{"paper":{"title":"The Grade Conjecture and Asymptotic Intersection Multiplicity","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Jesse S. Beder","submitted_at":"2012-02-16T05:10:48Z","abstract_excerpt":"Given a finitely generated module $M$ over a local ring $A$ of characteristic $p$ with $\\pd M < \\infty$, we study the asymptotic intersection multiplicity $\\chi_\\infty(M, A/\\underline{x})$, where $\\underline{x} = (x_1, \\ldots, x_r)$ is a system of parameters for $M$. We show that there exists a system of parameters such that $\\chi_\\infty$ is positive if and only if $\\dim \\Ext^{d-r}(M, A) = r$, where $d = \\dim A$ and $r = \\dim M$. We use this to prove several results relating to the Grade Conjecture, which states that $\\grade M + \\dim M = \\dim A$ for any module $M$ with $\\pd M < \\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.3513","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"}