{"paper":{"title":"The Betti poset in monomial resolutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Sonja Mapes, Timothy B. P. Clark","submitted_at":"2014-07-22T00:52:10Z","abstract_excerpt":"Let $P$ be a finite partially ordered set with unique minimal element $\\hat{0}$. We study the Betti poset of $P$, created by deleting elements $q\\in P$ for which the open interval $(\\hat{0}, q)$ is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form $(\\hat{0},p)\\subset P$ and corresponding open intervals in the Betti poset. Our motivating application is that the Betti poset of a monomial ideal's lcm-lattice encodes both its $\\mathbb{Z}^{d}$-graded Betti numbers and the structure of its minimal free resolution. In the case of ri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.5702","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"}