An Elementary Approach to Homological Algebra

An Elementary Approach to Homological Algebra

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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 more

Product details

  • Hardback | 328 pages
  • 161 x 241.3 x 23.9mm | 585.14g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • 1584884002
  • 9781584884002
  • 2,386,340

Review quote

"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 Reviewsshow more

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 INDEXshow more

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