{"paper":{"title":"Reductive group schemes, the Greenberg functor, and associated algebraic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alexander Stasinski","submitted_at":"2010-03-18T14:50:36Z","abstract_excerpt":"Let $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\\mathcal{F}$ associates to $\\mathbf{G}$ a linear algebraic group $G:=(\\mathcal{F}\\mathbf{G})(k)$ over $k$, such that $G\\cong\\mathbf{G}(A)$. We prove that if $\\mathbf{G}$ is a reductive group scheme over $A$, and $\\mathbf{T}$ is a maximal torus of $\\mathbf{G}$, then $T$ is a Cartan subgroup of $G$, and every Cartan subgroup of $G$ is obtained uniquely in this way. The proof is based on establishing a Nullstellensatz analogue for sm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3598","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"}