{"paper":{"title":"Betti numbers of multigraded modules of generic type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Alexandre Tchernev, Hara Charalambous","submitted_at":"2010-04-30T08:51:28Z","abstract_excerpt":"Let $R=\\Bbbk[x_1,...,x_m]$ be the polynomial ring over a field $\\Bbbk$ with the standard $\\mathbb Z^m$-grading (multigrading), let $L$ be a Noetherian multigraded $R$-module, let $\\beta_{i,\\alpha}(L)$ the $i$th (multigraded) Betti number of $L$ of multidegree $\\a$. We introduce the notion of a generic (relative to $L$) multidegree, and the notion of multigraded module of generic type. When the multidegree $\\a$ is generic (relative to $L$) we provide a Hochster-type formula for $\\beta_{i,\\alpha}(L)$ as the dimension of the reduced homology of a certain simplicial complex associated with $L$. Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5472","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"}