{"paper":{"title":"The Brauer group and the Brauer-Manin set of products of varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Alexei N. Skorobogatov, Yuri G. Zarhin","submitted_at":"2011-12-14T01:10:40Z","abstract_excerpt":"Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $\\mathbb Q$, and let $\\ov X$ and $\\ov Y$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of $\\Br(\\ov X)\\oplus \\Br(\\ov Y)$ has finite index in the Galois invariant subgroup of $\\Br(\\ov X\\times\\ov Y)$. This implies that the cokernel of the natural map $\\Br(X)\\oplus\\Br(Y)\\to\\Br(X\\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer-Manin set of the product"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3089","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"}