{"paper":{"title":"A matrix description for $K_1$ of graded rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.KT","authors_text":"Zuhong Zhang","submitted_at":"2013-07-26T07:20:44Z","abstract_excerpt":"The current paper is dedicated to the study of the classical $K_1$ groups of graded rings. Let $A$ be a $\\Gamma$ graded ring with identity $1$, where the grading $\\Gamma$ is an abelian group. We associate a category with suspension to the $\\Gamma$ graded ring $A$. This allows us to construct the group valued functor $K_1$ of graded rings. It will be denoted by $K_1^{gr}$. It is not only an abelian group but also a $\\mathbb Z[\\Gamma]$-module. From the construction, it follows that there exists \"locally\" a matrix description of $K_1^{gr}$ of graded rings. The matrix description makes it possible"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6940","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"}