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 Külshammer concerned with Donovan’s conjecture remains valid over \(\mathcal {O}\).

中文翻译：

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皮卡德阶阶和Külshammer减法

设\（（K，\ mathcal {O}，k）\）为p模系统，并假定k是代数闭合的。我们证明如果Λ是可分K-代数中的\（\ mathcal {O} \）-阶，则\（\ text {Pic} _ {\ mathcal {O}}（{\ Lambda}）\）携带k上的代数群的结构。作为对有限群的模块化表示理论的一种应用，我们证明了与Donovan猜想有关的Külshammer约简定理在\（\ mathcal {O} \）上仍然有效。