{"paper":{"title":"Depth of edge rings arising from finite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Akihiro Higashitani, Augustine B. O'Keefe, Kyouko Kimura, Takayuki Hibi","submitted_at":"2010-09-08T08:02:11Z","abstract_excerpt":"Let $G$ be a finite graph and $K[G]$ the edge ring of $G$. Based on the technique of Gr\\\"obner bases and initial ideals, it will be proved that, given integers $f$ and $d$ with $7 \\leq f \\leq d$, there exists a finite graph $G$ on $[d]={1,...,d}$ with $\\depth K[G] = f$ and with $\\Krull-dim K[G] = d$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.1472","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"}