The Picard group of an order and K\"ulshammer 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 $\textrm{Pic}_{\mathcal O}(\Lambda)$ 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 K\"ulshammer concerned with Donovan's conjecture remains valid over $\mathcal O$.