ALGEBRAIC K-THEORY

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, 𝕂-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 SetsClassifying Spaces of CategoriesQuillen K-theoryThe Agreement with Classical K-theoryK-theory of ringsG-Theory of schemesK-theory of Projective BundlesWork of Swan on Quadric HypersurfacesEpilogue: K-theoryEpilogue: Hermitian K-theoryEpilogue: Triangulated CategoriesAppendices:Category Theory and Exact CategoriesHomotopy TheoryCW 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"