{"paper":{"title":"Splitting Algebras: Koszul, Cohen-Macaulay and Numerically Koszul","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Brad Shelton, Tyler Kloefkorn","submitted_at":"2012-07-05T19:12:50Z","abstract_excerpt":"We study a finite dimensional quadratic graded algebra R defined from a finite ranked poset. This algebra has been central to the study of the splitting algebra of the poset, A, as introduced by Gelfand, Retakh, Serconek and Wilson . The algebra A is known to be quadratic when the poset satisfies a combinatorial condition known as uniform, and R is the quadratic dual of an associated graded algebra of A. We prove that R is Koszul and the poset is uniform if and only if the poset is Cohen-Macaulay. Koszulity of R implies Koszulity of A. We also show that when R is Koszul, the cohomology of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.1330","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"}