On the geometry of lattices and finiteness of Picard groups

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 Ext1Λ(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