Satya Mandal 
ALGEBRAIC K-THEORY 
The Homotopy Approach of Quillen and an Approach from Commutative Algebra

In this book the author takes a pedagogic approach to Algebraic K-theory. He tried to find the shortest route possible, with complete details, to arrive at the homotopy approach of Quillen [Q] to Algebraic K-theory, with a simple goal to produce a self-contained and comprehensive pedagogic document in Algebraic K-theory, that is accessible to upper level graduate students. That is precisely what this book faithfully executes and achieves.

The contents of this book can be divided into three parts — (1) The main body (Chapters 2-8), (2) Epilogue Chapters (Chapters 9, 10, 11) and (3) the Background and preliminaries (Chapters A, B, C, 1). The main body deals with Quillen’s definition of K-theory and the K-theory of schemes. Chapters 2, 3, 5, 6, and 7 provide expositions of the paper of Quillen [Q], and chapter 4 is on agreement of Classical K-theory and Quillen K-theory. Chapter 8 is an exposition of the work of Swan [Sw1] on K-theory of quadrics.

The Epilogue chapters can be viewed as a natural progression of Quillen’s work and methods. These represent significant benchmarks and include Waldhausen K-theory, Negative K-theory, Hermitian K-theory, ????-theory spectra, Grothendieck-Witt theory spectra, Triangulated categories, Nori-Homotopy and its relationships with Chow-Witt obstructions for projective modules. In most cases, the proofs are improvisation of methods of Quillen [Q].

The background, preliminaries and tools needed in chapters 2-11, are developed in chapters A on Category Theory and Exact Categories, B on Homotopy, C on CW Complexes, and 1 on Simplicial Sets.

Contents:

• Simplicial Sets


• Classifying Spaces of Categories


• Quillen K-theory


• The Agreement with Classical K-theory


• K-theory of rings


• G-Theory of schemes


• K-theory of Projective Bundles


• Work of Swan on Quadric Hypersurfaces


• Epilogue: K-theory


• Epilogue: Hermitian K-theory


• Epilogue: Triangulated Categories


• Appendices:
  
  
  
  • Category Theory and Exact Categories
  
  
  • Homotopy Theory
  
  
  • CW Complexes
  
  

Readership: Graduate students and researchers in Algebra.

Key Features:

• Currently, there is no book on Higher Algebraic K-theory that is suitable for non-experts. This book fills the vacuum


• The book is unique because it is the only readable book in the literature, for a upper level graduate student and a broader math community


• The book will serve as a major reference book which covers more than fifty years of slow developments


• The emphasis is on the complete details, precise definitions and complete proofs of all the materials leading up to Quillen K-theory and beyond; not just ‘surfing’