On the algebraic Ktheory of formal power series
Abstract
Let R be a discrete unital ring, and let M be an Rbimodule. We extend Waldhausen's equivalence from the suspension of the Nil Ktheory 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 Ktheory of parametrized endomorphism of R with coefficients in M to the Ktheory 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 Ktheory of paramatrized endomorphisms converges to, is the inverse limit of the Ktheory 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 Ktheory, as a functor of augmented Ralgebras, takes on augmented Ralgebras which are tensor algebras on a connected Rbimodule.
 Publication:

arXiv eprints
 Pub Date:
 October 2010
 arXiv:
 arXiv:1010.6040
 Bibcode:
 2010arXiv1010.6040L
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Algebraic Topology;
 19D50;
 19D35;
 19D55;
 16E20