On the geometry of lattices and finiteness of Picard groups

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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 ExtΛ1 (L,L) = 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.

Original languageEnglish
Pages (from-to)219-233
Number of pages15
JournalJournal Fur Die Reine Und Angewandte Mathematik
Volume2022
Issue number782
DOIs
Publication statusPublished - 12 Nov 2021

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