The Picard Group of an Order and Külshammer Reduction

Florian Eisele

doi:10.1007/s10468-020-09957-x

The Picard Group of an Order and Külshammer Reduction

Florian Eisele

Pure Mathematics

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Abstract

Let (K, O, k) be a p-modular system and assume k is algebraically closed. We show that if Λ is an O-order in a separable K-algebra, then Pic _O(Λ) 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 O.

Original language	English
Pages (from-to)	505-518
Number of pages	14
Journal	Algebras and Representation Theory
Volume	24
Issue number	2
Early online date	16 Mar 2020
DOIs	https://doi.org/10.1007/s10468-020-09957-x
Publication status	Published - Apr 2021

Keywords

Finite groups
Modular Representation theory
Orders
Representation theory

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Eisele2021_Article_ThePicardGroupOfAnOrderAndKülsFinal published version, 440 KBLicence: CC BY

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title = "The Picard Group of an Order and K{\"u}lshammer Reduction",

abstract = "Let (K, O, k) be a p-modular system and assume k is algebraically closed. We show that if Λ is an O-order in a separable K-algebra, then Pic O(Λ) 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{\"u}lshammer concerned with Donovan{\textquoteright}s conjecture remains valid over O.",

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AB - Let (K, O, k) be a p-modular system and assume k is algebraically closed. We show that if Λ is an O-order in a separable K-algebra, then Pic O(Λ) 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 O.

KW - Finite groups

KW - Modular Representation theory

KW - Orders

KW - Representation theory

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The Picard Group of an Order and Külshammer Reduction

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