{"paper":{"title":"Homological invariants of powers of fiber products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hop D. Nguyen, Thanh Vu","submitted_at":"2018-03-11T19:10:20Z","abstract_excerpt":"Let $R$ and $S$ be polynomial rings of positive dimensions over a field $k$. Let $I\\subseteq R, J\\subseteq S$ be non-zero homogeneous ideals none of which contains a linear form. Denote by $F$ the fiber product of $I$ and $J$ in $T=R\\otimes_k S$. We compute homological invariants of the powers of $F$ using the data of $I$ and $J$. Under the assumption that either $\\text{char}~ k=0$ or $I$ and $J$ are monomial ideals, we provide explicit formulas for the depth and regularity of powers of $F$. In particular, we establish for all $s\\ge 2$ the intriguing formula $\\text{depth}(T/F^s)=0$. If moreove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.04016","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"}