The Picard group of an order and Külshammer reduction

Abstract

Let $(K,\mathcal{O},k)$ be a $p$-modular system and assume $k$ is algebraically closed. We show that if $\Lambda$ is an $\mathcal{O}$-order in a separable $K$-algebra, then ${\text{Pic}}_{\mathcal{O}}(\Lambda )$ carries the structure of an algebraic group over $k$. As an application to the modular representation theory of finite groups, we show that a reduction theorem by Külshammer concerned with Donovan's conjecture remains valid over $\mathcal{O}$.