{"paper":{"title":"Bigraded structures and the depth of blow-up algebras","license":"","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Gemma Colom\\'e-Nin, Juan Elias","submitted_at":"2004-05-18T07:03:05Z","abstract_excerpt":"Let $R$ be a Cohen-Macaulay local ring, and let $I\\subset R$ be an ideal with minimal reduction $J$. In this paper we attach to the pair $I$, $J$ a non-standard bigraded module $\\Sigma^{I,J}$.\n The study of the bigraded Hilbert function of $\\SIJ$ allows us to prove a improved version of Wang's conjecture and a weak version of Sally's conjecture, both on the depth of the associated graded ring $gr_I(R)$. The module $\\SIJ$ can be considered as a refinement of the Sally's module previously introduced by W. Vasconcelos."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0405344","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"}