Gorenstein homological algebra is an important area of mathematics, with applications in commutative and noncommutative algebra, model category theory, representation theory, and algebraic geometry. While in classical homological algebra the existence of the projective, injective, and flat resolutions over arbitrary rings are well known, things are a little different when it comes to Gorenstein homological algebra. The main open problems in this area deal with the existence of the Gorenstein injective, Gorenstein projective, and Gorenstein flat resolutions. Gorenstein Homological Algebra is especially suitable for graduate students interested in homological algebra and its applications.
Table of Contents
1 Modules - projective, injective, at modules
2 Gorenstein projective, injective and at modules
3 Gorenstein projective resolutions
4 Gorenstein injective resolutions
5 Gorenstein at precovers and preenvelopes
6 Connections with Tate (co)homology
7 Totally acyclic complexes
8 Generalizations of the Gorenstein modules
9 Gorenstein projective, injective, at complexes, dg-projective, dg-injective, dg-at complexes
Alina Iacob is a professor of mathematics at Georgia Southern University. Her primary research interests are homological and communicative algebra.
This book is a very necessary contribution to the bibliography on relative homological algebra, specifically to the so-called Gorenstein homological algebra. With a very didactic style, the author introduces us to this complex matter from the beginning, reaching the most current and profound results. It emphasizes the use of a classical, as well as modern mathematical language, including the use of derived categories and cotorsion pairs. It is a good first contact with this topic, which makes it very interesting for graduate students, although more knowledgeable readers may find fresh insights from reading the book.
This book is distinguished from others that deal with related topics in the introduction of Gorenstein projective, injective and flat complexes, as well as dg-projective, dg-injective and dg-flat complexes. It also presents the theory of some generalizations of Gorenstein modules, such as the Gorenstein AC-projective, injective and flat modules. A central topic is the existence of Gorenstein projective, injective and flat (pre)covers and (pre) envelopes and their generalizations to complexes. Because the author is an expert in Tate's cohomology, this particular topic is presented for both modules and complexes in an especially attractive and novel way. The Gorenstein homological methods have proved to be very useful in characterizing various classes of rings like Iwanaga Gorenstein, regular or coherent rings.
Written by one of the world’s acknowledged experts in the field, Gorenstein Homological Algebra is an excellent resource for algebraists and mathematicians from other disciplines interested in deepening their knowledge of Gorenstein homological algebra.
-Juan Ramon Garcia Rozas, Universidad de Almeria
Gorenstein homological algebra is undoubtedly the most developed and important part of contemporary relative homological algebra. Over the past two decades, the theory was developed smoothly over Gorenstein rings, and parts of it even for more general rings. The major current open problems concern the questions of whether, or to what extent, can the theory be extended to arbitrary rings. Besides these major problems, the book addresses relations between Gorenstein homological algebra and Tate (co)homology, and finishes with a generalization of the theory from module categories to categories of complexes of modules.
Professor Iacob’s book presents an up-to-date overview of the current status of Gorenstein homological algebra. It is essentially self-contained, making the presentation accessible to graduate students of algebra and representation theory.
-Jan Trlifaj, Charles Uiversity, Prague
This book provides the latest development and trends in Gorenstein homological algebra, ring theory, module theory, and so on. It is very useful for specialists to get an overview on various areas in relative homological algebra, and also for younger researchers looking for fields to investigate.
-Nanqing Ding, Nanjing University, China