Introduction to Algebraic K - Theory

Abstract

This thesis is an introduction to algebraic K-theory, focusing on the study of rings, although we will give some geometric interpretations and some relations to topological K-theory. We will study the K_0 group. The first two chapters of the thesis include some basic notions of rings and modules, and category theory, especially regarding abelian categories. Afterwards, we present projective modules and their properties, that will be used to define the K_0 group in Chapter 4. This chapter will include basic definitions and the main theorems of the Grothendieck group of a ring. We will also show some relations with vector bundles and topological K-theory. The last chapter introduces the K_0 group of a category with exact sequences and the G_0 group. It includes the proofs of three important abstract theorems about K_0 and the Fundamental Theorem of G_0.