{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ML3U3TNUTXTY4OZF5IYE6ZFYFG","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":"ac56ba006a772ff58bafc72c6b6bd31d0cb122b346f6262dcac9b93a0e0f9e18","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-11-11T10:14:38Z","title_canon_sha256":"1322bb83abe1d67ae1d968a7b3788e168dfc510be6830073a15905dc28b72aaf"},"schema_version":"1.0","source":{"id":"1611.03644","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.03644","created_at":"2026-05-18T00:20:56Z"},{"alias_kind":"arxiv_version","alias_value":"1611.03644v2","created_at":"2026-05-18T00:20:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03644","created_at":"2026-05-18T00:20:56Z"},{"alias_kind":"pith_short_12","alias_value":"ML3U3TNUTXTY","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"ML3U3TNUTXTY4OZF","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"ML3U3TNU","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:67cd91837fe647944432fb65fc0a2e2e662c488857e31744528faf2a1773b940","target":"graph","created_at":"2026-05-18T00:20:56Z","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":"We study commutative complex $K$-theory, a generalised cohomology theory built from spaces of ordered commuting tuples in the unitary groups. We show that the spectrum for commutative complex $K$-theory is stably equivalent to the $ku$-group ring of $BU(1)$ and thus obtain a splitting of its representing space $B_{com}U$ as a product of all the terms in the Whitehead tower for $BU$, $B_{com}U\\simeq BU\\times BU\\langle 4\\rangle \\times BU\\langle 6\\rangle \\times \\dots .$ As a consequence of the spectrum level identification we obtain the ring of coefficients for this theory. Using the rational Hop","authors_text":"Simon Gritschacher","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-11-11T10:14:38Z","title":"The spectrum for commutative complex $K$-theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03644","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:ba23168b750c1a559b77c13ef9ba4d315440f787b331e94513a0744b56906008","target":"record","created_at":"2026-05-18T00:20:56Z","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":"ac56ba006a772ff58bafc72c6b6bd31d0cb122b346f6262dcac9b93a0e0f9e18","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-11-11T10:14:38Z","title_canon_sha256":"1322bb83abe1d67ae1d968a7b3788e168dfc510be6830073a15905dc28b72aaf"},"schema_version":"1.0","source":{"id":"1611.03644","kind":"arxiv","version":2}},"canonical_sha256":"62f74dcdb49de78e3b25ea304f64b829aa5ecd7f721c49387b256acabfe5b905","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62f74dcdb49de78e3b25ea304f64b829aa5ecd7f721c49387b256acabfe5b905","first_computed_at":"2026-05-18T00:20:56.987010Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:56.987010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/ViK1ePAJiQGeYO/08bGxN8HkN/dIG1bgJNwjprj0HUuWlhEsrrh5G2OjnZPcMnrpYs06BUe74bua+YILTPBAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:56.987563Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.03644","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba23168b750c1a559b77c13ef9ba4d315440f787b331e94513a0744b56906008","sha256:67cd91837fe647944432fb65fc0a2e2e662c488857e31744528faf2a1773b940"],"state_sha256":"53b2e5a731410e9e14e93ba2438b33510eedc578e89b71e2681979a0a6b37073"}