Description
Book Synopsis
This monograph is devoted to a new class of non-commutative rings, skew Poincaré–Birkhoff–Witt (PBW) extensions. Beginning with the basic definitions and ring-module theoretic/homological properties, it goes on to investigate finitely generated projective modules over skew PBW extensions from a matrix point of view. To make this theory constructive, the theory of Gröbner bases of left (right) ideals and modules for bijective skew PBW extensions is developed. For example, syzygies and the Ext and Tor modules over these rings are computed. Finally, applications to some key topics in the noncommutative algebraic geometry of quantum algebras are given, including an investigation of semi-graded Koszul algebras and semi-graded Artin–Schelter regular algebras, and the noncommutative Zariski cancellation problem.
The book is addressed to researchers in noncommutative algebra and algebraic geometry as well as to graduate students and advanced undergraduate students.
Trade Review
“The text under review attempts to summarize the breadth of research on skew PBW extensions since that initial paper. … Readers familiar with the area of skew PBW extensions may find this book a convenient reference.” (Jason Gaddis, Mathematical Reviews, June, 2023)
“This very well-written book … . there are a few new techniques and approaches in the book via which the authors attack successfully the proofs of some classical results from this branch. Besides, certain innovations in the presentation of some specific moments in the subject are also demonstrated.” (Peter Danchev, zbMATH 1489.16002, 2022)
Table of Contents
Preface.- I Ring and Module-Theoretic Properties of Skew PBW Extensions.- II Projective Modules Over Skew PBW Extensions.- III Matrix and Gröbner Methods for Skew PBW Extensions.- IV Applications: The Noncommutative AlgebraicGeometry of Skew PBW Extensions.- References.
