Introduction to Algebraic K-Theory

This book gives a basic introduction to the algebraic K-theory space K(R) without assuming much knowledge in category theory or homotopy theory.

The work is divided into two parts. Part I introduces all the necessary tools from category theory and simplicial homotopy theory which one needs to define the K-theory space. Part II defines K(R) and studies the basic properties of these spaces.

While many textbooks cover the classical material in algebraic K-theory, this book offers an accessible introduction to this active area of research for graduate students. One can also view these lecture notes as an introduction to the use of categories in homotopy theory, or rather the interaction between the two areas of mathematics.

Basic introduction to algebraic K-theory Suitable for a semester long course and includes exercises Self contained and accessible for graduate students

Autorentext

Ib Madsen is a Professor of Mathematics at the University of Copenhagen. He is one of the leading algebraic topologists and has made significant progress in geometric topology and algebraic K-theory over the last 50 years. Together with Bökstedt and Hsiang, he invented Topological Cyclic Homology and introduced trace methods, which are among the main tools for computing algebraic K-theory. Along with Hesselholt, he made fundamental computations in algebraic K-theory. Ib Madsen has authored many publications in prestigious journals and several books.

Irakli Patchkoria is a Senior Lecturer at the University of Aberdeen, working in algebraic topology. He specializes in (equivariant) stable homotopy theory and has worked for many years on Topological Cyclic and Hochschild Homology and their applications. In particular, he has worked on computations of the real THH and TC, Witt vectors with coefficients, algebraic K-theory, and the de Rham-Witt complex.

Inhalt

Part I: Categories and homotopy theory.-Categories an dspaces.-Adjunctions and homotopies.-Geometric realization and CW-complexes.-Limits and colimits.-Cartesian closed categories and the k-topology.-Homotopy fibers and Quillen's Theorem A and B.-Part II: Exact categories and algebraic K-theory.-Introduction and thegroups K0(R) and K1(R).-The K-theory space.-The additivity theorem.-Cofinality and resolution theorems.-Quillen's Q-construction.-Localization of categories.- The homology of K(C).-The K-theory localization theorem.- A The spectral sequence of a functor .-B Pullbacks and pushouts.-C Further Reading.-Exercises.