{"paper":{"title":"On the rate of graded modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Maryam Jahangiri, Rasoul Ahangari Maleki","submitted_at":"2014-10-30T10:56:18Z","abstract_excerpt":"Let $K$ be a field, $R$ a standard graded $K$-algebra and $M$ be a finitely generated graded $R$-module. The rate of $M$, $rate_R(M)$, is a measure of the growth of the shifts in the minimal graded free resolution of $M$. In this paper, we find upper bounds for this invariant. More precisely, let $(A,\\mathfrak{n})$ be a regular local ring and $I\\subseteq \\mathfrak{n} ^t$ be an ideal of $A$, where $t\\geq 2$. We prove that if $(B=A/I, \\mathfrak{m} =\\mathfrak{n} /I)$ is a Cohen-Macaulay local ring with multiplicity $e(B)= \\binom{h+t-1}{h}$, where $h=embdim(B)-dim B$, then $rat(gr_{\\mathfrak{m}}(B"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8325","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"}