### On the p-typical curves in Quillen's K-theory

Several years ago, Bloch introduced the complex C_{*}(A;p) of
*p*-typical curves in K-theory and outlined its connection to
the crystalline cohomology of Berthelot-Grothendieck. However, to
verify this connection Bloch restricted his attention to the symbolic
part of K-theory, since only this admitted a detailed study at the
time. In this paper we evaluate C***(A;p) in terms of
Bokstedt's topological Hochschild homology. Using this we show that if
A is a smooth algebra over a perfect field k of positive
characteristic *p*, then C***(A;p) is isomorphic to the
de Rham-Witt complex of Bloch-Deligne-Illusie. This confirms the
outlined relationship between p-typical curves in K-theory and
crystalline cohomology in the smooth case. In the singular case,
however, we get something new: we calculate C_{*}(A;p)
for the ring k[t]/(t^{2}) of dual numbers over k and show that,
by contrast to crystalline cohomology, its cohomology groups are finitely
generated modules over the ring of *p*-typical Witt vectors W(k).

Lars Hesselholt <larsh@math.nagoya-u.ac.jp>