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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1505.07819","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-05-28T19:37:43Z","cross_cats_sorted":[],"title_canon_sha256":"cfa48f16cb4167e8a017a7b4cf8bbd230e44789487bc4ec126e2dd1bc9ab2013","abstract_canon_sha256":"2a9884bc2c4640b558dd2c875c090c0efeab71388b7821b721497c6204fa35c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:00:55.754932Z","signature_b64":"naThjQyKFFO3TvJdKEXonnMKBlgKz6/OwORt3MkvDGEfnnFoJtiEDRB7ZcfVwcbKgCyjMuLXpa5NQy3b59WUBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce31c9bc710b13606ff511cc8bd38d6c07e496472937eb19abdd1fe2c255c52b","last_reissued_at":"2026-05-18T02:00:55.754280Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:00:55.754280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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