Classical and involutive invariants of Krull domains
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 present volume would provide an excellent alternative to readers interested in the basic theory of Krull domains and their multiplicative ideal theory. And as this book goes well beyond that theory, not only with the treatment of involutive invariants but with an extensive presentation of recent theory of the classical invariants, it presumably will now be the standard reference for researchers and place of access to the subject for students.
scientific article; zbMATH DE number 1375523