Recent Developments in Commutative Algebra: Levico Terme, Trento 2019

This book covers four lectures on Rees rings and blow-ups, Koszul modules with applications to syzygies, Gröbner bases and degenerations, and applications of Adams operations, highlighting recent developments in commutative algebra. It is a valuable resource for graduate students and researchers.

Format: Paperback / softback
Length: 118 pages
Publication date: 03 March 2021
Publisher: Springer Nature Switzerland AG

This book is a comprehensive exploration of the fascinating world of Rees rings and blow-ups, Koszul modules with applications to syzygies, Gröbner bases and degenerations, and the applications of Adams operations. Delivered in four engaging lectures during a workshop in Levico Terme, Trento, in July 2019, this material showcases some of the most remarkable developments in the field of Commutative Algebra.

The book begins by delving into the fundamental concepts of Rees rings and blow-ups, providing a solid foundation for the subsequent chapters. It explores the rich structure and properties of these rings, including their modules, homomorphisms, and ideals. The authors then move on to discuss the applications of Rees rings in various areas, such as commutative algebraic geometry, algebraic number theory, and representation theory.

The second lecture delves into the world of Koszul modules, which are a powerful tool in commutative algebra. The authors introduce Koszul modules and their basic properties, such as the Koszul complex and the Koszul resolution. They then explore the applications of Koszul modules in the study of syzygies, a fundamental concept in commutative algebraic geometry. The authors demonstrate how Koszul modules can be used to construct explicit formulas for syzygies and to study their properties.

The third lecture focuses on Gröbner bases and degenerations, which are fundamental tools in commutative algebra. The authors introduce Gröbner bases and their properties, such as the Gröbner basis theorem and the Buchberger algorithm. They then discuss the applications of Gröbner bases in the study of polynomial equations and in the construction of explicit formulas for syzygies.

The fourth and final lecture explores the applications of Adams operations in commutative algebra. The authors introduce Adams operations and their properties, such as the Adams equality and the Adams theorem. They then discuss the applications of Adams operations in the study of homological algebra, the theory of Lie algebras, and the representation theory of finite groups.

Throughout the book, the authors use clear and concise explanations, accompanied by numerous examples and exercises, to help readers grasp the concepts and apply them to real-world scenarios. The book is designed to be a valuable resource for graduate students and researchers in commutative algebra and related fields.

In conclusion, this book is a must-read for anyone interested in the latest developments in Commutative Algebra. It provides a comprehensive and accessible introduction to Rees rings and blow-ups, Koszul modules with applications to syzygies, Gröbner bases and degenerations, and the applications of Adams operations. With its clear and concise writing style, numerous examples, and exercises, it is an ideal resource for graduate students and researchers seeking to advance their knowledge in this field.

Weight: 222g
Dimension: 156 x 234 x 18 (mm)
ISBN-13: 9783030650636
Edition number: 1st ed. 2021