{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:PPEBPXTHXO2FYQBPJCMHZFTDQC","short_pith_number":"pith:PPEBPXTH","schema_version":"1.0","canonical_sha256":"7bc817de67bbb45c402f48987c966380bc1c8d241f2797efd820a043a3d3d7a3","source":{"kind":"arxiv","id":"2404.13776","version":1},"attestation_state":"computed","paper":{"title":"Hopf algebras, Steinberg modules, and the unstable cohomology of $SL_n(\\mathbb Z)$ and $GL_n(\\mathbb Z)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.KT","math.NT"],"primary_cat":"math.AT","authors_text":"Avner Ash, Jeremy Miller, Peter Patzt","submitted_at":"2024-04-21T21:11:56Z","abstract_excerpt":"We prove that the direct sum of all homology groups of the integral general linear groups with Steinberg module coefficients form a commutative Hopf algebra, in particular a free graded commutative algebra. We use this to construct new infinite families of unstable cohomology classes of $SL_n(\\mathbb Z)$."},"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":"2404.13776","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2024-04-21T21:11:56Z","cross_cats_sorted":["math.KT","math.NT"],"title_canon_sha256":"d8bde1fb0867b19b386dc6f341d5bd86dedfea9bdc2d0f851b0322dab7d5ba22","abstract_canon_sha256":"b2f669571a231798de780f2bb9110e089505131001c7411bbaffcd4e57441d15"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:10:36.974248Z","signature_b64":"XDqrs1d0D+4/Cf+v9v5bo6R6RHLMwu6YNCwARRss4Xiscu4MXaT6KMHm1T6N6lit5bLoCQaQWzO142CYPfvdBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7bc817de67bbb45c402f48987c966380bc1c8d241f2797efd820a043a3d3d7a3","last_reissued_at":"2026-07-05T08:10:36.973869Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:10:36.973869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hopf algebras, Steinberg modules, and the unstable cohomology of $SL_n(\\mathbb Z)$ and $GL_n(\\mathbb Z)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.KT","math.NT"],"primary_cat":"math.AT","authors_text":"Avner Ash, Jeremy Miller, Peter Patzt","submitted_at":"2024-04-21T21:11:56Z","abstract_excerpt":"We prove that the direct sum of all homology groups of the integral general linear groups with Steinberg module coefficients form a commutative Hopf algebra, in particular a free graded commutative algebra. We use this to construct new infinite families of unstable cohomology classes of $SL_n(\\mathbb Z)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.13776","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.13776/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2404.13776","created_at":"2026-07-05T08:10:36.973924+00:00"},{"alias_kind":"arxiv_version","alias_value":"2404.13776v1","created_at":"2026-07-05T08:10:36.973924+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.13776","created_at":"2026-07-05T08:10:36.973924+00:00"},{"alias_kind":"pith_short_12","alias_value":"PPEBPXTHXO2F","created_at":"2026-07-05T08:10:36.973924+00:00"},{"alias_kind":"pith_short_16","alias_value":"PPEBPXTHXO2FYQBP","created_at":"2026-07-05T08:10:36.973924+00:00"},{"alias_kind":"pith_short_8","alias_value":"PPEBPXTH","created_at":"2026-07-05T08:10:36.973924+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.23863","citing_title":"The Goncharov Lie coalgebra of a field","ref_index":4,"is_internal_anchor":false},{"citing_arxiv_id":"2605.06499","citing_title":"A projective resolution of the symplectic Steinberg module","ref_index":13,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PPEBPXTHXO2FYQBPJCMHZFTDQC","json":"https://pith.science/pith/PPEBPXTHXO2FYQBPJCMHZFTDQC.json","graph_json":"https://pith.science/api/pith-number/PPEBPXTHXO2FYQBPJCMHZFTDQC/graph.json","events_json":"https://pith.science/api/pith-number/PPEBPXTHXO2FYQBPJCMHZFTDQC/events.json","paper":"https://pith.science/paper/PPEBPXTH"},"agent_actions":{"view_html":"https://pith.science/pith/PPEBPXTHXO2FYQBPJCMHZFTDQC","download_json":"https://pith.science/pith/PPEBPXTHXO2FYQBPJCMHZFTDQC.json","view_paper":"https://pith.science/paper/PPEBPXTH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2404.13776&json=true","fetch_graph":"https://pith.science/api/pith-number/PPEBPXTHXO2FYQBPJCMHZFTDQC/graph.json","fetch_events":"https://pith.science/api/pith-number/PPEBPXTHXO2FYQBPJCMHZFTDQC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PPEBPXTHXO2FYQBPJCMHZFTDQC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PPEBPXTHXO2FYQBPJCMHZFTDQC/action/storage_attestation","attest_author":"https://pith.science/pith/PPEBPXTHXO2FYQBPJCMHZFTDQC/action/author_attestation","sign_citation":"https://pith.science/pith/PPEBPXTHXO2FYQBPJCMHZFTDQC/action/citation_signature","submit_replication":"https://pith.science/pith/PPEBPXTHXO2FYQBPJCMHZFTDQC/action/replication_record"}},"created_at":"2026-07-05T08:10:36.973924+00:00","updated_at":"2026-07-05T08:10:36.973924+00:00"}