On the geometry of lattices and finiteness of Picard groups

2019/07/31 by Eisele, Florian · 2 citations
#16G30 #20C11 #20C20 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1908.00129

Abstract

Let (K,\mathcal O, k) be a p-modular system with k algebraically closed and \mathcal O unramified, and let Λ be an \mathcal O-order in a separable K-algebra. We call a Λ-lattice L rigid if \rm Ext¹_Λ(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 \mathcal O are always finite.

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