{"paper":{"title":"Powers of the Phantom Ideal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CT","authors_text":"Ivo Herzog, Xianhui Fu","submitted_at":"2013-12-18T21:38:42Z","abstract_excerpt":"It is proved that if G is a finite group, then the order of G is a proper upper bound for the phantom number of G. More specifically, if k is a field whose characteristic divides the order of G, and $\\Phi$ is the ideal of phantom morphisms in the stable category k[G]-$\\underline{\\rm Mod}$ of modules over the group algebra k[G], then $\\Phi^{n-1} = 0,$ where n is the nilpotency index of the Jacobson radical J of k[G]. If $R$ is a semiprimary ring, with $J^n =0,$ and $\\Phi$ denotes the phantom ideal in the module category R-Mod, then $\\Phi^n$ is the ideal of morphisms that factor through a projec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5348","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":""},"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"}