{"paper":{"title":"Generalized Hilbert Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Claudia Polini, Yu Xie","submitted_at":"2012-02-18T21:23:26Z","abstract_excerpt":"Let $M$ be a finite module and let $I$ be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of $I$ on $M$ using the 0th local cohomology functor. We show that our definition re-conciliates with that of Ciuperc$\\breve{{\\rm a}}$. By generalizing Singh's formula (which holds in the case of $\\lambda(M/IM)<\\infty$), we prove that the generalized Hilbert coefficients $j_0,..., j_{d-2}$ are preserved under a general hyperplane section, where $d={\\rm dim}\\,M$. We also keep track of the behavior of $j_{d-1}$. Then we apply these results to study the generalized"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4106","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"}