[Binghamton Geometry/Topology Seminar] May 12

[Lucas Sabalka]

                   BINGHAMTON GEOMETRY/TOPOLOGY SEMINAR

This week, we will have our final seminar of the semester as well as our
fourth and final lecturer in our *Dean's Speaker Series in Geometry and
Topology*, Ayelet Lindenstrauss of Indiana.


*Date:*  Thursday, May 12, 2011
*Time:*  2:50-3:50pm
*Place:*  Library North 2205 followed by coffee/tea in the Anderson Reading
Room.

*Speaker:*  Ayelet Lindenstrauss (Indiana)
Title:  K-theory of formal power series
** *
Abstract:  *(joint with Randy McCarthy)

We study the algebraic K-theory of parametrized endomorphisms of a unital
ring R with coefficients in a simplicial R-bimodule M, and compare it with
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 can be thought of
as Witt vectors with coefficients in M, and 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).
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