{"paper":{"title":"Cohomology of Burnside Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.GR"],"primary_cat":"math.RA","authors_text":"Benen Harrington","submitted_at":"2019-08-16T20:31:16Z","abstract_excerpt":"Let $G$ be a finite group and $A(G)$ its Burnside ring. For $H \\subset G$ let $\\mathbb{Z}_H$ denote the $A(G)$-module corresponding to the mark homomorphism associated to $H$. When the order of $G$ is square-free we give a complete description of the $A(G)$-modules $\\textrm{Ext}^l_{A(G)}(\\mathbb{Z}_H, \\mathbb{Z}_J)$ and $\\textrm{Tor}^{A(G)}_l(\\mathbb{Z}_H, \\mathbb{Z}_J)$ for any $H, J \\subset G$ and $l \\geq 0$. We show that if the order of $G$ is not square-free then there exist $H, J \\subset G$ such that $\\textrm{Ext}^l_{A(G)}(\\mathbb{Z}_H, \\mathbb{Z}_J)$ and $\\textrm{Tor}^{A(G)}_l(\\mathbb{Z}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06156","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/1908.06156/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"}