{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:Z2LWI3PBBW7M3YHIDVOPUILA2W","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":"4202b8847fdcdda2a72ed973b0836c1f3dbc7ba43c79c3716694ab744f2c6e0d","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-10-13T14:22:35Z","title_canon_sha256":"c3298e252f9e28d3dc8fa6e3766e6851b1dadb73d2f0c2196f984809f939b9f5"},"schema_version":"1.0","source":{"id":"1110.2956","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.2956","created_at":"2026-05-18T00:53:51Z"},{"alias_kind":"arxiv_version","alias_value":"1110.2956v2","created_at":"2026-05-18T00:53:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.2956","created_at":"2026-05-18T00:53:51Z"},{"alias_kind":"pith_short_12","alias_value":"Z2LWI3PBBW7M","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"Z2LWI3PBBW7M3YHI","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"Z2LWI3PB","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:e839cac38479090f27072c6dd078a47930d5daa3df3ed0f0e1ce7fa16e27024d","target":"graph","created_at":"2026-05-18T00:53:51Z","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":"Using an analogy between the Brauer groups in algebra and the Whitehead groups in topology, we first use methods of algebraic K-theory to give a natural definition of Brauer spectra for commutative rings, such that their homotopy groups are given by the Brauer group, the Picard group and the group of units. Then, in the context of structured ring spectra, the same idea leads to two-fold non-connected deloopings of the spectra of units.","authors_text":"Markus Szymik","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-10-13T14:22:35Z","title":"Brauer spaces for commutative rings and structured ring spectra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2956","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:dbfb9ddcff8cfe024ca22d8286ccf19b33ea3b22867105ec5635d27047b7844a","target":"record","created_at":"2026-05-18T00:53:51Z","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":"4202b8847fdcdda2a72ed973b0836c1f3dbc7ba43c79c3716694ab744f2c6e0d","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-10-13T14:22:35Z","title_canon_sha256":"c3298e252f9e28d3dc8fa6e3766e6851b1dadb73d2f0c2196f984809f939b9f5"},"schema_version":"1.0","source":{"id":"1110.2956","kind":"arxiv","version":2}},"canonical_sha256":"ce97646de10dbecde0e81d5cfa2160d5b482a2b3adb9a03a0637c24a610a4123","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce97646de10dbecde0e81d5cfa2160d5b482a2b3adb9a03a0637c24a610a4123","first_computed_at":"2026-05-18T00:53:51.759957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:51.759957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o7VCZbR4n8ZAMwgdwD9jxEgsD+CO05msrsgnnb52YrX3vOhkPnoOfkQXMbHr/sum3/uyt7YwZY7xXeDUHmUYBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:51.760466Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.2956","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dbfb9ddcff8cfe024ca22d8286ccf19b33ea3b22867105ec5635d27047b7844a","sha256:e839cac38479090f27072c6dd078a47930d5daa3df3ed0f0e1ce7fa16e27024d"],"state_sha256":"c746265e7084f0c4bf3a9e7a15377af47edc677e85500046ab9288630947ca97"}