The Approximation Theorem and the K-Theory of Generalized Free Products
R. Schwänzl, Ross E. Staffeldt · Transactions of the American Mathematical Society · 1995
this paper to show how to derive the main theorems of [8] as applications of results and methods of [9]. Using different methods, Pierre Vogel [7] has also reconsidered results of [8]. We first develop language so that essentially the same fibration used in [8] may be derived from a general fibration theorem (Theorem 1.6.4 of [9, page 354]) developed as part of the overall approach to abstract algebraic K-theory described in [9]. This is Proposition 2.1 below. Our main contribution is the description of how the approximation theorem (Theorem 1.6.7 of [9, page 354]) may then be used to interpret terms in the fibration. These results are Theorems 2 and 3 stated at the end of the section. We are able to replace the technical manouvering required in the original proofs with arguments that follow a standard pattern and are more conceptual. The paper also provides an introduction to a few of the ideas we will use in [5], where we generalize the situation to the case of simplicial rings. One of the goals of [5] is to set up a framework which will also allow us to handle decomposition problems in the