{"paper":{"title":"K-theory of Azumaya algebras over schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"R. Hazrat, R. Hoobler","submitted_at":"2009-11-07T10:30:27Z","abstract_excerpt":"Let $X$ be a connected, noetherian scheme and $\\mathcal{A}$ be a sheaf of Azumaya algebras on $X$ which is a locally free $\\mathcal{O}_{X}$-module of rank $a$. We show that the kernel and cokernel of $K_{i}(X) \\to K_{i}(\\mathcal{A}) $ are torsion groups with exponent $a^{m}$ for some $m$ and any $i\\geq 0$, when $X$ is regular or $X$ is of dimension $d$ with an ample sheaf (in this case $m\\leq d+1$). As a consequence, $K_{i}(X,\\mathbb Z/m)\\cong K_{i}(\\mA,\\mathbb Z/m)$, for any $m$ relatively prime to $a$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.1406","kind":"arxiv","version":2},"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"}