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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."},"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":"1301.4928","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-01-21T17:10:34Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"0f97c27da21e6e10152c7bd206b2da1d27fe9cf89b05c1378f287f6d7da68f40","abstract_canon_sha256":"c04c5a8afd7b9b7af6fb6accd25a57d7e9ad3247871fed531cec5ee871da3ee4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:30.097616Z","signature_b64":"+Ja2UC18wym5BJtRWT5+o3snPQLcnie179+elpf0IeZ0y0clqSAwgNrOwgzESXmM3tDnwPdGEjUs65NbDwuoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37570ce2c20076336eb5bd5e14ad8e8ef46fcf5233d27769f0f97686b75c5dc7","last_reissued_at":"2026-05-18T00:44:30.097122Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:30.097122Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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