{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZTURQAV4WB2WXFC32RRBTCUISR","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":"c9b24df525d77181b672fafe40ce019d491eebafa7f8198bf667404bffec0b66","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-09-02T09:23:42Z","title_canon_sha256":"78ec93c6168496eb62706b98bb195e7eadb4edefafccd6f827df6b5040333f0c"},"schema_version":"1.0","source":{"id":"1609.00516","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.00516","created_at":"2026-05-18T00:16:45Z"},{"alias_kind":"arxiv_version","alias_value":"1609.00516v4","created_at":"2026-05-18T00:16:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.00516","created_at":"2026-05-18T00:16:45Z"},{"alias_kind":"pith_short_12","alias_value":"ZTURQAV4WB2W","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZTURQAV4WB2WXFC3","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZTURQAV4","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:6813fd76b7433925365409d411bafaa30374e7149d5ac10ef5c7b9675df82b33","target":"graph","created_at":"2026-05-18T00:16:45Z","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":"Grothendieck proved that any finite epimorphism of noetherian schemes factors into a finite sequence of effective epimorphisms. We define the complexity of a flat groupoid $R\\rightrightarrows X$ with finite stabilizer to be the length of the canonical sequence of the finite map $R\\to X\\times\\_{X/R} X$, where $X/R$ is the Keel-Mori geometric quotient. For groupoids of complexity at most 1, we prove a theorem of descent along the quotient $X\\to X/R$ and a theorem of quotient of a groupoid by a normal subgroupoid. We expect that the complexity could play an important role in the finer study of qu","authors_text":"David Rydh (KTH), Gabriel Zalamansky (IMJ-PRG), Matthieu Romagny (IRMAR)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-09-02T09:23:42Z","title":"The complexity of a flat groupoid"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00516","kind":"arxiv","version":4},"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:f02a9702b8104b09c3d785fa198c41f070d0a798cd2e53330452582c933d58ed","target":"record","created_at":"2026-05-18T00:16:45Z","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":"c9b24df525d77181b672fafe40ce019d491eebafa7f8198bf667404bffec0b66","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-09-02T09:23:42Z","title_canon_sha256":"78ec93c6168496eb62706b98bb195e7eadb4edefafccd6f827df6b5040333f0c"},"schema_version":"1.0","source":{"id":"1609.00516","kind":"arxiv","version":4}},"canonical_sha256":"cce91802bcb0756b945bd462198a889461104d1bea8eeb162c58399dd0471d04","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cce91802bcb0756b945bd462198a889461104d1bea8eeb162c58399dd0471d04","first_computed_at":"2026-05-18T00:16:45.766619Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:45.766619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VwmLBF9z0SXxKqFRI/szZ2BWC+t5JayC6+co9zXc09Xj0xum4zlccGAhMP8QZk75zUA3r6XyPsU+qIcJel4ICQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:45.767337Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.00516","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f02a9702b8104b09c3d785fa198c41f070d0a798cd2e53330452582c933d58ed","sha256:6813fd76b7433925365409d411bafaa30374e7149d5ac10ef5c7b9675df82b33"],"state_sha256":"8701af2424cf0a224395657b57d9c05f94a0cee9a938920eb48c03f5f3a79940"}