{"paper":{"title":"On the geometry of lattices and finiteness of Picard groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.RA"],"primary_cat":"math.RT","authors_text":"Florian Eisele","submitted_at":"2019-07-31T22:42:28Z","abstract_excerpt":"Let $(K,\\mathcal O, k)$ be a $p$-modular system with $k$ algebraically closed and $\\mathcal O$ unramified, and let $\\Lambda$ be an $\\mathcal O$-order in a separable $K$-algebra. We call a $\\Lambda$-lattice $L$ rigid if ${\\rm Ext}^1_{\\Lambda}(L,L)=0$, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the $\\Lambda$-lattices of a given dimension into \"varieties of lattices\", we show that there are only finitely many rigid $\\Lambda$-lattices $L$ of any given dimension. As a consequence we show that if the first Hochschild cohomology of $\\Lambda$ van"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.00129","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/1908.00129/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"}