Classical and involutive invariants of Krull domains
type: Item
instance of: scholarly article
title: Classical and involutive invariants of Krull domains
zbMATH Open document ID: 0967.13013
published in: K-Monographs in Mathematics
publication date: 1999-12-07T00:00:00Z
review text: This book presents a very complete, expository, and essentially self-contained treatment of Krull domains and their invariants, the latter being their class groups, Picard groups, and Brauer groups. These associated groups are ``classical'' in the authors' terminology when the modules and algebras from whose classes they are constructed have no additional structure, and ``involutive'' when the modules and algebras also carry an appropriate involution. While much of the theory of Krull domains and their classical invariants has appeared in monographs in the literature, this book is the first to include the theory of the involutive invariants as well. The book consists of an introduction, five chapters, and a 105 item bibliography. The first chapter, ``Krull domains and their modules'', introduces the basic notions of Krull domains and lattices, and some of the basic technology of torsion theories and quotient categories. -- Chapter two, ``Classical invariants'', introduces the class group, the Picard group, and the Brauer group, as well as their cohomological and \(K\)-theory interpretations. -- Chapter three, ``Involutions'', is about modules and algebras with involutions (in the module case, these are termed hermitian modules). Besides the basics, they also present a discussion of hermitian Picard groups and hermitian Morita theory. -- Chapter four, ``Involutive Brauer groups'', introduces these objects and presents exact sequences to calculate them. It also discusses their geometric interpretation and relates the authors' version of the exact sequences to the original versions, which were geometrically based. -- The final chapter, ``Functorial behaviour'', discusses the behavior of the invariants under change of rings, in particular under Galois type extensions, developing the necessary descent theory and cohomological interpretations. Even if \textit{R. Fossum}'s book ``The divisor class group of a Krull domain'' (1973; Zbl 0256.13001) were still in print, the pr…
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 13C20
Mathematics Subject Classification ID: 16K50
zbMATH DE Number: 1375523
zbMATH Keywords: Krull domain
zbMATH Keywords: class group
zbMATH Keywords: Picard group
zbMATH Keywords: Brauer group
zbMATH Keywords: involution
author: M. Victoria Reyes Sánchez
author: Alain Verschoren
reviewed by: Andy R. Magid
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1375523