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# An Elementary Approach to Homological Algebra

By

## L.R. Vermani

ISBN 9781584884002
Published May 28, 2003 by CRC Press
328 Pages

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## Book Description

Homological algebra was developed as an area of study almost 50 years ago, and many books on the subject exist. However, few, if any, of these books are written at a level appropriate for students approaching the subject for the first time.

An Elementary Approach to Homological Algebra fills that void. Designed to meet the needs of beginning graduate students, it presents the material in a clear, easy-to-understand manner. Complete, detailed proofs make the material easy to follow, numerous worked examples help readers understand the concepts, and an abundance of exercises test and solidify their understanding.

Often perceived as dry and abstract, homological algebra nonetheless has important applications in many important areas. The author highlights some of these, particularly several related to group theoretic problems, in the concluding chapter. Beyond making classical homological algebra accessible to students, the author's level of detail, while not exhaustive, also makes the book useful for self-study and as a reference for researchers.

MODULES
Modules
Free Modules
Exact Sequences
Homomorphisms
Tensor Product of Modules
Direct and Inverse Limits
Pull Back and Push Out
CATEGORIES AND FUNCTORS
Categories
Functors
The Functors Hom and Tensor
PROJECTIVE AND INJECTIVE MODULES
Projective Modules
Injective Modules
Baer's Criterion
An Embedding Theorem
HOMOLOGY OF COMPLEXES
Ker-Coker Sequence
Connecting Homomorphism: The General Case
Homotopy
DERIVED FUNCTORS
Projective Resolutions
Injective Resolutions
Derived Functors
TORSION AND EXTENSION FUNCTORS
Derived Functors--Revisited
Torsion and Extension Functors
Some Further Properties of Torsion Functors
Tor and Direct Limits
THE FUNCTOR ExtnR
Ext1 and Extensions
Baer Sum of Extensions
Some Further Properties of Extension Functors
HEREDITARY AND SEMIHEREDITARY RINGS
Hereditary Rings and Dedekind Domains
Invertible Ideals and Dedekind Rings
Semihereditary and Prufer Rings
UNIVERSAL COEFFICIENT THEOREM
Universal Coefficient Theorem for Homology
Universal Coefficient Theorem for Cohomology
The Kunneth Formula: A Special Case
DIMENSIONS OF MODULES AND RINGS
Projectively and Injectively Equivalent Modules
Dimensions of Modules and Rings
Global Dimension of Rings
Global Dimension of Noetherian Rings
Global Dimension of Artin Rings
COHOMOLOGY OF GROUPS
Homology and Cohomology Groups
Some Examples
The Groups H0(G,A) and H0(G,A)
The Groups H1(G,A) and H1(G,A)
Homology and Cohomology of Direct Sums
The Bar Resolution
Second Cohomology Group and Extensions
Some Homomorphisms
Some Exact Sequences
SOME APPLICATIONS
An Exact Sequence
Outer Automorphisms of p-Groups
A Theorem of Magnus
BIBLIOGRAPHY
INDEX

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## Reviews

"The book is a self-contained one and there are … complete proofs for all the results. In our opinion, it is a highly recommended introductory book in Homological Algebra for everyone interested in this subject."
- Zentralblatt MATH, 1045

"… [T]he author on the one hand has included the underlying basics, model and category theory, which are developed from scratch. On the other hand the elementary notions and results of homological algebra are treated in great detail and often their importance within that theory as well as in applications is shown. … All in all the author has managed … admirably to reach the goal he has set [for] himself."
- Monatshefte fur Mathematik

"When I next teach an introductory course on homological algebra at my institution, I will certainly consider adopting this book as the textbook for the course.
-Mathematical Reviews