{"paper":{"title":"Higher-rank graphs and the graded $K$-theory of Kumjian-Pask algebras","license":"http://creativecommons.org/licenses/by/4.0/","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.","cross_cats":["math.OA","math.RA"],"primary_cat":"math.KT","authors_text":"David Pask, Promit Mukherjee, Roozbeh Hazrat, Sujit Kumar Sardar","submitted_at":"2025-07-26T09:20:11Z","abstract_excerpt":"This paper lays out the foundations of graded $K$-theory for Leavitt algebras associated with higher-rank graphs, also known as Kumjian-Pask algebras, establishing it as a potential tool for their classification.\n  For a row-finite $k$-graph $\\Lambda$ without sources, we show that there exists a $\\mathbb{Z}[\\mathbb{Z}^k]$-module isomorphism between the graded zeroth (integral) homology $H_0^{gr}(\\mathcal{G}_\\Lambda)$ of the infinite path groupoid $\\mathcal{G}_\\Lambda$ and the graded Grothendieck group $K_0^{gr}(KP_\\mathsf{k}(\\Lambda))$ of the Kumjian-Pask algebra $KP_\\mathsf{k}(\\Lambda)$, whic"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4e01e0aeb296156cdaa7b2b2016684b80b5204655c4e77d6227f86b6dbd488df"},"source":{"id":"2507.19879","kind":"arxiv","version":2},"verdict":{"id":"ce8b554c-79dc-44c5-a1e6-f76c5bc70e27","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T03:33:04.548863Z","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).","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","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).","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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.19879/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":47,"sample":[{"doi":"","year":null,"title":"G. Abrams, P. Ara, M. Siles Molina: Leavitt Path Algebras, Lecture Notes in Mathematics, vol. 2191, Springer Verlag,","work_id":"838f5b8d-ec1a-47be-9572-3bc21784bfaa","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"G. Abrams, E. Ruiz, M. Tomforde: Recasting the Hazrat conjecture: Relating shift equivalence to graded Morita equivalence, Algebr. Represent. Theory 27 (2024), 1477–1511. 2, 4, 34, 36, 42","work_id":"834b2bee-c637-4960-9366-15fcf3cdf0b6","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"P. Ara, J. Bosa, E. Pardo, A. Sims: The groupoids of adaptable separated graphs and their semigroups , Int. Math. Res. Not. IMRN (2021), 15444–15496. 3, 28","work_id":"7d968c32-a018-4f3a-8aed-b8f2c6484deb","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"P. Ara, C. B¨ onicke, J. Bosa, K. Li: The type semigroup, comparison, and almost finiteness for ample groupoids , Ergod. Th. & Dynam. Sys. 43(2) (2023), 361–400. 3, 26","work_id":"54dbd213-3584-489d-9729-bd123666d0a5","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"P. Ara, R. Hazrat, H. Li, A. 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","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":47,"snapshot_sha256":"512ea57910ab621b0c7b26b27aa9cf50274095fa7be8021b07e4bf5d6e8c4640","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"9a9fa1686a48c58057a5256b7a3cc0d74f781c51d12b1a3dc527edc144710e10"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}