{"paper":{"title":"FI-modules and the cohomology of modular representations of symmetric groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.GT"],"primary_cat":"math.RT","authors_text":"Rohit Nagpal","submitted_at":"2015-05-16T17:11:20Z","abstract_excerpt":"An FI-module $V$ over a commutative ring $\\bf{k}$ encodes a sequence $(V_n)_{n \\geq 0}$ of representations of the symmetric groups $(\\mathfrak{S}_n)_{n \\geq 0}$ over $\\bf{k}$. In this paper, we show that for a \"finitely generated\" FI-module $V$ over a field of characteristic $p$, the cohomology groups $H^t(\\mathfrak{S}_n, V_n)$ are eventually periodic in $n$. We describe a recursive way to calculate the period and the periodicity range and show that the period is always a power of $p$. As an application, we show that if $\\mathcal{M}$ is a compact, connected, oriented manifold of dimension $\\ge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04294","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"}