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On the geometry of lattices and finiteness of Picard groups

Eisele, F. ORCID: 0000-0001-8267-2094 (2021). On the geometry of lattices and finiteness of Picard groups. Journal fur die Reine und Angewandte Mathematik, doi: 10.1515/crelle-2021-0064

Abstract

Let (K, O, k) be a p-modular system with k algebraically closed and O unramified, and let Λ be an O-order in a separable K-algebra. We call a Λ-lattice L rigid if Ext Λ 1 = 0, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Λ-lattices of a given dimension into "varieties of lattices", we show that there are only finitely many rigid Λ-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Λ vanishes, then the Picard group and the outer automorphism group of Λ are finite. In particular, the Picard groups of blocks of finite groups defined over O are always finite.

Publication Type: Article
Additional Information: © 2021 Walter de Gruyter GmbH, Berlin/Boston. This article has been published in Journal für die reine und angewandte Mathematik (Crelles Journal), DOI: doi.org/10.1515/crelle-2021-0064. This work is licensed under the Creative Commons Attribution 4.0 International License.
Subjects: Q Science > QA Mathematics
Departments: School of Mathematics, Computer Science & Engineering > Mathematics
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