{"paper":{"title":"On The Telescopic Picard Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Shai Keidar","submitted_at":"2024-12-10T18:09:27Z","abstract_excerpt":"We prove that for any prime $p$ and height $n \\ge 1$, the telescopic Picard group $\\mathrm{Pic}(\\mathrm{Sp}_{Tn})$ contains a subgroup of the form $\\mathbb{Z}_p \\times \\mathbb{Z}/a_p(p^n-1)$, where $a_p = 1$ if $p = 2$ and $a_p = 2$ if $p$ is odd. Using Kummer theory, we obtain an $(\\mathbb{F}_{p^n}^\\times \\rtimes \\mathbb{Z}/n)$-Galois extension of $\\mathbb{S}_{T(n)}$, obtaining the first example of a lift of a non-Abelian Galois extension of the $K(n)$-local sphere to the telescopic world, at arbitrary positive height and prime. Our proof proceeds by setting up a higher categorical framework "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.07716","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/2412.07716/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"}