{"paper":{"title":"Laurent coefficients and Ext of finite graded modules","license":"","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Judith D. Sally, Luchezar L. Avramov, Ragnar-Olaf Buchweitz","submitted_at":"1994-09-23T00:00:00Z","abstract_excerpt":"Let $R=\\bigoplus_{n\\ges0}R_n$ be a graded commutative ring generated over a field $K=R_0$ by homogeneous elements $x_1,\\dots,x_e$ of positive degrees $d_1,\\dots,d_e$. The Hilbert-Serre Theorem shows that for each finite graded $R$--module $M=\\bigoplus_{n\\in\\BZ}M_n$ the {\\it Hilbert series\\/} $\\sum_{n\\in\\BZ}(\\rank_K M_n)t^n$ is the Laurent expansion around $0$ of a rational function\n  $$ H_M(t)=\\frac{q_M(t)}{\\prod_{i=1}^e(1-t^{d_i})} $$\n with $q_M(t)\\in\\BZ[t,\\ti]$.  We demonstrate that Laurent expansions $\\left[M\\right]_z$ of $H_M(t)$ around other points $z$ of the extended complex plane $\\over"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9409208","kind":"arxiv","version":1},"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"}