Algebra 3: Homological Algebra and Its Applications

This book is a comprehensive guide to homological algebra, covering abstract theory of derived functors, sheaf co-homology, and etale and l-adic co-homology. It is valuable for graduate and higher undergraduate students specializing in mathematics, with prerequisite knowledge of algebra, linear algebra, topology, and calculus.

Format: Paperback / softback
Length: 300 pages
Publication date: 01 March 2022
Publisher: Springer Verlag, Singapore

This comprehensive third volume in a four-part series delves into essential aspects of homological algebra, encompassing a wide range of topics. It begins with an abstract theory of derived functors, moves on to explore sheaf co-homology, and provides an introduction to etale and l-adic co-homology. The book is organized into four chapters, each dedicated to discussing homology theory within an abelian category, accompanied by significant and fundamental applications in geometry, topology, algebraic geometry, and group theory. This text is particularly valuable for advanced graduate and upper-level undergraduate students with a specialization in any area of mathematics. The author has made every effort to ensure that the book stands on its own by introducing the necessary concepts and results. A solid foundation in algebra, linear algebra, topology, and calculus of several variables is considered advantageous.

Abstract Theory of Derived Functors:
The third volume in this series begins with an in-depth exploration of the abstract theory of derived functors. This foundational concept plays a crucial role in homological algebra, providing a framework for studying homomorphisms and homological operations. The author delves into the theoretical aspects and applications of derived functors, including their properties, relations, and their use in various areas of mathematics.

Sheaf Co-Homology:
The book then moves on to discuss sheaf co-homology, a branch of homological algebra that studies the homology of sheaves on a topological space. Sheaves are mathematical objects that represent collections of functions on a space, and sheaf co-homology provides a way to analyze the structure and properties of these collections. The author provides a comprehensive introduction to sheaf co-homology, covering topics such as sheaf cohomology, sheaf homology, and the fundamental theorem of sheaf co-homology.

Etale and L-adic Co-Homology:
The third volume concludes with an introduction to etale and l-adic co-homology, two branches of homological algebra that study the homology of schemes over fields with characteristic zero. These branches of homology have played a significant role in the development of modern mathematics, particularly in the study of algebraic geometry and number theory. The author provides a clear and concise explanation of these topics, including the definition of schemes, the construction of co-homology groups, and their applications to various problems in mathematics.

Applications in Geometry, Topology, Algebraic Geometry, and Group Theory:
Throughout the book, the author showcases the applications of homology theory in various branches of mathematics. These applications range from geometric problems such as the study of manifolds and homology groups to topological issues such as the classification of spaces and the study of homotopy groups. In algebraic geometry, homology theory is used to study the properties of algebraic varieties, such as their moduli spaces and the Betti numbers. Additionally, homology theory finds applications in group theory, where it is used to study the structure of groups and the classification of finite groups.

Value to Graduate and Higher Undergraduate Students:
This book is designed to cater to the needs of graduate and higher undergraduate students specializing in any branch of mathematics. The author has carefully crafted the text to be self-contained, introducing all the relevant concepts and results required for a comprehensive understanding of homological algebra. The book assumes a basic knowledge of algebra, linear algebra, topology, and calculus of several variables, which serves as a foundation for the subsequent discussions.

Conclusion:
In conclusion, this third volume in the four-part series on homological algebra is a valuable resource for advanced students and researchers in mathematics. It provides a comprehensive and up-to-date introduction to homological algebra, covering a wide range of topics and applications. The book's clear and concise writing style, coupled with its extensive examples and exercises, makes it accessible to students with a solid background in mathematics. Whether you are interested in algebraic geometry, topology, or group theory, this book will undoubtedly enhance your understanding of homological algebra and its applications.

Weight: 486g
Dimension: 235 x 155 (mm)
ISBN-13: 9789813363281
Edition number: 1st ed. 2021