{"paper":{"title":"Lower central series and free resolutions of hyperplane arrangements","license":"","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.AG","authors_text":"Alexander I. Suciu, Henry K. Schenck","submitted_at":"2001-09-11T03:28:33Z","abstract_excerpt":"If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\\k)$ is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $\\phi_k$, of the lower central series quotients of $\\pi_1(M)$ can be computed from the Betti numbers, $b_{ii}=\\dim_{\\k} \\Tor^A_i(\\k,\\k)_i$, of the linear strand in a (minimal) free resolution of $\\k$ over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, $b'_{ij}=\\dim_{\\k} \\Tor^E_i(A,\\k)_j$, of a (minimal) resolution of $A$ over the exterior algebra $E$.\n  From this analysis,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0109070","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"}