Classical and Involutive Invariants of Krull Domains

ISBN-13: 9789401064941 / Angielski / Miękka / 2013 / 260 str.

M.V.Reyes Sanchez;A. Verschoren

Opis

This monograph is devoted to Krull domains and its invariants. The book shows how a serious study of invariants of Krull domains necessitates input from various fields of mathematics, including rings and module theory, commutative algebra, K-theory, cohomology theory, localization theory and algebraic geometry. About half of the book is dedicated to so-called involutive invariants, such as the involutive Brauer group, and is essentially the first to cover these topics. In a structured and methodical way, the work presents a large quantity of results previously scattered throughout the literature. Audience: This volume is recommended as a first introduction to this rapidly developing subject, but will also be useful as a state-of-the-art reference work, both to students at graduate and postgraduate levels and to researchers in commutative rings and algebra, algebraic K-theory, algebraic geometry, and associative rings.

Spis treści

1. Krull domains and their modules.- 1.1 Krull domains.- 1.2 Lattices.- 1.3 Divisorial lattices.- 1.4 The modified tensor product.- 1.5 A torsion-theoretic point of view.- 2. Classical invariants.- 2.1 The class group.- 2.2 The Brauer group.- 2.3 Enters cohomology.- 2.4 The long exact sequence.- 2.5 A K-theoretic point of view.- 3 Involutions.- 3.1 The categories C*.- 3.2 Algebras with involution.- 3.3 Involutions of trivial Azumaya algebras.- 3.4 Hermitian Picard groups.- 3.5 A Morita duality point of view.- 4 Involutive Brauer groups.- 4.1 Saltman’s theorem.- 4.2 The involutive Brauer group.- 4.3 Exact sequences.- 4.4 Cohomological interpretation.- 4.5 A geometric point of view.- 5 Functorial behaviour.- 5.1 Change of base ring.- 5.2 Divisorial descent.- 5.3 Separability and divisorial Galois theory.- 5.4 Norms.- 5.5 An Amitsur cohomology point of view.