An Elementary Approach to Homological Algebra: 1st Edition (Hardback) book cover

An Elementary Approach to Homological Algebra

1st Edition

By L.R. Vermani

CRC Press

328 pages

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pub: 2003-05-28
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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.

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

Table of Contents

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

About the Series

Monographs and Surveys in Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT022000
MATHEMATICS / Number Theory