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An Introduction to Homological Algebra

Book Description

The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described. This book is suitable for second or third year graduate students. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. By making these tools more accessible, the book helps to break down the technological barrier between experts and casual users of homological algebra.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Acknowledgments
  5. Contents
  6. Introduction
  7. 1 Chain Complexes
    1. 1.1 Complexes of R-Modules
    2. 1.2 Operations on Chain Complexes
    3. 1.3 Long Exact Sequences
    4. 1.4 Chain Homotopies
    5. 1.5 Mapping Cones and Cylinders
    6. 1.6 More on Abelian Categories
  8. 2 Derived Functors
    1. 2.1 δ-Functors
    2. 2.2 Projective Resolutions
    3. 2.3 Injective Resolutions
    4. 2.4 Left Derived Functors
    5. 2.5 Right Derived Functors
    6. 2.6 Adjoint Functors and Left/Right Exactness
    7. 2.7 Balancing Tor and Ext
  9. 3 Tor and Ext
    1. 3.1 Tor for Abelian Groups
    2. 3.2 Tor and Flatness
    3. 3.3 Ext for Nice Rings
    4. 3.4 Ext and Extensions
    5. 3.5 Derived Functors of the Inverse Limit
    6. 3.6 Universal Coefficient Theorems
  10. 4 Homological Dimension
    1. 4.1 Dimensions
    2. 4.2 Rings of Small Dimension
    3. 4.3 Change of Rings Theorems
    4. 4.4 Local Rings
    5. 4.5 Koszul Complexes
    6. 4.6 Local Cohomology
  11. 5 Spectral Sequences
    1. 5.1 Introduction
    2. 5.2 Terminology
    3. 5.3 The Leray-Serre Spectral Sequence
    4. 5.4 Spectral Sequence of a Filtration
    5. 5.5 Convergence
    6. 5.6 Spectral Sequences of a Double Complex
    7. 5.7 Hyperhomology
    8. 5.8 Grothendieck Spectral Sequences
    9. 5.9 Exact Couples
  12. 6 Group Homology and Cohomology
    1. 6.1 Definitions and First Properties
    2. 6.2 Cyclic and Free Groups
    3. 6.3 Shapiro’s Lemma
    4. 6.4 Crossed Homomorphisms and H1
    5. 6.5 The Bar Resolution
    6. 6.6 Factor Sets and H2
    7. 6.7 Restriction, Corestriction, Inflation, and Transfer
    8. 6.8 The Spectral Sequence
    9. 6.9 Universal Central Extensions
    10. 6.10 Covering Spaces in Topology
    11. 6.11 Galois Cohomology and Profinite Groups
  13. 7 Lie Algebra Homology and Cohomology
    1. 7.1 Lie Algebras
    2. 7.2 g-Modules
    3. 7.3 Universal Enveloping Algebras
    4. 7.4 H1 and H1
    5. 7.5 The Hochschild-Serre Spectral Sequence
    6. 7.6 H2 and Extensions
    7. 7.7 The Chevalley-Eilenberg Complex
    8. 7.8 Semisimple Lie Algebras
    9. 7.9 Universal Central Extensions
  14. 8 Simplicial Methods in Homological Algebra
    1. 8.1 Simplicial Objects
    2. 8.2 Operations on Simplicial Objects
    3. 8.3 Simplicial Homotopy Groups
    4. 8.4 The Dold-Kan Correspondence
    5. 8.5 The Eilenberg-Zilber Theorem
    6. 8.6 Canonical Resolutions
    7. 8.7 Cotriple Homology
    8. 8.8 André-Quillen Homology and Cohomology
  15. 9 Hochschild and Cyclic Homology
    1. 9.1 Hochschild Homology and Cohomology of Algebras
    2. 9.2 Derivations, Differentials, and Separable Algebras
    3. 9.3 H2, Extensions, and Smooth Algebras
    4. 9.4 Hochschild Products
    5. 9.5 Morita Invariance
    6. 9.6 Cyclic Homology
    7. 9.7 Group Rings
    8. 9.8 Mixed Complexes
    9. 9.9 Graded Algebras
    10. 9.10 Lie Algebras of Matrices
  16. 10 The Derived Category
    1. 10.1 The Category K(A)
    2. 10.2 Triangulated Categories
    3. 10.3 Localization and the Calculus of Fractions
    4. 10.4 The Derived Category
    5. 10.5 Derived Functors
    6. 10.6 The Total Tensor Product
    7. 10.7 Ext and RHom
    8. 10.8 Replacing Spectral Sequences
    9. 10.9 The Topological Derived Category
  17. Appendix A: Category Theory Language
    1. A.1 Categories
    2. A.2 Functors
    3. A.3 Natural Transformations
    4. A.4 Abelian Categories
    5. A.5 Limits and Colimits
    6. A.6 Adjoint Functors
  18. References
  19. Index