On the algebraic K-theory of formal power series

TL;DR

This paper extends Waldhausen's equivalence to the K-theory of tensor algebras over a ring R, establishing a connection with formal power series and analyzing the convergence of the Goodwillie Taylor tower for parametrized endomorphisms.

Contribution

It generalizes Waldhausen's equivalence to tensor algebras and provides a detailed analysis of the Goodwillie Taylor tower convergence for K-theory of parametrized endomorphisms.

Findings

Map induces equivalence on finite stages of the Goodwillie Taylor tower.

When M is connected, the map is an equivalence.

The inverse limit of K-theory of finite truncations describes the invariant W(R;M).

Abstract

Let R be a discrete unital ring, and let M be an R-bimodule. We extend Waldhausen's equivalence from the suspension of the Nil K-theory of R with coefficients in M to the K theory of the tensor algebra T_R(M), and get a map from the suspension of the K-theory of parametrized endomorphism of R with coefficients in M to the K-theory of the ring of formal power series in M over R. This map induces an equivalence on the finite stages in the Goodwillie Taylor tower of the functors. When M is connected, this map is an equivalence. For general M, we use the map to show that the suspension of the the invariant W(R;M), which is what the Goodwillie Taylor tower of the K-theory of paramatrized endomorphisms converges to, is the inverse limit of the K-theory of finite truncations of T_R(M), quotiented out by increasing powers of the augmentation ideal. This map also gives us the values that the Goodwillie Taylor tower of K-theory, as a functor of augmented R-algebras, takes on augmented R-algebras which are tensor algebras on a connected R-bimodule.