Lower Algebraic K-Theory of Rings

Abstract

This thesis is a first step towards a controlled algebraic K-theory. We give explicit formulas for the proof of Fundamental Theorem of Algebraic K-Theory. As a consequence, we provide explicit estimates on the control of propagation.

The first part of this thesis is an introduction to K0 and K1-groups of rings, where we develop necessary background materials.

In the second part of this thesis, we prove the Fundamental Theorem of Algebraic K-Theory by elementary means and give explicit formulas. A detailed discussion of propagation control is given at the end of this part.

In the last part of this thesis, we introduce negative algebraic K-theory and prove its Fundamental Theorem of Algebraic K-Theory.

This work is intended as a first step towards quantitative computations for lower algebraic K-theory.