Simplicial homotopy group model for K2 of a ring, A

Abstract

We construct an isomorphism between the group K2(R) from classical, algebraic K-Theory for a ring R and a simplicial homotopy group constructed using simplicial homotopy theory based on that same ring R. First I describe the basic aspects of simplicial homotopy theory. Special attention is paid to the use of category theory, which will be applied to the construction of a simplicial set. K-Theory for K0(R), K1(R) and K2(R) is then described before we set to work describing explicitly the nature of isomorphisms for K0(R) and K1(R) based on previous work. After introducing some theory related to K-Theory, some considerations and corrections on previous work motivate more new theory that helps the isomorphism with K2(R). Such theory is developed, mainly with regards to finitely generated projective modules over R and then elementary matrices with entries from R, culminating in the description of the Steinberg Relations that are central to the understanding of K2(R) in terms of homotopy classes. We then use new considerations on the previous work to show that a map whose image is constructed through this article is an isomorphism since it is the composition of isomorphisms.