{"paper":{"title":"The classifying topos of a group scheme and invariants of symmetric bundles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Baptiste Morin, Martin J. Taylor, Philippe Cassou-Nogu\\`es, Ted Chinburg","submitted_at":"2013-01-21T17:10:34Z","abstract_excerpt":"Let $Y$ be a scheme in which 2 is invertible and let $V$ be a rank $n$ vector bundle on $Y$ endowed with a non-degenerate symmetric bilinear form $q$. The orthogonal group ${\\bf O}(q)$ of the form $q$ is a group scheme over $Y$ whose cohomology ring $H^*(B_{{\\bf O}(q)},{\\bf Z}/2{\\bf Z})\\simeq A_Y[HW_1(q),..., HW_n(q)]$ is a polynomial algebra over the \\'etale cohomology ring $A_Y:=H^*(Y_{et},{\\bf Z}/2{\\bf Z})$ of the scheme $Y$. Here the $HW_i(q)$'s are Jardine's universal Hasse-Witt invariants and $B_{{\\bf O}(q)}$ is the classifying topos of ${\\bf O}(q)$ as defined by Grothendieck and Giraud."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.4928","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"}