The Picard Group of an Order and Külshammer Reduction

Let $(K,\mathcal {O},k)$ be a p-modular system and assume k is algebraically closed. We show that if Λ 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 Kulshammer concerned with Donovan’s conjecture remains valid over $\mathcal {O}$.