{"paper":{"title":"Length and multiplicity of the local cohomology with support in a hyperplane arrangement","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Toshinori Oaku","submitted_at":"2015-09-06T13:52:41Z","abstract_excerpt":"Let $R$ be the polynomial ring in $n$ variables with coefficients in a field $K$ of characteristic zero. Let $D_n$ be the $n$-th Weyl algebra over $K$. Suppose that $f \\in R$ defines a hyperplane arrangement in the affine space $K^n$. Then the length and the multiplicity of the 1st local cohomology group $H^1_{(f)}(R)$ as left $D_n$-module coincide and are explicitly expressed in terms of the Poincar\\'e polynomial or the M\\\"obius function of the arrangement."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01813","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"}