# On the algebraic K-theory of formal power series

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- Type
- Preprint
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- Submission Date
- Identifiers
- arXiv ID: 1010.6040
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- arXiv
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## 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. See more