{"paper":{"title":"Cohomology for Frobenius kernels of $SL_2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.QA"],"primary_cat":"math.RT","authors_text":"Nham V. Ngo","submitted_at":"2012-09-07T22:20:44Z","abstract_excerpt":"Let $(SL_2)_r$ be the $r$-th Frobenius kernels of the group scheme $SL_2$ defined over an algebraically field of characteristic $p>2$. In this paper we give for $r\\ge 1$ a complete description of the cohomology groups for $(SL_2)_r$. We also prove that the reduced cohomology ring $\\opH^\\bullet((SL_2)_r,k)_{\\red}$ is Cohen-Macaulay. Geometrically, we show for each $r\\ge 1$ that the maximal ideal spectrum of the cohomology ring for $(SL_2)_r$ is homeomorphic to the fiber product $G\\times_B\\fraku^r$. Finally, we adapt our calculations to obtain analogous results for the cohomology of higher Frobe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1662","kind":"arxiv","version":3},"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"}