On the geometry of lattices and finiteness of Picard groups

Abstract

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 $\Lambda$ vanishes, then the Picard group and the outer automorphism group of $\Lambda$ are finite. In particular the Picard groups of blocks of finite groups defined over $\mathcal O$ are always finite.