{"paper":{"title":"Cohen--Macaulaynees for symbolic power ideals of edge ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Giancarlo Rinaldo, Ken-ichi Yoshida, Naoki Terai","submitted_at":"2012-03-09T01:12:54Z","abstract_excerpt":"Let $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G) \\subseteq S$ denote the edge ideal of a graph $G$. We show that the $\\ell$th symbolic power $I(G)^{(\\ell)}$ is a Cohen-Macaulay ideal (i.e., $S/I(G)^{(\\ell)}$ is Cohen-Macaulay) for some integer $\\ell \\ge 3$ if and only if $G$ is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers $I(G)^{(\\ell)}$ are Cohen-Macaulay ideals. Similarly, we characterize graphs $G$ for which $S/I(G)^{(\\ell)}$ has (FLC).\n  As an application, we show that an edge ideal $I(G)$ is complete intersecti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1967","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"}