{"paper":{"title":"Intersections of Leray complexes and regularity of monomial ideals","license":"","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Gil Kalai, Roy Meshulam","submitted_at":"2006-01-30T23:38:05Z","abstract_excerpt":"For a simplicial complex X and a field K, let h_i(X)=\\dim \\tilde{H}_i(X;K). It is shown that if X,Y are complexes on the same vertex set, then for all k h_{k-1}(X\\cap Y) \\leq \\sum_{\\sigma \\in Y} \\sum_{i+j=k} h_{i-1}(X[\\sigma])\\cdot h_{j-1}(\\lk(Y,\\sigma)) . A simplicial complex X is d-Leray over K, if h_i(Y)=0 for all induced subcomplexes Y \\subset X and i \\geq d. Let L_K(X) denote the minimal d such that X is d-Leray over K. The above theorem implies that if X,Y are simplicial complexes on the same vertex set then L_K(X \\cap Y) \\leq L_K(X) +L_K(Y). Reformulating this inequality in commutative "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0601745","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0601745/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}