Seminar

(About joint work with Randy McCarthy) We use the algebraic K-theory of parametrized endomorphisms of a unital ring R with coefficients in a simplicial R-bimodule M to describe the algebraic K-theory of the ring  of formal power series in M over R.

Waldhausen defined an equivalence from the suspension of the reduced Nil K-theory of R with coefficients in M to the reduced algebraic K-theory  of the tensor algebra T_R(M). Extending Waldhausen's map from nilpotent endomorphisms to all endomorphisms, our map has to land in the ring of formal power series rather than in the tensor algebra, and is no longer in general an equivalence (it is an equivalence when the bimodule M is connected). Nevertheless, the map shows a close connection between its source and its target: it induces an equivalence on the Goodwillie  Taylor towers of the two (as functors of M, with R fixed), and allows us  to give a formula for the suspension of the invariant W(R;M) (which is  what the Goodwillie Taylor tower of the source functor converges to) as  the inverse limit, as n goes to infinity, of the reduced algebraic  K-theory of T_R(M)/ (M^n).