This work aims to inspire readers as well as develop their understanding of syzygies. Starting with the basic background, it also includes numerous detailed examples and open problems and conjectures that point to challenging new areas for future research.

The study of free resolutions is a core and beautiful area in Commutative Algebra. The main goal of this book is to inspire the readers and develop their intuition about syzygies and Hilbert functions. Many examples are given in order to illustrate ideas and key concepts.

A valuable feature of the book is the inclusion of open problems and conjectures; these provide a glimpse of exciting, and often challenging, research directions in the field. Three types of problems are presented: Conjectures, Problems, and Open-Ended Problems. The latter do not describe specific problems but point to interesting directions for exploration.

The first part of the monograph contains basic background material on graded free resolutions. Further coverage of topics includes syzygies over a polynomial ring, resolutions over quotient rings, lex ideals and Hilbert functions, compression, resolutions of monomial ideals, and syzygies of toric ideals. With a clear and self-contained exposition this text is intended for advanced graduate students and postdoctorates; it will be also of interest to senior mathematicians.

Focuses on the graded aspect of syzygies, while most of the standard texts in commutative algebra mainly consider local rings Provides a substantial amount of open problems and conjectures A wealth of examples Contains basic background material on free resolutions making the text accessible to beginners Includes supplementary material: sn.pub/extras

Inhalt

Graded Free Resolutions.- Hilbert Functions.- Monomial Resolutions.- Syzygies of Toric Ideals