The Picard group of an order and K\"ulshammer reduction

TL;DR

This paper proves that the Picard group of an order in a separable algebra over a p-modular system has an algebraic group structure and applies this to validate a reduction theorem related to Donovan's conjecture in modular representation theory.

Contribution

It establishes the algebraic group structure of the Picard group of an order and extends Külshammer's reduction theorem to the setting of orders over f6.

Findings

Picard group of an order forms an algebraic group over an algebraically closed field

Külshammer's reduction theorem remains valid over f6 for orders in separable algebras

Application to modular representation theory and Donovan's conjecture

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 $\mathcal O$.