{"paper":{"title":"Linearity Defects of Face Rings","license":"","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Kohji Yanagawa, Ryota Okazaki","submitted_at":"2006-07-31T06:23:36Z","abstract_excerpt":"Let $S = K[x_1, ..., x_n ]$ be a polynomial ring over a field $K$, and $E = K < y_1, ..., y_n >$ an exterior algebra. The \"linearity defect\" $ld_E(N)$ of a finitely generated graded $E$-module $N$ measures how far $N$ departs from \"componentwise linear\". It is known that $ld_E(N) < \\infty$ for all $N$. But the value can be arbitrary large, while the similar invariant $ld_S(M)$ for an $S$-module $M$ is alway at most $n$.\n  We show that if $I_\\Delta$ (resp. $J_\\Delta$) is the squarefree monomial ideal of $S$ (resp. $E$) corresponding to a simplicial complex $\\Delta$ on ${1, >..., n}$, then $ld_E"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607780","kind":"arxiv","version":3},"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/0607780/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"}