{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:ML3U3TNUTXTY4OZF5IYE6ZFYFG","short_pith_number":"pith:ML3U3TNU","schema_version":"1.0","canonical_sha256":"62f74dcdb49de78e3b25ea304f64b829aa5ecd7f721c49387b256acabfe5b905","source":{"kind":"arxiv","id":"1611.03644","version":2},"attestation_state":"computed","paper":{"title":"The spectrum for commutative complex $K$-theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Simon Gritschacher","submitted_at":"2016-11-11T10:14:38Z","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1611.03644","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-11-11T10:14:38Z","cross_cats_sorted":[],"title_canon_sha256":"1322bb83abe1d67ae1d968a7b3788e168dfc510be6830073a15905dc28b72aaf","abstract_canon_sha256":"ac56ba006a772ff58bafc72c6b6bd31d0cb122b346f6262dcac9b93a0e0f9e18"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:56.987563Z","signature_b64":"/ViK1ePAJiQGeYO/08bGxN8HkN/dIG1bgJNwjprj0HUuWlhEsrrh5G2OjnZPcMnrpYs06BUe74bua+YILTPBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62f74dcdb49de78e3b25ea304f64b829aa5ecd7f721c49387b256acabfe5b905","last_reissued_at":"2026-05-18T00:20:56.987010Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:56.987010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The spectrum for commutative complex $K$-theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Simon Gritschacher","submitted_at":"2016-11-11T10:14:38Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03644","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1611.03644","created_at":"2026-05-18T00:20:56.987102+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.03644v2","created_at":"2026-05-18T00:20:56.987102+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03644","created_at":"2026-05-18T00:20:56.987102+00:00"},{"alias_kind":"pith_short_12","alias_value":"ML3U3TNUTXTY","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"ML3U3TNUTXTY4OZF","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"ML3U3TNU","created_at":"2026-05-18T12:30:32.724797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ML3U3TNUTXTY4OZF5IYE6ZFYFG","json":"https://pith.science/pith/ML3U3TNUTXTY4OZF5IYE6ZFYFG.json","graph_json":"https://pith.science/api/pith-number/ML3U3TNUTXTY4OZF5IYE6ZFYFG/graph.json","events_json":"https://pith.science/api/pith-number/ML3U3TNUTXTY4OZF5IYE6ZFYFG/events.json","paper":"https://pith.science/paper/ML3U3TNU"},"agent_actions":{"view_html":"https://pith.science/pith/ML3U3TNUTXTY4OZF5IYE6ZFYFG","download_json":"https://pith.science/pith/ML3U3TNUTXTY4OZF5IYE6ZFYFG.json","view_paper":"https://pith.science/paper/ML3U3TNU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.03644&json=true","fetch_graph":"https://pith.science/api/pith-number/ML3U3TNUTXTY4OZF5IYE6ZFYFG/graph.json","fetch_events":"https://pith.science/api/pith-number/ML3U3TNUTXTY4OZF5IYE6ZFYFG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ML3U3TNUTXTY4OZF5IYE6ZFYFG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ML3U3TNUTXTY4OZF5IYE6ZFYFG/action/storage_attestation","attest_author":"https://pith.science/pith/ML3U3TNUTXTY4OZF5IYE6ZFYFG/action/author_attestation","sign_citation":"https://pith.science/pith/ML3U3TNUTXTY4OZF5IYE6ZFYFG/action/citation_signature","submit_replication":"https://pith.science/pith/ML3U3TNUTXTY4OZF5IYE6ZFYFG/action/replication_record"}},"created_at":"2026-05-18T00:20:56.987102+00:00","updated_at":"2026-05-18T00:20:56.987102+00:00"}