf5dcd008cdabcebbc161478bbb20ee5","abstract_canon_sha256":"0a531f4bacb156010e5b6000f5474e9e28ae95aab8e6ba7fc042154db3a5051e"},"schema_version":"1.0"},"canonical_sha256":"c6bb11423cfcd0b5929aba05cda0a4bbe1c5869c1c5250c26e85e327f47f32b7","source":{"kind":"arxiv","id":"1011.1183","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1183","created_at":"2026-05-18T03:19:22Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1183v6","created_at":"2026-05-18T03:19:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1183","created_at":"2026-05-18T03:19:22Z"},{"alias_kind":"pith_short_12","alias_value":"Y25RCQR47TIL","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"Y25RCQR47TILLEU2","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"Y25RCQR4","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:Y25RCQR47TILLEU2XIC43IFEXP","target":"record","payload":{"canonical_record":{"source":{"id":"1011.1183","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-04T15:09:34Z","cross_cats_sorted":["math.RA","math.RT"],"title_canon_sha256":"57d3435b4449047307d47c3a5d9cd6dddf5dcd008cdabcebbc161478bbb20ee5","abstract_canon_sha256":"0a531f4bacb156010e5b6000f5474e9e28ae95aab8e6ba7fc042154db3a5051e"},"schema_version":"1.0"},"canonical_sha256":"c6bb11423cfcd0b5929aba05cda0a4bbe1c5869c1c5250c26e85e327f47f32b7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:22.741184Z","signature_b64":"0FFIEoB70PzCxlwmGfiBUw5YYQFrjCGMTglhywF2p476kE6Ixja0aTDVFFzMrolu4nqiDtwF+2q+odWwkNdvBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6bb11423cfcd0b5929aba05cda0a4bbe1c5869c1c5250c26e85e327f47f32b7","last_reissued_at":"2026-05-18T03:19:22.740484Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:22.740484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.1183","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:19:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yFADBgM50INInw+4h34dS4sTlUqdyKHDOTuu28fQjAUtP9GX9LeHWBZRHbN67cD2jk6gypzshFXFUFo1BMlPDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T09:34:05.628569Z"},"content_sha256":"5c985cbffe18d78e4997e2c159a78a9fb5f982ad13ec5143714ab65264ad4a21","schema_version":"1.0","event_id":"sha256:5c985cbffe18d78e4997e2c159a78a9fb5f982ad13ec5143714ab65264ad4a21"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:Y25RCQR47TILLEU2XIC43IFEXP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On unipotent algebraic G-groups and 1-cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA","math.RT"],"primary_cat":"math.GR","authors_text":"David I. Stewart","submitted_at":"2010-11-04T15:09:34Z","abstract_excerpt":"In this paper we consider non-abelian 1-cohomology for groups with coefficients in other groups. We prove versions of the `five lemma' arising from this situation. We go on to show that a connected unipotent algebraic group Q acted on morphically by a connected algebraic group G admits a filtration with successive quotients having the structure of G-modules. From these results we deduce extensions to results due to Cline, Parshall, Scott and van der Kallen. Firstly, if G is a connected, reductive algebraic group with Borel subgroup B and Q a unipotent algebraic G-group, we show the restriction"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1183","kind":"arxiv","version":6},"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":"2026-05-18T03:19:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"phxk5uHN7aB7ODH7CYsegF/mz4xA4ZbEHg25bISsc/n+PfXtTevTBxLIv/4BKmeUcSEDX4XDWBQOOlfzxvKkCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T09:34:05.628909Z"},"content_sha256":"8d363d9d81dca8a60944a3e93509fa232683fa537f7927d72283c420213508e5","schema_version":"1.0","event_id":"sha256:8d363d9d81dca8a60944a3e93509fa232683fa537f7927d72283c420213508e5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y25RCQR47TILLEU2XIC43IFEXP/bundle.json","state_url":"https://pith.science/pith/Y25RCQR47TILLEU2XIC43IFEXP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y25RCQR47TILLEU2XIC43IFEXP/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":"2026-06-30T09:34:05Z","links":{"resolver":"https://pith.science/pith/Y25RCQR47TILLEU2XIC43IFEXP","bundle":"https://pith.science/pith/Y25RCQR47TILLEU2XIC43IFEXP/bundle.json","state":"https://pith.science/pith/Y25RCQR47TILLEU2XIC43IFEXP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y25RCQR47TILLEU2XIC43IFEXP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:Y25RCQR47TILLEU2XIC43IFEXP","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":"0a531f4bacb156010e5b6000f5474e9e28ae95aab8e6ba7fc042154db3a5051e","cross_cats_sorted":["math.RA","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-04T15:09:34Z","title_canon_sha256":"57d3435b4449047307d47c3a5d9cd6dddf5dcd008cdabcebbc161478bbb20ee5"},"schema_version":"1.0","source":{"id":"1011.1183","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1183","created_at":"2026-05-18T03:19:22Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1183v6","created_at":"2026-05-18T03:19:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1183","created_at":"2026-05-18T03:19:22Z"},{"alias_kind":"pith_short_12","alias_value":"Y25RCQR47TIL","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"Y25RCQR47TILLEU2","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"Y25RCQR4","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:8d363d9d81dca8a60944a3e93509fa232683fa537f7927d72283c420213508e5","target":"graph","created_at":"2026-05-18T03:19:22Z","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":"In this paper we consider non-abelian 1-cohomology for groups with coefficients in other groups. We prove versions of the `five lemma' arising from this situation. We go on to show that a connected unipotent algebraic group Q acted on morphically by a connected algebraic group G admits a filtration with successive quotients having the structure of G-modules. From these results we deduce extensions to results due to Cline, Parshall, Scott and van der Kallen. Firstly, if G is a connected, reductive algebraic group with Borel subgroup B and Q a unipotent algebraic G-group, we show the restriction","authors_text":"David I. Stewart","cross_cats":["math.RA","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-04T15:09:34Z","title":"On unipotent algebraic G-groups and 1-cohomology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1183","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":"sha256:5c985cbffe18d78e4997e2c159a78a9fb5f982ad13ec5143714ab65264ad4a21","target":"record","created_at":"2026-05-18T03:19:22Z","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":"0a531f4bacb156010e5b6000f5474e9e28ae95aab8e6ba7fc042154db3a5051e","cross_cats_sorted":["math.RA","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-04T15:09:34Z","title_canon_sha256":"57d3435b4449047307d47c3a5d9cd6dddf5dcd008cdabcebbc161478bbb20ee5"},"schema_version":"1.0","source":{"id":"1011.1183","kind":"arxiv","version":6}},"canonical_sha256":"c6bb11423cfcd0b5929aba05cda0a4bbe1c5869c1c5250c26e85e327f47f32b7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c6bb11423cfcd0b5929aba05cda0a4bbe1c5869c1c5250c26e85e327f47f32b7","first_computed_at":"2026-05-18T03:19:22.740484Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:22.740484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0FFIEoB70PzCxlwmGfiBUw5YYQFrjCGMTglhywF2p476kE6Ixja0aTDVFFzMrolu4nqiDtwF+2q+odWwkNdvBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:22.741184Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.1183","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c985cbffe18d78e4997e2c159a78a9fb5f982ad13ec5143714ab65264ad4a21","sha256:8d363d9d81dca8a60944a3e93509fa232683fa537f7927d72283c420213508e5"],"state_sha256":"9e8f590d967ef3bf0406d5fb1e515b6262211ff301e76368e8bd9d7e7064da56"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vcy1fKofJFQYmbjFC6Aim0nf53mfwkZpIGMd0WccX61dwrG7D1SLLl5O0dTLw+/7sO0mWAHqIzPAD2xC42SDBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T09:34:05.630799Z","bundle_sha256":"10d1d79220c702203d2d97cf9bbd934c98f9682ef06b7f689f3cd4520f6856a7"}}