Topics in Algebraic K-theory

Abstract

The objective of this thesis is to study the algebraic K-theory of exact categories. In algebraic K-theory we construct a sequence of groups, called K n , which are invari- ants of a given exact category. We look at two different constructions of K n , Quillen’s Q-construction of the K-groups of an exact category as the homotopy groups of a topological space and Wladhausen’s S-construction of the K-groups as the stable ho- motopy groups of a spectrum, and show that they are equivalent. The S-construction is then used to prove the main aim of this thesis, the additivity theorem. The ad- ditivity theorem then helps us prove fundamental results about the K-groups. The main results considered are, the cofinality theorem and resolution theorem for exact categories, and the devissage theorem and localisation theorem for abelian categories.