{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:L6TE3ZSYAE5NDEMR5FJIDYKTH6","short_pith_number":"pith:L6TE3ZSY","canonical_record":{"source":{"id":"1007.3780","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-22T01:19:46Z","cross_cats_sorted":[],"title_canon_sha256":"380030cb36dfba2bac173f16767f7d79cacf7aad0492bbc9af5fc04072c9b120","abstract_canon_sha256":"86de18d44fe820625559f929281964f417d57986eb355f62c6b4061316b98787"},"schema_version":"1.0"},"canonical_sha256":"5fa64de658013ad19191e95281e1533f8208c5a143ca7e8b5a0e0bbc2950fe70","source":{"kind":"arxiv","id":"1007.3780","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.3780","created_at":"2026-05-18T03:34:25Z"},{"alias_kind":"arxiv_version","alias_value":"1007.3780v2","created_at":"2026-05-18T03:34:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.3780","created_at":"2026-05-18T03:34:25Z"},{"alias_kind":"pith_short_12","alias_value":"L6TE3ZSYAE5N","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"L6TE3ZSYAE5NDEMR","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"L6TE3ZSY","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:L6TE3ZSYAE5NDEMR5FJIDYKTH6","target":"record","payload":{"canonical_record":{"source":{"id":"1007.3780","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-22T01:19:46Z","cross_cats_sorted":[],"title_canon_sha256":"380030cb36dfba2bac173f16767f7d79cacf7aad0492bbc9af5fc04072c9b120","abstract_canon_sha256":"86de18d44fe820625559f929281964f417d57986eb355f62c6b4061316b98787"},"schema_version":"1.0"},"canonical_sha256":"5fa64de658013ad19191e95281e1533f8208c5a143ca7e8b5a0e0bbc2950fe70","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:25.801063Z","signature_b64":"SRWJJ3ieUOsGlX/sP4HsI82niQzRBL2mbS9kz8pvgFwi5yke2HZ2VLDqzbwB9qvXBo44RPbTrErWz7haIJolDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5fa64de658013ad19191e95281e1533f8208c5a143ca7e8b5a0e0bbc2950fe70","last_reissued_at":"2026-05-18T03:34:25.800646Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:25.800646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.3780","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":"2026-05-18T03:34:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YuJY7iO6niiCk5Q+XFW/iR/RL7+2dlLvFpFixPBQ8JZZ1AG01dqeVpSmol1isJrGHGN7HHN1tIGW1M8B6FQoDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T20:03:21.325659Z"},"content_sha256":"b866dfb4ce3d76e4984ca6450915f0af8f32a3c24a2ed499b1d3064b37b8a617","schema_version":"1.0","event_id":"sha256:b866dfb4ce3d76e4984ca6450915f0af8f32a3c24a2ed499b1d3064b37b8a617"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:L6TE3ZSYAE5NDEMR5FJIDYKTH6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Equivariant pretheories and invariants of torsors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Kirill Zainoulline, Stefan Gille","submitted_at":"2010-07-22T01:19:46Z","abstract_excerpt":"In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.3780","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":"2026-05-18T03:34:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CykNRHF2Fch+/qCkA1wCZ3skqsAJOanImUsFhe93d8e0J0dMweDBOebrhoNFu7v6s+IEso5aiLjWKPcs7jKmBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T20:03:21.326303Z"},"content_sha256":"18f6eac672978aefc815d9744009710927ec328d71267b6239d6a5891b25202a","schema_version":"1.0","event_id":"sha256:18f6eac672978aefc815d9744009710927ec328d71267b6239d6a5891b25202a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L6TE3ZSYAE5NDEMR5FJIDYKTH6/bundle.json","state_url":"https://pith.science/pith/L6TE3ZSYAE5NDEMR5FJIDYKTH6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L6TE3ZSYAE5NDEMR5FJIDYKTH6/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":"2026-06-05T20:03:21Z","links":{"resolver":"https://pith.science/pith/L6TE3ZSYAE5NDEMR5FJIDYKTH6","bundle":"https://pith.science/pith/L6TE3ZSYAE5NDEMR5FJIDYKTH6/bundle.json","state":"https://pith.science/pith/L6TE3ZSYAE5NDEMR5FJIDYKTH6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L6TE3ZSYAE5NDEMR5FJIDYKTH6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:L6TE3ZSYAE5NDEMR5FJIDYKTH6","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":"86de18d44fe820625559f929281964f417d57986eb355f62c6b4061316b98787","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-22T01:19:46Z","title_canon_sha256":"380030cb36dfba2bac173f16767f7d79cacf7aad0492bbc9af5fc04072c9b120"},"schema_version":"1.0","source":{"id":"1007.3780","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.3780","created_at":"2026-05-18T03:34:25Z"},{"alias_kind":"arxiv_version","alias_value":"1007.3780v2","created_at":"2026-05-18T03:34:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.3780","created_at":"2026-05-18T03:34:25Z"},{"alias_kind":"pith_short_12","alias_value":"L6TE3ZSYAE5N","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"L6TE3ZSYAE5NDEMR","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"L6TE3ZSY","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:18f6eac672978aefc815d9744009710927ec328d71267b6239d6a5891b25202a","target":"graph","created_at":"2026-05-18T03:34:25Z","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 the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an ","authors_text":"Kirill Zainoulline, Stefan Gille","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-22T01:19:46Z","title":"Equivariant pretheories and invariants of torsors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.3780","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:b866dfb4ce3d76e4984ca6450915f0af8f32a3c24a2ed499b1d3064b37b8a617","target":"record","created_at":"2026-05-18T03:34:25Z","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":"86de18d44fe820625559f929281964f417d57986eb355f62c6b4061316b98787","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-22T01:19:46Z","title_canon_sha256":"380030cb36dfba2bac173f16767f7d79cacf7aad0492bbc9af5fc04072c9b120"},"schema_version":"1.0","source":{"id":"1007.3780","kind":"arxiv","version":2}},"canonical_sha256":"5fa64de658013ad19191e95281e1533f8208c5a143ca7e8b5a0e0bbc2950fe70","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5fa64de658013ad19191e95281e1533f8208c5a143ca7e8b5a0e0bbc2950fe70","first_computed_at":"2026-05-18T03:34:25.800646Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:25.800646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SRWJJ3ieUOsGlX/sP4HsI82niQzRBL2mbS9kz8pvgFwi5yke2HZ2VLDqzbwB9qvXBo44RPbTrErWz7haIJolDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:25.801063Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.3780","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b866dfb4ce3d76e4984ca6450915f0af8f32a3c24a2ed499b1d3064b37b8a617","sha256:18f6eac672978aefc815d9744009710927ec328d71267b6239d6a5891b25202a"],"state_sha256":"962d463ee5927f920669c9c42f1414365182cdba6c2cc6cf8dd26d0b6d57e36a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qqVr+NF9rTRneL0ewGYiK3z9oT6gw3V5SHWsbWqel40PcmFmwoXCFrzLxO9T5zxnigAW8//o9dyjAM0FOS0pBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T20:03:21.330276Z","bundle_sha256":"f364d78d048c2befae1f6b6a6f3b2ad4398326905615c1e2947147b7fa5d9620"}}