{"paper":{"title":"Permutation modules and Chow motives of geometrically rational surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Stefan Gille","submitted_at":"2015-05-28T19:37:43Z","abstract_excerpt":"We prove that the Chow motive with integral coefficient of a geometrically rational surfaces~$S$ over a perfect field~$k$ is zero dimensional if and only if the Picard group of~$\\bar{k}\\times_{k}S$, where~$\\bar{k}$ is an algebraic closure of~$k$, is a direct summand of a $\\Gal (\\bar{k}/k)$-permutation module, and~$S$ possesses a zero cycle of degree one. As shown by Colliot-Th\\'el\\`ene in a letter to the author (which we have reproduced in the appendix) this is in turn equivalent to~$S$ having a zero cycle of degree~$1$ and $\\CH_{0}(k(S)\\times_{k}S)$ being torsion free."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.07819","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"}