{"paper":{"title":"The Type Defect of a Simplicial Complex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Hailong Dao, Jay Schweig","submitted_at":"2017-04-05T02:15:47Z","abstract_excerpt":"Fix a field $k$. When $\\Delta$ is a simplicial complex on $n$ vertices with Stanley-Reisner ideal $I_\\Delta$, we define and study an invariant called the $\\textit{type defect}$ of $\\Delta$. Except when $\\Delta$ is of a single simplex, the type defect of $\\Delta$, $\\textrm{td}(\\Delta)$, is the difference $ \\dim_k \\textrm{Tor}_c^S(S/ I_\\Delta,k) - c$, where $c$ is the codimension of $\\Delta$ and $S = k[x_1, \\ldots x_n]$. We show that this invariant admits surprisingly nice properties. For example, it is well-behaved when one glues two complexes together along a face. Furthermore, $\\Delta$ is Coh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.01243","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":""},"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"}