On the geometry of lattices and finiteness of Picard groups

Abstract

Let ( K , 𝒪 , k ) {(K,\mathcal{O},k)} be a p-modular system with k algebraically closed and 𝒪 {\mathcal{O}} unramified, and let Λ be an 𝒪 {\mathcal{O}} -order in a separable K-algebra. We call a Λ-lattice L rigid if Ext Λ 1 ⁡ ( L , L ) = 0 {{\operatorname{Ext}}^{1}_{\Lambda}(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 {\mathcal{O}} are always finite.

Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

“…Indeed, in the pro-p case Theorem 1.2 replaces Stallings' theory of ends, crucial in the proof of the original result. Recently, the greater generality of coefficient ring has yielded applications of Theorem 1.2 in the calculation of Picard groups of blocks of finite groups [6,16,21].…”