{"paper":{"title":"Homological properties of simple modules over Leavitt path algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Simple modules over Leavitt path algebras associated to cycles and irreducible polynomials have explicitly constructed projective resolutions, from which the dimensions of their extension spaces are computed.","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alberto Tonolo, Francesca Mantese","submitted_at":"2026-04-13T09:47:32Z","abstract_excerpt":"Let $K$ be any field, and let $E$ be any graph. We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra $L_K(E)$ associated to cycles and irreducible polynomials. Then we study the dimension of the $K$-vector space of the extensions between two such simple modules."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra L_K(E) associated to cycles and irreducible polynomials. Then we study the dimension of the K-vector space of the extensions between two such simple modules.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The explicitness of the resolutions and the resulting dimension formulas hold for arbitrary graphs E and any field K, without additional restrictions on the cycles or polynomials that would invalidate the constructions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Explicit projective resolutions are built for simple modules over Leavitt path algebras tied to cycles and irreducible polynomials, followed by computation of extension dimensions between them.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Simple modules over Leavitt path algebras associated to cycles and irreducible polynomials have explicitly constructed projective resolutions, from which the dimensions of their extension spaces are computed.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a5ea3bedfaa7509b9ea4935822a3fbbc7fd287e924175c6c3cd225b6af201437"},"source":{"id":"2604.11241","kind":"arxiv","version":2},"verdict":{"id":"01dec7a6-8f6e-4404-94b4-a45f411f7f6a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:56:38.596207Z","strongest_claim":"We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra L_K(E) associated to cycles and irreducible polynomials. Then we study the dimension of the K-vector space of the extensions between two such simple modules.","one_line_summary":"Explicit projective resolutions are built for simple modules over Leavitt path algebras tied to cycles and irreducible polynomials, followed by computation of extension dimensions between them.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The explicitness of the resolutions and the resulting dimension formulas hold for arbitrary graphs E and any field K, without additional restrictions on the cycles or polynomials that would invalidate the constructions.","pith_extraction_headline":"Simple modules over Leavitt path algebras associated to cycles and irreducible polynomials have explicitly constructed projective resolutions, from which the dimensions of their extension spaces are computed."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11241/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"}