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Sims: Graded Steinberg algebras and their representations , Algebra Number Theory 12(1) (2018), 131–172. 3, 8, 9","work_id":"cbcd47ac-7ee3-41ac-9a0f-14f8dc086375","year":2018}],"snapshot_sha256":"512ea57910ab621b0c7b26b27aa9cf50274095fa7be8021b07e4bf5d6e8c4640"},"source":{"id":"2507.19879","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-19T03:33:04.548863Z","id":"ce8b554c-79dc-44c5-a1e6-f76c5bc70e27","model_set":{"reader":"grok-4.3"},"one_line_summary":"Establishes a Z[Z^k]-module isomorphism between graded H_0 of the groupoid and graded K_0 of the Kumjian-Pask algebra for k-graphs, shows preservation under graph moves, and provides a lifting criterion for homomorphisms.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"For row-finite k-graphs without sources, the graded zeroth homology of the infinite path groupoid is isomorphic as a Z[Z^k]-module to the graded Grothendieck group of the Kumjian-Pask algebra, preserving positive cones.","strongest_claim":"For a row-finite k-graph Λ without sources, there exists a Z[Z^k]-module isomorphism between the graded zeroth (integral) homology H_0^{gr}(G_Λ) of the infinite path groupoid G_Λ and the graded Grothendieck group K_0^{gr}(KP_k(Λ)) of the Kumjian-Pask algebra KP_k(Λ), which respects the positive cones (i.e., the talented monoids).","weakest_assumption":"The k-graph Λ is row-finite and has no sources; this assumption is used to define the infinite path groupoid G_Λ and to ensure the Kumjian-Pask algebra is well-behaved for the homology and K-theory constructions (abstract, first paragraph)."}},"verdict_id":"ce8b554c-79dc-44c5-a1e6-f76c5bc70e27"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d63124143ff642427ca9c9ce6f19b953c3c0d9948087145ed3a5b350db81d4eb","target":"record","created_at":"2026-05-20T14:03:19Z","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":"2e5200302cbbd3b56a6c54e4f374470a735f939a8bd86b82fdbc621f1b49a3bd","cross_cats_sorted":["math.OA","math.RA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.KT","submitted_at":"2025-07-26T09:20:11Z","title_canon_sha256":"f882df143a87c233b8a958132d9badc2224064e690c369c9712aed00efe98469"},"schema_version":"1.0","source":{"id":"2507.19879","kind":"arxiv","version":2}},"canonical_sha256":"dee64c1deb812fb293594e54471e9707b6fc07160e49297b163eb333e22f8d25","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dee64c1deb812fb293594e54471e9707b6fc07160e49297b163eb333e22f8d25","first_computed_at":"2026-05-20T14:03:19.795296Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T14:03:19.795296Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GStru+Jkov3Hda8rXhdnJRSu7mXd/JNpmslDQ/FuxJk1A45V/U5+FFB5nJ+ssZhIXx14q7oSPTqO95d8lcsTAw==","signature_status":"signed_v1","signed_at":"2026-05-20T14:03:19.795846Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.19879","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d63124143ff642427ca9c9ce6f19b953c3c0d9948087145ed3a5b350db81d4eb","sha256:5308b586620bde20b020c5a713a2b268076e4065b0a39bfcc4bcf41c415d92a4"],"state_sha256":"92679f46befb4342197540184d33fb89bac38de00fc15d7897911b895acbd960"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"837CMUOYrP4RLXe6qaNyXs0w72sCGRlqnPPnDmuW7V0YZ0hzK8YUyfr5FU0LxtDF7xmEZ9TGu94aXcIWU6i+Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T10:24:30.690459Z","bundle_sha256":"871ea8773ce98b88d8d13dc823417db9275f2dd1ab7d318571b4afbb9728f665"}}