(PDF) Algebra 3: Homological Algebra and Its Applications

Algebra 3: Homological Algebra and Its Applications PDF

Algebra has played a central and a decisive role in formulating and solving the problems in all branches of mathematics, science, and engineering. My earlier plan was to write a series of three volumes on algebra covering a wide spectrum to cater the need of students and researchers at various levels. The two initial volumes have already appeared. However, looking at the size and the contents to be covered, we decided to split the third volume into two volumes, Algebra 3 and Algebra 4.

Algebra 3 concentrates on the homological algebra together with its important applications in mathematics, whereas Algebra 4 is about Lie algebras, Chevalley groups, and their representation theory.

Homological algebra has played and is playing a pivotal role in understanding and classifying (up to certain equivalences) the mathematical structures such as topological, geometrical, arithmetical, and the algebraic structures by associating computable algebraic invariants to these structures. Indeed, it has also shown its deep intrinsic presence in dealing with the problems in physics, in particular, in string theory and quantum theory. The present volume, Algebra 3, the third volume in the series, is devoted to introduce the homological methods and to have some of its important applications in geometry, topology, algebraic geometry, algebra, and representation theory. It contains category theory, abelian categories, and homology theory in abelian categories, the n-fold extension functors EXT^n( _ , _ ),the torsion functors TORn( _ , _ ), the theory of derived functors, simplicial and singular homology theories with their applications, co-homology of groups, sheaf theory, sheaf co-homology, some amount of algebraic geometry, E’tale sheaf theory and co-homology, and the ‘-adic co-homology with a demonstration showing its application in the representation theory. The book can act as a text for graduate and advance graduate students specializing in mathematics.

There is no essential prerequisite to understand this book except some basics in algebra (as in Algebra 1 and Algebra 2) together with some amount of calculus and topology. An attempt to follow the logical ordering has been made throughout the book.

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Contents of Algebra 3: Homological Algebra and Its Applications book:

  • 1 Homological Algebra
  • 1.1 Categories and Functors
  • 1.2 Abelian Categories
  • 1.3 Category of Chain Complexes and Homology
  • 1.4 Extensions and the Functor EXT
  • 2 Homological Algebra 2, Derived Functors
  • 2.1 Resolutions and Extensions
  • 2.2 Tensor and Tor Functors
  • 2.3 Abstract Theory of Derived Functors
  • 2.4 Kunneth Formula
  • 2.5 Spectral Sequences
  • 3 Homological Algebra 3: Examples and Applications
  • 3.1 Polyhedrons and Simplicial Homology
  • 3.2 Applications
  • 3.3 Co-homology of Groups
  • 3.4 Calculus and Co-homology
  • 4 Sheaf Co-homology and Its Applications
  • 4.1 Presheaves and Sheaves
  • 4.2 Sheaf Co-homology and Cech Co-homology
  • 4.3 Algebraic Varieties
  • 4.4 Schemes
  • 4.5 Weil Conjectures and l-adic Co-homology

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